{"version":"physics.v1","nodes":[{"id":1,"code":"00","level":1,"parent_id":null,"name":"Foundations: Mathematics & Vectors for Physics","description":"JEE chapter 00 (outside the current JEE syllabus): Foundations: Mathematics & Vectors for Physics. Concepts: Pre-calculus tools (algebra, trigonometry, approximations, geometry); Differentiation and integration for physics; Vectors; Graph reading.","sort":0,"aliases":["Basic Mathematics & Vectors for Physics","Integration","Basic Maths (Pre-Calculus)","Differentiation","Application of Derivatives"]},{"id":2,"code":"00.01","level":2,"parent_id":1,"name":"Pre-calculus tools (algebra, trigonometry, approximations, geometry)","description":"Parent node for pure math tools used inside physics problems: trigonometry, approximations, logarithms, quadratics, mensuration and coordinate geometry. Tag here when the question is essentially a math exercise embedded in a physics wrapper (e.g., solving for an angle, simplifying an expression) rather than testing a specific physics law.","sort":1,"aliases":["mathematical tools for physics","Basic Maths (Pre-Calculus)"]},{"id":3,"code":"00.01.01","level":3,"parent_id":2,"name":"Trigonometric ratios, identities and equations","description":"Questions using sin/cos/tan values, standard identities (sin²θ+cos²θ=1, compound and double-angle formulas) or solving trig equations. Typical content: finding the angle where two quantities are equal, simplifying expressions like sin(θ+φ), or evaluating tan θ = μ in friction/incline setups.","sort":2,"aliases":[]},{"id":4,"code":"00.01.02","level":3,"parent_id":2,"name":"Sine rule, cosine rule and similar triangles","description":"Questions applying the sine rule (a/sinA = b/sinB), cosine rule (c² = a²+b²−2ab·cosC) or similar-triangle ratios to geometric figures. Common in force-triangle problems, relative velocity triangles, ladder/rod geometry, and finding an unknown side or angle in a vector diagram.","sort":3,"aliases":[]},{"id":5,"code":"00.01.03","level":3,"parent_id":2,"name":"Small-angle and binomial approximations","description":"Questions using sinθ ≈ tanθ ≈ θ and cosθ ≈ 1−θ²/2 for small angles, or binomial expansion (1+x)ⁿ ≈ 1+nx for x≪1. Typical content: pendulum small-angle reduction, percentage change in g with height (Δg/g ≈ −2Δh/R), or approximating a quantity when a parameter changes slightly.","sort":4,"aliases":[]},{"id":6,"code":"00.01.04","level":3,"parent_id":2,"name":"Logarithms and exponential functions","description":"Questions involving log rules (ln(ab)=ln a+ln b), exponentials e^(±kt), or converting between log and exponential forms. Typical content: radioactive decay N=N₀e^(−λt) half-life calculations, RC discharging, taking logs to linearize y=axⁿ, or solving for time when a quantity falls to a fraction of its value.","sort":5,"aliases":[]},{"id":7,"code":"00.01.05","level":3,"parent_id":2,"name":"Quadratic and simultaneous equations","description":"Questions solved by the quadratic formula, discriminant analysis, or eliminating variables between two equations. Typical content: projectile time from a quadratic in t, finding when a particle is at a given position, or simultaneous equations for two unknown tensions/currents.","sort":6,"aliases":[]},{"id":8,"code":"00.01.06","level":3,"parent_id":2,"name":"Mensuration and coordinate geometry (areas, volumes, lines, circles)","description":"Questions needing areas/volumes (sphere, cylinder, cone, shell) or coordinate geometry (distance formula, equation of a line or circle x²+y²=r²). Typical content: mass or moment of inertia of a uniform solid, center of mass of composite shapes, flux through a curved surface, or locus of a particle moving in a circle.","sort":7,"aliases":[]},{"id":9,"code":"00.02","level":2,"parent_id":1,"name":"Differentiation and integration for physics","description":"Parent node for calculus used in physics: derivatives, integrals, maxima/minima, area under curves, and elementary differential equations. Tag here when the question's core skill is performing or interpreting differentiation/integration rather than a specific chapter concept.","sort":8,"aliases":["calculus for physics","derivatives","chain rule","maxima and minima","definite integrals","area under curve","Differentiation","Integration","Application of Derivatives"]},{"id":10,"code":"00.02.01","level":3,"parent_id":9,"name":"Differentiation: rules and standard derivatives","description":"Questions requiring derivative rules (power, product, quotient, chain) and standard derivatives of xⁿ, sin x, cos x, eˣ, ln x. Typical content: getting velocity v=dx/dt or acceleration from position, finding dV/dr, or differentiating a given function of time in kinematics/SHM.","sort":9,"aliases":[]},{"id":11,"code":"00.02.02","level":3,"parent_id":9,"name":"Maxima and minima using derivatives","description":"Questions asking for maximum/minimum of a quantity by setting dy/dx=0 and checking the second derivative. Typical content: projectile range maximized at 45°, minimum distance between two moving particles, maximum power delivery, or least length/time in an optimization setup.","sort":10,"aliases":[]},{"id":12,"code":"00.02.03","level":3,"parent_id":9,"name":"Integration: standard integrals and techniques","description":"Questions requiring standard integrals (∫xⁿdx, ∫sin x dx, ∫dx/(a²+x²)) or techniques like substitution, by parts, and partial fractions. Typical content: finding position from velocity, work from a variable force F(x), or total charge from a time-dependent current.","sort":11,"aliases":[]},{"id":13,"code":"00.02.04","level":3,"parent_id":9,"name":"Definite integral as area under a curve","description":"Questions where a definite integral is evaluated as the area under a curve, often read off a graph. Typical content: displacement as area under v–t, work as area under F–x, impulse as area under F–t, including triangular/semi-circular/trapezoidal graph areas.","sort":12,"aliases":[]},{"id":14,"code":"00.02.05","level":3,"parent_id":9,"name":"Simple differential equations (variable separable, exponential solutions)","description":"Questions setting up and solving simple ODEs by separation of variables or recognizing exponential solutions. Typical content: dN/dt=−λN for decay, m dv/dt=−kv for terminal velocity, capacitor charging q=Cε(1−e^(−t/RC)), or deriving time constants from a rate equation.","sort":13,"aliases":[]},{"id":15,"code":"00.03","level":2,"parent_id":1,"name":"Vectors","description":"Parent node for vector algebra: identifying vectors, addition/resolution into components, and dot/cross products. Tag here when the question tests vector manipulation itself (resultants, components, products) rather than a specific physics law that uses vectors.","sort":14,"aliases":["scalars and vectors","vector algebra","dot and cross products","vector addition and resolution","unit vectors"]},{"id":16,"code":"00.03.01","level":3,"parent_id":15,"name":"Scalars and vectors","description":"Questions asking to classify quantities as scalars or vectors, or dealing with magnitude, direction, unit vectors, and position vectors. Typical content: 'which of the following is a vector', writing a vector in î, ĵ, k̂ form, or finding magnitude and direction cosines.","sort":15,"aliases":[]},{"id":17,"code":"00.03.02","level":3,"parent_id":15,"name":"Vector addition and resolution","description":"Questions using triangle/parallelogram law, R=√(A²+B²+2ABcosθ), or resolving vectors into perpendicular components. Typical content: resultant of two/three forces, equilibrium conditions ΣF=0, resolving weight on an incline, or finding a component of a vector along a given direction.","sort":16,"aliases":[]},{"id":18,"code":"00.03.03","level":3,"parent_id":15,"name":"Dot and cross products of vectors","description":"Questions using A·B=ABcosθ or A×B=ABsinθ n̂, including component forms and properties (perpendicular/parallel conditions). Typical content: work as F·d, torque as r×F, angle between two vectors, area of a parallelogram spanned by vectors, or checking orthogonality.","sort":17,"aliases":[]},{"id":19,"code":"00.04","level":2,"parent_id":1,"name":"Graph reading","description":"Parent node for extracting information from graphs: slopes, intercepts, areas, curve shapes, linearization, and sketching. Tag here when the question is centered on interpreting or constructing a graph rather than on the underlying physics formula.","sort":18,"aliases":[]},{"id":20,"code":"00.04.01","level":3,"parent_id":19,"name":"Slope and intercept of straight-line graphs","description":"Questions reading slope (m) and intercept (c) from a straight-line graph y=mx+c to extract physical constants. Typical content: g from slope of T² vs L, velocity from slope of x–t, acceleration from slope of v–t, or identifying what slope/intercept represents in a given plot.","sort":19,"aliases":[]},{"id":21,"code":"00.04.02","level":3,"parent_id":19,"name":"Area under graphs and its physical meaning","description":"Questions where the area under a graph carries physical meaning and must be computed or compared. Typical content: displacement from v–t area, work from F–x area, impulse from F–t area, energy from P–t area, or using area to compare quantities between two processes.","sort":20,"aliases":[]},{"id":22,"code":"00.04.03","level":3,"parent_id":19,"name":"Reading non-linear graphs: parabola, hyperbola, exponential","description":"Questions identifying or interpreting parabolic (y∝x²), hyperbolic (y∝1/x), or exponential curves from their shape or data. Typical content: recognizing x∝t² for uniform acceleration, PV=constant isotherm shape, exponential decay of current, or matching a given curve to the correct relation.","sort":21,"aliases":[]},{"id":23,"code":"00.04.04","level":3,"parent_id":19,"name":"Linearization: choosing plots to obtain a straight line","description":"Questions asking which plot of given variables yields a straight line, and how slope/intercept give constants. Typical content: T² vs L for a pendulum, 1/v vs 1/u for lenses, log y vs log x for power laws, or v² vs s to find acceleration from experimental data.","sort":22,"aliases":[]},{"id":24,"code":"00.04.05","level":3,"parent_id":19,"name":"Curve sketching: shifting and scaling of standard graphs","description":"Questions sketching or identifying transformed graphs: shifts y=f(x−a), scalings y=af(bx), reflections of standard curves. Typical content: sketching sin(ωt+φ) vs sin(ωt), shifted parabolas, wave snapshots at different times, or predicting how a graph changes when a parameter is doubled.","sort":23,"aliases":[]},{"id":25,"code":"01","level":1,"parent_id":null,"name":"Units, Dimensions & Error Analysis","description":"JEE chapter 01: Units, Dimensions & Error Analysis. Concepts: Measurement of length, mass and time; SI units; Dimensional analysis; Error analysis; Experimental skills and measuring instruments; Significant figures; Physical World: scope of physics and fundamental forces.","sort":24,"aliases":["Units and Measurements","Physical World"]},{"id":26,"code":"01.01","level":2,"parent_id":25,"name":"Measurement of length, mass and time","description":"Measurement topics: vernier callipers and screw gauge readings, order-of-magnitude estimates, parallax distances, the ranges of lengths/masses/times in nature, and the oil-film method for molecular size.","sort":25,"aliases":["measurement","order-of-magnitude estimation","Fermi estimation","parallax method"]},{"id":27,"code":"01.01.01","level":3,"parent_id":26,"name":"Order-of-magnitude (Fermi) estimation","description":"Fermi estimation problems: estimate the number of molecules in a water drop, mass of air in a room, number of hairs on a head, or distance to the Moon using rough assumptions and powers of ten.","sort":26,"aliases":[]},{"id":28,"code":"01.01.02","level":3,"parent_id":26,"name":"Parallax method for measuring large distances","description":"Measuring large distances by parallax: D = b/θ with a known baseline b (Earth's orbital diameter for nearby stars), converting a parallax angle given in arcseconds into metres or parsecs.","sort":27,"aliases":[]},{"id":29,"code":"01.01.03","level":3,"parent_id":26,"name":"Range of lengths, masses and time intervals in nature","description":"Matching objects to orders of magnitude: sizes from a nucleus (~10⁻¹⁵ m) to the observable universe (~10²⁶ m), masses from the electron (~10⁻³⁰ kg) to the universe, and times from nuclear (~10⁻²² s) to the age of the universe (~10¹⁷ s).","sort":28,"aliases":[]},{"id":30,"code":"01.01.04","level":3,"parent_id":26,"name":"Estimation of very small distances and molecular size (oil-film method)","description":"Estimating molecular size from an oil drop of volume V spreading into a monolayer of area A: the film thickness t = V/A equals the molecular diameter (classic oleic-acid experiment).","sort":29,"aliases":[]},{"id":31,"code":"01.02","level":2,"parent_id":25,"name":"SI units","description":"SI unit content: the seven base units, derived units built from them, and the correct conventions for writing unit names and symbols.","sort":30,"aliases":["International System of Units"]},{"id":32,"code":"01.02.01","level":3,"parent_id":31,"name":"The international system of units","description":"The SI system itself: the seven base quantities (length, mass, time, electric current, temperature, amount of substance, luminous intensity) and notation rules such as never pluralising symbols or capitalising units derived from names.","sort":31,"aliases":[]},{"id":33,"code":"01.02.02","level":3,"parent_id":31,"name":"Base units","description":"The seven SI base units — metre, kilogram, second, ampere, kelvin, mole, candela — with their symbols and definitions; questions ask to identify base units or count how many base units enter a given quantity.","sort":32,"aliases":[]},{"id":34,"code":"01.02.03","level":3,"parent_id":31,"name":"Derived units","description":"Derived units expressed through base units: newton = kg·m/s², joule, watt, pascal, coulomb, radian, etc.; questions give a defining formula and ask for the SI unit of the quantity.","sort":33,"aliases":[]},{"id":35,"code":"01.03","level":2,"parent_id":25,"name":"Dimensional analysis","description":"Dimensional analysis umbrella: writing dimensional formulae, testing equation homogeneity, deriving functional forms up to constants, and converting numerical values between unit systems.","sort":34,"aliases":[]},{"id":36,"code":"01.03.01","level":3,"parent_id":35,"name":"Dimensional formulae","description":"Writing [M^a L^b T^c] forms for quantities like Planck's constant, viscosity, surface tension or the gravitational constant, or extracting dimensions from a defining equation such as F = 6πηrv to get [η].","sort":35,"aliases":[]},{"id":37,"code":"01.03.02","level":3,"parent_id":35,"name":"Checking equations by dimensions","description":"Principle of dimensional homogeneity: check which of several given equations can be physically correct, reject wrong ones like v = at², and note that dimensionless factors (½, 2π) cannot be fixed by dimensions.","sort":36,"aliases":[]},{"id":38,"code":"01.03.03","level":3,"parent_id":35,"name":"Dimensional analysis as a solution strategy","description":"Using dimensions to derive or scale relations: e.g. T ∝ √(l/g) for a pendulum, v ∝ √(P/ρ) for sound speed, or finding how a quantity changes when the quantities it depends on are altered, up to a dimensionless constant.","sort":37,"aliases":[]},{"id":39,"code":"01.03.04","level":3,"parent_id":35,"name":"Conversion of units across systems (MKS, CGS, SI)","description":"Unit conversion via n₁u₁ = n₂u₂ between SI, CGS and FPS systems: e.g. 1 N = 10⁵ dyne, 1 J = 10⁷ erg, expressing G or pressure in CGS units, and converting numerical values when the unit changes.","sort":38,"aliases":[]},{"id":40,"code":"01.04","level":2,"parent_id":25,"name":"Error analysis","description":"Error analysis umbrella: least count and reading errors of instruments, absolute, relative and percentage errors, and how errors combine when measured quantities are added, multiplied or raised to powers.","sort":39,"aliases":["errors in measurement","propagation of errors","absolute, relative and percentage error"]},{"id":41,"code":"01.04.01","level":3,"parent_id":40,"name":"Errors in measurement","description":"Computing errors: mean absolute and fractional error from repeated readings, propagation rules ΔZ/Z = ΔA/A + ΔB/B for products and n·ΔA/A for powers, and distinguishing systematic from random errors.","sort":40,"aliases":[]},{"id":42,"code":"01.04.02","level":3,"parent_id":40,"name":"Absolute, relative and percentage error","description":"Questions computing absolute error (|measured − mean|), relative error (Δa/a_mean) and percentage error (Δa/a × 100%) from repeated readings, e.g. period of a pendulum measured n times, or % error in density from measured mass and radius.","sort":41,"aliases":[]},{"id":43,"code":"01.04.03","level":3,"parent_id":40,"name":"Propagation of errors","description":"Questions applying error combination rules: absolute errors add for sums/differences, relative errors add for products/quotients, and relative error multiplies by the power for Z = A^p B^q / C^r; typical archetype: find % error in Z given % errors in A, B, C.","sort":42,"aliases":[]},{"id":44,"code":"01.04.04","level":3,"parent_id":40,"name":"Systematic and random errors","description":"Questions classifying errors as systematic (zero error, calibration fault, parallax, bias) versus random (unpredictable scatter reduced by averaging), or computing mean absolute error from a set of repeated observations.","sort":43,"aliases":[]},{"id":45,"code":"01.05","level":2,"parent_id":25,"name":"Experimental skills and measuring instruments","description":"Parent node for laboratory measurement questions: choosing the right instrument, least count, reading main and fractional scales, and correcting readings; includes vernier callipers, screw gauge, stopwatch, thermometer, spring balance, metre scale.","sort":44,"aliases":["experimental physics","vernier calliper","screw gauge","least count","lab experiments","JEE Main experimental skills"]},{"id":46,"code":"01.05.01","level":3,"parent_id":45,"name":"Vernier callipers: least count and reading","description":"Questions on vernier callipers: least count = 1 MSD − 1 VSD (or MSD/number of divisions), reading = main scale + vernier coincidence × LC; typical setups measure diameter of a cylinder/rod/sphere from given MSD and VSD data.","sort":45,"aliases":[]},{"id":47,"code":"01.05.02","level":3,"parent_id":45,"name":"Screw gauge: pitch, least count and reading","description":"Questions on screw gauge: pitch, least count = pitch/number of circular scale divisions, reading = MSR + circular scale reading × LC; typical setups measure wire thickness or sphere diameter, sometimes with circular scale divisions and half-pitch (0.5 mm) cases.","sort":46,"aliases":[]},{"id":48,"code":"01.05.03","level":3,"parent_id":45,"name":"Zero error and zero correction in measuring instruments","description":"Questions on positive/negative zero error of vernier callipers or screw gauge and zero correction (true reading = observed − zero error), e.g. circular scale shows +3 divisions at closed jaws — find corrected diameter.","sort":47,"aliases":[]},{"id":49,"code":"01.05.04","level":3,"parent_id":45,"name":"Least count of common laboratory instruments (stopwatch, thermometer, spring balance)","description":"Questions on least counts of everyday lab instruments: stopwatch (~0.1 s or 0.01 s), thermometer (0.1–1 °C), spring balance (g per division), metre scale (1 mm); archetype: smallest value the instrument can reliably measure, or count of divisions between marked values.","sort":48,"aliases":[]},{"id":50,"code":"01.06","level":2,"parent_id":25,"name":"Significant figures","description":"Parent node for significant figures: counting significant digits, rounding rules, sig-fig handling in arithmetic, and scientific notation; questions test how precisely a measured or computed value should be reported.","sort":49,"aliases":[]},{"id":51,"code":"01.06.01","level":3,"parent_id":50,"name":"Rules for counting significant figures","description":"Questions counting significant figures using the rules: nonzero digits significant, zeros between nonzero digits significant, leading zeros not significant, trailing zeros significant only after a decimal point; e.g. how many sig figs in 0.00520, 2.308, 6.320×10².","sort":50,"aliases":[]},{"id":52,"code":"01.06.02","level":3,"parent_id":50,"name":"Rounding off numbers","description":"Questions rounding a number to a specified number of significant figures or decimal places using the rule: drop digit ≥5 rounds up, <5 leaves unchanged; e.g. round 2.745 to 3 sig figs or 15.275 g to 1 decimal place.","sort":51,"aliases":[]},{"id":53,"code":"01.06.03","level":3,"parent_id":50,"name":"Significant figures in arithmetic operations","description":"Questions applying sig-fig rules in arithmetic: multiplication/division keeps the fewest significant figures, addition/subtraction keeps the fewest decimal places; e.g. evaluate (3.24 × 0.060)/2.1 or 2.5 + 3.42 with correct final precision.","sort":52,"aliases":[]},{"id":54,"code":"01.06.04","level":3,"parent_id":50,"name":"Scientific notation and significant figures","description":"Questions expressing measurements in scientific notation a×10ⁿ with the correct number of significant figures, or converting between notation forms; e.g. write 0.0003450 as 3.450×10⁻⁴ and state its sig figs.","sort":53,"aliases":[]},{"id":55,"code":"01.07","level":2,"parent_id":25,"name":"Physical World: scope of physics and fundamental forces","description":"Parent node for the Physical World chapter: scope and excitement of physics, fundamental forces of nature, and links between physics, technology and society; mostly conceptual/factual and assertion–reason style questions.","sort":54,"aliases":["Physics, technology and society","Nature of physics","Scope of Physics and Fundamental Forces"]},{"id":56,"code":"01.07.01","level":3,"parent_id":55,"name":"Fundamental forces in nature","description":"Questions on the four fundamental forces — gravitational, electromagnetic, strong nuclear, weak nuclear — their relative strengths (~10⁻³⁹ : 10⁻² : 10⁻¹³ : 1 style comparisons) and ranges; archetypes: rank forces, identify the force governing a given phenomenon (planet binding, beta decay, nucleus stability).","sort":55,"aliases":[]},{"id":57,"code":"01.07.02","level":3,"parent_id":55,"name":"Physics, technology and society","description":"Factual questions linking physics discoveries to technology and society: semiconductors and electronics, lasers and communication, nuclear fission/fusion and energy, steam engine and thermodynamics, silicon chips; often match-the-following or 'which discovery led to which application'.","sort":56,"aliases":[]},{"id":58,"code":"01.07.03","level":3,"parent_id":55,"name":"Scope and excitement of physics","description":"Factual/conceptual questions on the scope of physics — from sub-nuclear (10⁻¹⁵ m) to the universe (10²⁶ m), principal domains like mechanics, thermodynamics, electrodynamics, optics, quantum and relativistic physics — and the ideas of unification and reductionism.","sort":57,"aliases":[]},{"id":59,"code":"01.07.04","level":3,"parent_id":55,"name":"Virtual particles and force range (Yukawa mechanism)","description":"Questions on the Yukawa picture: forces mediated by virtual particle exchange, with range r ≈ ħ/(mc); archetype: estimate the range of nuclear force from the pion mass (~1.5 fm) or explain why massive mediators give short-range forces.","sort":58,"aliases":[]},{"id":60,"code":"02","level":1,"parent_id":null,"name":"Kinematics","description":"JEE chapter 02: Kinematics. Concepts: Position, displacement, velocity and acceleration; Relative velocity in one dimension; Uniformly accelerated motion (kinematic equations); Kinematics with variable acceleration; Motion graphs; Circular kinematics; Kinematics in two dimensions (position, velocity and acceleration vectors); Relative velocity in two dimensions; Projectile motion.","sort":59,"aliases":["Motion in a Straight Line","Motion in a Plane"]},{"id":61,"code":"02.01","level":2,"parent_id":60,"name":"Position, displacement, velocity and acceleration","description":"Parent node for basic 1D kinematics quantities: questions that define or compute position, displacement, path length, average/instantaneous velocity and acceleration for a particle moving along a straight line. Tag here only when the question mixes these quantities and no specific subconcept fits.","sort":60,"aliases":["basic kinematics quantities","position, velocity and acceleration","motion in a straight line"]},{"id":62,"code":"02.01.01","level":3,"parent_id":61,"name":"Acceleration","description":"Questions computing average acceleration (a = Δv/Δt) or instantaneous acceleration (dv/dt) in 1D, including sign conventions, deceleration, and statements like 'velocity decreases at 2 m/s every second'. Typical items: particle's velocity changes from u to v in time t, or v(t) given and a at an instant asked.","sort":61,"aliases":[]},{"id":63,"code":"02.01.02","level":3,"parent_id":61,"name":"Average and instantaneous velocity","description":"Questions computing average velocity (total displacement / total time) over an interval versus instantaneous velocity (dx/dt at an instant), often from a given x(t) function like x = 3t² − 4t. Classic archetype: show average velocity over [t1,t2] equals instantaneous velocity at some midpoint for uniform acceleration.","sort":62,"aliases":[]},{"id":64,"code":"02.01.03","level":3,"parent_id":61,"name":"Position, path length and displacement","description":"Questions distinguishing distance (path length) from displacement: a particle moves along a line, reverses direction (e.g., goes 4 m east then 3 m west), and total path length and net displacement are asked. Also position vector on x-axis, x(t) given, displacement between two times.","sort":63,"aliases":[]},{"id":65,"code":"02.01.04","level":3,"parent_id":61,"name":"Rectilinear motion (general 1D motion)","description":"Catch-all for general straight-line motion problems that don't fit a specific subconcept: given x(t) or a relation between v and t/x, find when the particle changes direction, its velocity at some instant, or total distance travelled. Use for mixed 1D questions not clearly about graphs, gravity, or relative velocity.","sort":64,"aliases":[]},{"id":66,"code":"02.01.05","level":3,"parent_id":61,"name":"Shortest-time path across regions of different speeds (Fermat principle)","description":"Problems minimizing travel time across two regions with different constant speeds (1D Fermat/Snell analogy): e.g., a person rows to a point on the opposite bank then walks, or light/particle crossing two media; answer uses Snell's-law-like condition sinθ₁/v₁ = sinθ₂/v₂ or boundary-point optimization by differentiation.","sort":65,"aliases":[]},{"id":67,"code":"02.02","level":2,"parent_id":60,"name":"Relative velocity in one dimension","description":"Parent node for relative velocity in one dimension: any question where two objects move along the same line and their motion is analyzed in one object's frame. Tag here only if no specific subconcept (formula, overtaking, collision, rain-man) fits.","sort":66,"aliases":["relative motion in one dimension","relative velocity 1d"]},{"id":68,"code":"02.02.01","level":3,"parent_id":67,"name":"Relative velocity: definition and formula (1D)","description":"Questions directly using v_AB = v_A − v_B (with sign convention along the line): find relative velocity of A with respect to B, time when their separation is a given value, or when one particle's velocity equals another's. Includes two particles with velocities in same or opposite directions.","sort":67,"aliases":[]},{"id":69,"code":"02.02.02","level":3,"parent_id":67,"name":"Overtaking and crossing problems (trains and cars)","description":"Train/car crossing and overtaking problems: train crossing a pole, platform, bridge, or another moving train; car overtaking a truck. Core relation: time = (sum of relevant lengths)/(relative speed), with relative speed = difference for same direction, sum for opposite direction.","sort":68,"aliases":[]},{"id":70,"code":"02.02.03","level":3,"parent_id":67,"name":"Head-on motion, collision and closest approach (1D)","description":"Two objects moving toward each other (head-on) or one chasing another: will they collide, time to collision, minimum separation / closest approach, or distance travelled by each before meeting. Uses relative velocity of approach (sum of speeds head-on) and initial gap.","sort":69,"aliases":[]},{"id":71,"code":"02.02.04","level":3,"parent_id":67,"name":"Rain-man problems (relative velocity of rain)","description":"Rain-man problems: rain falls vertically with speed v_r, man walks/runs with speed v_m; find the angle at which rain appears to fall (tanθ = v_m/v_r) for umbrella tilt, or the apparent speed of rain relative to the man. Variants: given observed angle, find man's speed or rain's speed.","sort":70,"aliases":[]},{"id":72,"code":"02.03","level":2,"parent_id":60,"name":"Uniformly accelerated motion (kinematic equations)","description":"Parent node for uniformly accelerated motion: any question using constant-acceleration kinematics. Tag here only if the question uses kinematic equations but doesn't fit a specific subconcept like free fall, nth second, or stopping distance.","sort":71,"aliases":["uniformly accelerated motion","kinematic equations for uniform acceleration","rectilinear motion","free fall","motion under gravity","vertically thrown body","stone dropped from height","nth-second distance"]},{"id":73,"code":"02.03.01","level":3,"parent_id":72,"name":"Kinematic equations for uniform acceleration","description":"Direct plug-in problems with v = u + at, s = ut + ½at², v² = u² + 2as: given any three of u, v, a, t, s find the rest. Includes sign-convention care (deceleration as negative a) and particle with given initial velocity and constant acceleration.","sort":72,"aliases":[]},{"id":74,"code":"02.03.02","level":3,"parent_id":72,"name":"Motion under gravity (free fall and vertical projection)","description":"Vertical motion under gravity (g ≈ 9.8 or 10 m/s²): bodies dropped, thrown up or down; max height u²/2g, time of ascent/descent, time of flight 2u/g, ball thrown up from a tower meeting another dropped, velocity on hitting ground. Includes two-body vertical meeting problems.","sort":73,"aliases":[]},{"id":75,"code":"02.03.03","level":3,"parent_id":72,"name":"Distance travelled in nth second","description":"Questions on distance travelled in the nth second: s_n = u + a(2n−1)/2, contrasted with total distance in n seconds (ut + ½at²). Typical: 'a particle covers 5 m in the 3rd second and 7 m in the 5th second, find u and a' or ratio-type items.","sort":74,"aliases":[]},{"id":76,"code":"02.03.04","level":3,"parent_id":72,"name":"Average velocity relations for uniform acceleration ((u+v)/2)","description":"Questions exploiting average velocity shortcuts for constant acceleration: v_avg = (u+v)/2 = s/t, displacement = average velocity × time, and relations like s ∝ odd numbers in successive seconds. Typical: 'body covers half the distance at speed v1 and half at v2' style checks or finding total time from u, v, s.","sort":75,"aliases":[]},{"id":77,"code":"02.03.05","level":3,"parent_id":72,"name":"Multi-stage motion: acceleration followed by deceleration","description":"Two-phase problems: a vehicle accelerates uniformly to a maximum speed then decelerates to rest (or to another speed), with continuity of velocity at the junction. Asked: maximum speed reached, total time, total distance, or minimum time to cover a distance given accel/decel limits.","sort":76,"aliases":[]},{"id":78,"code":"02.03.06","level":3,"parent_id":72,"name":"Stopping distance and reaction time","description":"Braking problems: driver reaction time t_r during which the car moves v·t_r, then braking distance v²/(2a) or v²/(2μg); total stopping distance = reaction distance + braking distance. Typical: 'car moving at 72 km/h stops in 20 m after brakes applied, find retardation' or how stopping distance scales with speed.","sort":77,"aliases":[]},{"id":79,"code":"02.04","level":2,"parent_id":60,"name":"Kinematics with variable acceleration","description":"Parent node for kinematics with non-constant acceleration: questions where a is a function of t, x, or v and calculus (integration or v dv/dx) is needed. Tag here only if no specific subconcept fits.","sort":78,"aliases":["non-uniform acceleration","time-varying and position-dependent acceleration"]},{"id":80,"code":"02.04.01","level":3,"parent_id":79,"name":"Kinematics with time-varying acceleration","description":"Acceleration given as a function of time (e.g., a = 6t − 4, or a piecewise a(t)): integrate to get v(t) and x(t) with initial conditions, then find velocity/displacement at an instant, or time to return to start. Also v given as f(t) and displacement by integration.","sort":79,"aliases":[]},{"id":81,"code":"02.04.02","level":3,"parent_id":79,"name":"Position-dependent acceleration","description":"Acceleration given as a function of position (e.g., a = kx, a = α − βx²): use v dv/dx = a(x) to find v(x), stopping distance, or velocity at a given position. Typical: 'particle starts at x=0 with v₀ and a = −kx, find where it stops'.","sort":80,"aliases":[]},{"id":82,"code":"02.04.03","level":3,"parent_id":79,"name":"Motion under gravity with air resistance (approximate models)","description":"Free fall / vertical motion with air resistance in simplified form: retarding acceleration given as proportional to v or v², or a constant drag value; find terminal velocity, time to reach terminal speed, or modified fall time/impact speed. Tag only when drag/air resistance is explicitly modelled, not ideal free fall.","sort":81,"aliases":[]},{"id":83,"code":"02.05","level":2,"parent_id":60,"name":"Motion graphs","description":"Parent node for graph-based kinematics questions: reading or drawing x–t, v–t (and occasionally a–t) graphs. Tag here only if the question is fundamentally graphical and doesn't fit the x–t or v–t subconcept.","sort":82,"aliases":["x-t graph","v-t graph","a-t graph","area under v-t graph"]},{"id":84,"code":"02.05.01","level":3,"parent_id":83,"name":"Position-time graphs","description":"Position–time graph questions: slope gives velocity, curvature sign gives acceleration; identify uniform motion, rest, direction reversal from an x–t plot, compare speeds of two objects from slopes, or find when two particles meet from intersecting x–t lines.","sort":83,"aliases":[]},{"id":85,"code":"02.05.02","level":3,"parent_id":83,"name":"Velocity-time graphs","description":"Velocity–time graph questions: slope gives acceleration, area under curve gives displacement (area magnitude gives distance); compute displacement/distance from triangular or trapezoidal v–t plots, find average velocity, or compare two motions shown on the same v–t axes.","sort":84,"aliases":[]},{"id":86,"code":"02.05.03","level":3,"parent_id":83,"name":"Acceleration-time graphs","description":"Questions giving an acceleration–time graph and asking for velocity change via area under the curve, identifying when the particle speeds up/slows down or reverses, or matching a(t) graphs to described motion (e.g., piecewise constant a, a ∝ t).","sort":85,"aliases":[]},{"id":87,"code":"02.05.04","level":3,"parent_id":83,"name":"Area under curves and slope interpretations","description":"Graph-reading questions across x–t, v–t and a–t plots: slope of x–t gives v, slope of v–t gives a, area under v–t gives displacement/distance, area under a–t gives Δv; typical tasks are finding average velocity, total distance with direction reversals, or identifying the correct graph for a given motion.","sort":86,"aliases":[]},{"id":88,"code":"02.06","level":2,"parent_id":60,"name":"Circular kinematics","description":"Parent node for circular-motion kinematics: angular variables (θ, ω, α), their linear counterparts, centripetal acceleration, and uniform vs non-uniform circular motion questions about particles on circular tracks, wheels, and rims.","sort":87,"aliases":["uniform circular motion","centripetal acceleration","angular velocity and angular acceleration","non-uniform circular motion (tangential and radial acceleration)"]},{"id":89,"code":"02.06.01","level":3,"parent_id":88,"name":"Uniform circular motion","description":"Questions on uniform circular motion (constant speed and ω): period T = 2πr/v, frequency, number of revolutions in a time interval, centripetal acceleration magnitude/direction, and positions or angles swept after given times.","sort":88,"aliases":[]},{"id":90,"code":"02.06.02","level":3,"parent_id":88,"name":"Angular displacement, angular velocity and angular acceleration","description":"Questions using θ(t), ω = dθ/dt, α = dω/dt and the rotational analogues of straight-line kinematics (ω = ω₀ + αt, θ = ω₀t + ½αt², ω² = ω₀² + 2αθ): spinning wheels speeding up/slowing down, given angular position functions, or average vs instantaneous angular velocity.","sort":89,"aliases":[]},{"id":91,"code":"02.06.03","level":3,"parent_id":88,"name":"Relation between linear and angular quantities (v = ωr, a = αr)","description":"Questions converting between linear and angular quantities via v = ωr, aₜ = αr, a_c = ω²r: rim speed of a wheel, belt/pulley contact points, two points at different radii on a rotating disc, or finding r from v and ω.","sort":90,"aliases":[]},{"id":92,"code":"02.06.04","level":3,"parent_id":88,"name":"Centripetal acceleration (a = v²/r)","description":"Questions computing centripetal acceleration a = v²/r = ω²r, its direction toward the centre, the net acceleration in uniform circular motion, or the force/acceleration needed to keep a particle on a circular path at given speed and radius.","sort":91,"aliases":[]},{"id":93,"code":"02.06.05","level":3,"parent_id":88,"name":"Non-uniform circular motion: tangential and radial acceleration components","description":"Questions where speed changes along a circle: splitting acceleration into tangential aₜ = dv/dt and radial v²/r components, magnitude a = √(aₜ² + a_c²), angle the total acceleration makes with velocity/radius, or speed given as a function of time or angle.","sort":92,"aliases":[]},{"id":94,"code":"02.06.06","level":3,"parent_id":88,"name":"Relative angular velocity between two moving bodies","description":"Questions using ω_rel = ω₁ − ω₂ for two bodies on the same or concentric circular tracks: time between successive meetings/overtakings, two particles starting together on a circle, or relative angular speed of clock hands.","sort":93,"aliases":[]},{"id":95,"code":"02.06.07","level":3,"parent_id":88,"name":"Circular motion with quadratic tangential drag","description":"Questions where tangential drag ∝ v² acts on a particle in circular motion: deceleration dv/dt = -kv² (or -kv²/r style setups), time or revolutions before stopping, and speed as a function of angle or time.","sort":94,"aliases":[]},{"id":96,"code":"02.07","level":2,"parent_id":60,"name":"Kinematics in two dimensions (position, velocity and acceleration vectors)","description":"Parent node for 2D vector kinematics: position r(t), velocity v(t) = dr/dt, acceleration a(t) given as vectors or components — finding speed, direction of motion, trajectory shape, or when v ⊥ a, v ∥ a.","sort":95,"aliases":["general 2D motion","curvilinear motion","motion in polar coordinates","trajectory from parametric position vector","motion in a plane"]},{"id":97,"code":"02.07.01","level":3,"parent_id":96,"name":"Curvilinear motion along a specified path","description":"Questions where the path is specified (e.g., y = f(x), a circle, y² = kx) and the particle moves along it with given speed or vₓ: finding v_y, velocity direction, acceleration components, or angle of velocity with the path/axis.","sort":96,"aliases":[]},{"id":98,"code":"02.07.02","level":3,"parent_id":96,"name":"General 2D parametric motion","description":"Questions giving x(t) and y(t) parametrically: computing velocity and acceleration vectors by differentiation, speed at a given instant, or eliminating t to identify the trajectory (line, parabola, circle).","sort":97,"aliases":[]},{"id":99,"code":"02.07.03","level":3,"parent_id":96,"name":"Motion in polar coordinates","description":"Questions in polar coordinates (r, θ): radial and transverse velocity/acceleration components (ṙ, rθ̇, r̈ − rθ̇², rθ̈ + 2ṙθ̇), particles moving with r = f(t), θ = g(t), or converting Cartesian data to polar motion.","sort":98,"aliases":[]},{"id":100,"code":"02.07.04","level":3,"parent_id":96,"name":"Path from a time-parametrized position vector","description":"Questions where a time-parametrized position vector r(t) = x(t)î + y(t)ĵ is given and the task is to identify the path/trajectory (straight line, parabola, circle) by eliminating t, or find where the particle crosses axes/has specific velocity or acceleration directions.","sort":99,"aliases":[]},{"id":101,"code":"02.07.05","level":3,"parent_id":96,"name":"Vector addition and resolution of vectors","description":"Questions on vector algebra in mechanics: resultant of two/three vectors via triangle/parallelogram law, resolving into components, unit vectors, magnitude and direction of a vector sum, minimum resultant, or angle between vectors using dot products.","sort":100,"aliases":[]},{"id":102,"code":"02.08","level":2,"parent_id":60,"name":"Relative velocity in two dimensions","description":"Parent node for relative velocity in a plane: computing v_AB = v_A − v_B for two moving objects, closest approach, interception/collision conditions, and classic river-boat and rain-man scenarios.","sort":101,"aliases":["relative motion in a plane","relative velocity 2d"]},{"id":103,"code":"02.08.01","level":3,"parent_id":102,"name":"Relative velocity in two dimensions (v_AB = v_A − v_B)","description":"Questions directly applying v_AB = v_A − v_B with vector components: velocity of A as seen by B, magnitude and direction of relative velocity, time/distance for one vehicle to overtake another on perpendicular or angled roads.","sort":102,"aliases":[]},{"id":104,"code":"02.08.02","level":3,"parent_id":102,"name":"River-boat problems: shortest time and shortest path crossing","description":"River-crossing questions: boat speed vs stream speed, heading angle for shortest time (aim straight across) vs shortest path (upstream drift zero), drift downstream, minimum crossing time, or condition for reaching the opposite point directly.","sort":103,"aliases":[]},{"id":105,"code":"02.08.03","level":3,"parent_id":102,"name":"Rain-man problems: apparent velocity of rain","description":"Rain-man questions: rain falling vertically or at an angle while a person walks/runs — apparent (relative) velocity of rain, umbrella angle from vertical, man's speed for rain to appear vertical, or finding true rain speed from two observed directions.","sort":104,"aliases":[]},{"id":106,"code":"02.08.04","level":3,"parent_id":102,"name":"Collision course and shortest distance between two moving objects","description":"Two ships, cars or swimmers moving with constant velocities; find whether they collide (relative velocity along the line joining them) or compute minimum separation and time of closest approach using v_rel = v1 - v2 and d_min = |r_rel x v_rel|/|v_rel|.","sort":105,"aliases":[]},{"id":107,"code":"02.08.05","level":3,"parent_id":102,"name":"Aircraft-wind relative velocity problems","description":"Plane's airspeed combined vectorially with wind velocity to get ground speed/track; questions ask the heading to fly due north, drift angle, resultant ground velocity, or time to reach a destination, plus analogous rain-vs-man umbrella problems.","sort":106,"aliases":[]},{"id":108,"code":"02.09","level":2,"parent_id":60,"name":"Projectile motion","description":"General projectile launched with speed u at angle theta under gravity: parabolic path from independent horizontal (uniform u cosθ) and vertical (uniform acceleration g) motions; parent node for all projectile question types.","sort":107,"aliases":["oblique projectile","horizontal projectile","projectile on an inclined plane","range, time of flight and maximum height","equation of trajectory"]},{"id":109,"code":"02.09.01","level":3,"parent_id":108,"name":"Projectile striking an inclined plane perpendicularly","description":"Projectile whose velocity at impact is perpendicular to an inclined plane of angle α; set the velocity component parallel to the incline to zero, giving relations like tanθ = cotα or time of flight t = u cos(θ+α)/g cosα.","sort":108,"aliases":[]},{"id":110,"code":"02.09.02","level":3,"parent_id":108,"name":"Projectile launched at an angle: time of flight, maximum height and range","description":"Standard angled-projection formulas: T = 2u sinθ/g, H = u²sin²θ/(2g), R = u²sin2θ/g; questions compute these quantities, use complementary-angle equal ranges (θ and 90°−θ), or find u/θ from given range and height.","sort":109,"aliases":[]},{"id":111,"code":"02.09.03","level":3,"parent_id":108,"name":"Horizontal projection from a height","description":"Body thrown horizontally with speed u from a cliff/tower of height h: time of fall t = √(2h/g), horizontal range x = u√(2h/g), and impact speed/direction from v_y = gt; trajectory is y = gx²/(2u²).","sort":110,"aliases":[]},{"id":112,"code":"02.09.04","level":3,"parent_id":108,"name":"Equation of trajectory and direction of velocity at a point","description":"Trajectory equation y = x tanθ − gx²/(2u²cos²θ); questions derive it, eliminate t from parametric equations, or find the velocity vector at a point via tanβ = v_y/v_x = (u sinθ − gt)/(u cosθ) and the speed there.","sort":111,"aliases":[]},{"id":113,"code":"02.09.05","level":3,"parent_id":108,"name":"Projectile motion on an inclined plane","description":"Projectile launched from an inclined plane of angle α, usually resolving motion along/perpendicular to the incline: time of flight T = 2u sin(θ−α)/(g cosα), range along incline R = 2u² sin(θ−α)cosθ/(g cos²α), and condition for maximum range.","sort":112,"aliases":[]},{"id":114,"code":"03","level":1,"parent_id":null,"name":"Laws of Motion & Friction","description":"JEE chapter 03: Laws of Motion & Friction. Concepts: Newton's Laws of Motion; Impulse and conservation of linear momentum; Friction; Dynamics of circular motion; Dynamics of Connected Bodies; Frames of Reference and Non-Inertial (Pseudo) Forces; Motion with Air Drag and Terminal Velocity; Variable-Mass Systems and Rocket Thrust.","sort":113,"aliases":["Laws of Motion"]},{"id":115,"code":"03.01","level":2,"parent_id":114,"name":"Newton's Laws of Motion","description":"Parent node for all of Newtonian dynamics: inertia, F = ma, action–reaction pairs, and their application to blocks, ropes, pulleys, inclines, and connected bodies. Any problem requiring ΣF = ma on a body or system maps here or to a subconcept.","sort":114,"aliases":["free-body diagrams","inertia","equilibrium of a particle (concurrent forces, Lami's theorem)"]},{"id":116,"code":"03.01.01","level":3,"parent_id":115,"name":"Tension in a chain sliding on a curved surface","description":"A uniform chain sliding off a smooth curved surface (hemisphere, cylinder, table edge): tension at a link arises from the centripetal requirement T = λv² (per unit length) plus gravity components. Questions ask for tension at the top link or at a point when a chain slides down a smooth sphere/cylinder.","sort":115,"aliases":[]},{"id":117,"code":"03.01.02","level":3,"parent_id":115,"name":"Variable tension in an accelerating rope","description":"Tension that varies along a massive rope or chain being accelerated: applying F = ma to a segment shows T(x) changes linearly along the rope, e.g., rope of mass m pulled by force F has tension F at the pulled end and less at the free end. Questions ask tension at a point a fraction along the rope, or in a hanging accelerated cable.","sort":116,"aliases":[]},{"id":118,"code":"03.01.03","level":3,"parent_id":115,"name":"Free-body diagrams","description":"Constructing free-body diagrams: isolating a body, drawing weight, normal, tension, friction, and applied forces correctly, including for connected systems, wedges, and pulleys. Questions may ask to identify missing/wrong forces on a diagram or count forces acting on a body in a given situation.","sort":117,"aliases":[]},{"id":119,"code":"03.01.04","level":3,"parent_id":115,"name":"Inertia and momentum","description":"Conceptual questions on inertia (resistance to change of state, mass as measure of inertia) and linear momentum p = mv: first-law statements, why passengers lurch in a braking bus, comparing momenta of two bodies, and qualitative distinctions between inertia, momentum, and force.","sort":118,"aliases":[]},{"id":120,"code":"03.01.05","level":3,"parent_id":115,"name":"Newton's laws of motion (fundamental principles)","description":"The three laws themselves: first law and inertia, second law F = dp/dt (and F = ma for constant mass), third law with correct identification of action–reaction pairs acting on different bodies. Conceptual items test law statements, frames of reference, and spotting invalid third-law pairs.","sort":119,"aliases":[]},{"id":121,"code":"03.01.06","level":3,"parent_id":115,"name":"Variational principles and Euler-Lagrange equation","description":"Advanced variational formulation: principle of least/stationary action, functional minimization, and the Euler–Lagrange equation d/dt(∂L/∂q̇) − ∂L/∂q = 0 used to derive equations of motion. Questions set up a Lagrangian for simple systems (pendulum, particle in potential) and extract the equation of motion.","sort":120,"aliases":[]},{"id":122,"code":"03.01.07","level":3,"parent_id":115,"name":"Catenary (equilibrium shape of a hanging chain)","description":"Equilibrium shape of a hanging chain/cable under its own weight: catenary y = a cosh(x/a), tension at the lowest point T₀ = μga, tension at supports, and sag calculations. Questions give chain length, span, and weight density and ask for sag or end tension, distinguishing catenary from parabola.","sort":121,"aliases":[]},{"id":123,"code":"03.01.08","level":3,"parent_id":115,"name":"Minimum-time descent and velocity reorientation problems","description":"Optimization problems in dynamics: brachistochrone-type minimum-time descent curves (cycloid) and problems where a body must rotate its velocity vector through an angle with limited acceleration, giving minimum time t = vΔθ/a (e.g., aircraft/car turning, or a particle reorienting velocity with bounded acceleration magnitude).","sort":122,"aliases":[]},{"id":124,"code":"03.02","level":2,"parent_id":114,"name":"Impulse and conservation of linear momentum","description":"Parent node for impulse and momentum: J = Δp, conservation of total momentum in isolated systems, and its use in collisions, explosions, recoil, and variable-mass situations. Any problem where forces are internal or impulsive maps here or to a subconcept.","sort":123,"aliases":["conservation of linear momentum","impulse","impulse-momentum theorem","recoil and explosion problems"]},{"id":125,"code":"03.02.01","level":3,"parent_id":124,"name":"Conservation of momentum","description":"Conservation of linear momentum: when net external force is zero, Σmᵢvᵢ is constant, applied to two-body separation, collisions, men walking on planks, and spring-released carts. Questions equate initial and final total momentum, often with velocity components or relative velocity constraints.","sort":124,"aliases":[]},{"id":126,"code":"03.02.02","level":3,"parent_id":124,"name":"Impulse-momentum theorem","description":"Impulse–momentum theorem J = ∫F dt = Δp = mΔv: computing change in momentum from a known force over time, or average force during a collision from Δp and contact time. Questions include ball bouncing off wall/bat, person landing and bending knees, and force during impact.","sort":125,"aliases":[]},{"id":127,"code":"03.02.03","level":3,"parent_id":124,"name":"Impulse from force-time graphs and variable forces","description":"Finding impulse as the area under a force–time graph (triangular, rectangular, sinusoidal, or piecewise pulses) and equating it to momentum change; also variable forces F(t) or F(x) integrated to get Δp. Questions ask for final velocity, peak force, or duration given a graph.","sort":126,"aliases":[]},{"id":128,"code":"03.02.04","level":3,"parent_id":124,"name":"Recoil and explosion problems (momentum conservation)","description":"Momentum conservation in recoil and explosions: gun recoil velocity (MV + mu = 0), bomb bursting into fragments with vector momentum balance, radioactive decay of a nucleus, and two carts pushed apart by a compressed spring. Questions ask recoil speeds, fragment velocity directions, or kinetic energy distribution.","sort":127,"aliases":[]},{"id":129,"code":"03.02.05","level":3,"parent_id":124,"name":"Boat-man problems (internal forces and momentum conservation)","description":"Boat–man (and plank–man) problems where internal forces move parts of a system but the center of mass stays fixed: m·x_man = M·x_boat displacement ratios, man walking from one end to the other, throwing mass from a boat, and related center-of-mass-shift calculations.","sort":128,"aliases":[]},{"id":130,"code":"03.03","level":2,"parent_id":114,"name":"Friction","description":"Parent node for friction: static and kinetic friction laws f ≤ μN, angle of repose, friction on inclines, two-block systems, and rolling resistance. Any question needing μ, normal force, or friction direction maps here or to a subconcept.","sort":129,"aliases":["static friction","kinetic friction","rolling friction","angle of repose"]},{"id":131,"code":"03.03.01","level":3,"parent_id":130,"name":"Capstan (belt-friction) equation","description":"Capstan/belt-friction equation T₂ = T₁e^{μθ} for a rope or belt wrapped around a rough cylinder through angle θ: a small holding force T₁ balances a large load T₂. Questions compute required holding force for a sailor/rope over a bollard, or number of turns needed for a given load ratio.","sort":130,"aliases":[]},{"id":132,"code":"03.03.02","level":3,"parent_id":130,"name":"Static friction","description":"Static friction as a self-adjusting force with f ≤ μ_sN, applied at impending motion where f = μ_sN: blocks on the verge of sliding, minimum force to start motion at optimal angle, angle of repose tanθ = μ_s, and ladder/stacked-block impending-slip conditions.","sort":131,"aliases":[]},{"id":133,"code":"03.03.03","level":3,"parent_id":130,"name":"Kinetic friction","description":"Kinetic friction f = μ_kN opposing relative sliding, treated as constant: deceleration and stopping distance of sliding blocks, work/heat dissipated by friction, acceleration of blocks on rough inclines, and relative slipping in two-block systems.","sort":132,"aliases":[]},{"id":134,"code":"03.03.04","level":3,"parent_id":130,"name":"Rolling friction","description":"Rolling friction (rolling resistance): the small resistive torque/force due to deformation, coefficient of rolling friction μ_r, why rolling requires far less force than sliding, and conditions for rolling without slipping versus slipping. Questions compare pull needed to roll versus drag a cylinder/wheel and compute retardation of a rolling body.","sort":133,"aliases":[]},{"id":135,"code":"03.03.05","level":3,"parent_id":130,"name":"Laws of dry friction (Coulomb's law)","description":"Questions applying Coulomb's laws: f_s ≤ μ_s N, f_k = μ_k N, μ_s > μ_k, angle of repose tanθ = μ_s, and friction independent of contact area. Typical tasks: find maximum applied force before slipping, friction during sliding, minimum μ to prevent slipping on inclines.","sort":134,"aliases":[]},{"id":136,"code":"03.03.06","level":3,"parent_id":130,"name":"Stacked blocks with layered friction (sliding vs tipping)","description":"Two or three blocks stacked with friction at each interface; find max horizontal force F (on top or bottom block) so blocks move together, or accelerations after slipping, using f_max = μN at each layer. Includes sliding-vs-tipping checks via torque balance about the bottom edge (block tips when F·h exceeds μMg·b/2).","sort":135,"aliases":[]},{"id":137,"code":"03.04","level":2,"parent_id":114,"name":"Dynamics of circular motion","description":"Parent node for F_net = mv²/r directed toward the center: resolving forces into radial and tangential components for bodies on circular paths. Covers generic questions computing v, ω, r, or the force supplying centripetal acceleration (tension, normal, gravity, friction).","sort":136,"aliases":["circular motion","centripetal force","banking of roads","conical pendulum","vertical circular motion","car on curved track"]},{"id":138,"code":"03.04.01","level":3,"parent_id":137,"name":"String winding around a cylinder (involute motion)","description":"Bob on a string unwinding from a fixed cylinder (involute path): free string length shortens as the bob swings down. Use energy conservation for speed at a given angle and radial force balance for tension; typical asks are speed, tension, or time to fully wind/unwind.","sort":137,"aliases":[]},{"id":139,"code":"03.04.02","level":3,"parent_id":137,"name":"Centripetal force in uniform circular motion","description":"Identifying and computing the centripetal force in uniform circular motion: a_c = v²/r = ω²r = 4π²r/T². Archetypes: stone on a string (T = mv²/r, plus mg at the bottom of a vertical circle), car on a curve, and which real force plays the centripetal role.","sort":138,"aliases":[]},{"id":140,"code":"03.04.03","level":3,"parent_id":137,"name":"Banking of roads","description":"Vehicle on a banked curve: frictionless optimum speed v = √(rg tanθ), and with friction the v_max/v_min formulas involving μ and banking angle. Questions give θ, r, μ and ask safe speed range or required banking angle.","sort":139,"aliases":[]},{"id":141,"code":"03.04.04","level":3,"parent_id":137,"name":"Conical pendulum","description":"Conical pendulum: bob revolving in a horizontal circle with string at angle θ, satisfying T cosθ = mg and T sinθ = mv²/r, giving tanθ = v²/(rg) and period 2π√(L cosθ/g). Questions ask tension, angle, speed, or time period given L and θ.","sort":140,"aliases":[]},{"id":142,"code":"03.04.05","level":3,"parent_id":137,"name":"Vertical circular motion (minimum speeds, slack string/rod conditions)","description":"Vertical circles: minimum top speed √(gr) for a string (slack when T < 0), v_top ≥ 0 for a rod, v_bottom = √(5gr) for a pendulum released from horizontal, and T_bottom − T_top = 6mg. Archetypes: bucket of water, bob on string/rod, condition for completing the loop.","sort":141,"aliases":[]},{"id":143,"code":"03.04.06","level":3,"parent_id":137,"name":"Friction-limited speed on level circular turns","description":"Level (unbanked) circular turns limited by friction: v_max = √(μrg), with friction supplying the centripetal force. Questions ask maximum speed without skidding, minimum μ for a given speed, or whether a car rounds a curve safely.","sort":142,"aliases":[]},{"id":144,"code":"03.05","level":2,"parent_id":114,"name":"Dynamics of Connected Bodies","description":"Parent node for bodies linked by strings/pulleys: draw FBDs, apply Newton's second law to each body plus constraint relations to find accelerations and tensions. Covers blocks on tables/inclines connected over pulleys, and elevator-hung pulley systems.","sort":143,"aliases":["pulleys, strings and wedges","Atwood machine systems"]},{"id":145,"code":"03.05.01","level":3,"parent_id":144,"name":"Compound Atwood machine systems","description":"Atwood-type setups with multiple/movable pulleys (pulley hanging from another string, compound arrangements). Solve using pulley constraint equations (e.g., a1 + a2 = 2a_pulley) plus force balance on massless pulleys to get each mass's acceleration and string tensions.","sort":144,"aliases":[]},{"id":146,"code":"03.05.02","level":3,"parent_id":144,"name":"Connected-system constraints (pulleys, strings, wedges)","description":"Writing constraint equations for connected systems: constant string length giving relations like a1 + a2 = 2a3, wedge constraints (acceleration components along incline surfaces matching), and normal-reaction coupling between wedge and block. Questions give a geometry and ask for acceleration ratios or tensions.","sort":145,"aliases":[]},{"id":147,"code":"03.06","level":2,"parent_id":114,"name":"Frames of Reference and Non-Inertial (Pseudo) Forces","description":"Parent node on inertial vs non-inertial frames: when Newton's laws fail and how to fix FBDs by adding pseudo force −ma_frame. Covers lift problems, blocks on accelerating surfaces, and pendulums in accelerating vehicles.","sort":146,"aliases":["Galilean relativity","centrifugal and Coriolis forces","fictitious forces"]},{"id":148,"code":"03.06.01","level":3,"parent_id":147,"name":"Galilean transformation and Newtonian invariance","description":"Galilean transformations x' = x − vt, u' = u − v, a' = a, with F = ma invariant across inertial frames. Questions test relative velocity computations and whether Newton's first/second law holds in a given moving frame.","sort":147,"aliases":[]},{"id":149,"code":"03.06.02","level":3,"parent_id":147,"name":"Non-inertial frames (centrifugal and Coriolis forces)","description":"Rotating (non-inertial) frames: centrifugal force mω²r outward and Coriolis force −2mω×v. Archetypes: effective gravity on rotating Earth, bead on a rotating rod, deflection of projectiles/winds, equilibrium in a rotating drum.","sort":148,"aliases":[]},{"id":150,"code":"03.06.03","level":3,"parent_id":147,"name":"Non-inertial frames and fictitious forces","description":"Linearly accelerating frames and fictitious forces: add −ma to FBDs, giving effective gravity g_eff = g + a (lift accelerating up) or g − a (down), and tilted pendulum/plumb line in an accelerating car at tanθ = a/g. Questions ask apparent weight, normal reactions, or pendulum angle.","sort":149,"aliases":[]},{"id":151,"code":"03.07","level":2,"parent_id":114,"name":"Motion with Air Drag and Terminal Velocity","description":"Parent node for motion with resistive force ∝ v or v²: setting up m dv/dt = mg − kv^n, terminal velocity, and approach to terminal speed. Archetypes: raindrops, skydivers, spheres falling through fluid.","sort":150,"aliases":["linear drag","quadratic drag"]},{"id":152,"code":"03.07.01","level":3,"parent_id":151,"name":"Motion with linear drag and terminal velocity","description":"Linear drag F = −bv: terminal velocity v_t = mg/b, exponential approach v(t) = v_t(1 − e^(−t/τ)) with time constant τ = m/b. Questions ask v_t, time to reach a fraction of v_t, or v as a function of time/distance.","sort":151,"aliases":[]},{"id":153,"code":"03.07.02","level":3,"parent_id":151,"name":"Vertical motion with quadratic air drag","description":"Quadratic drag F = −cv²: terminal speed v_t = √(mg/c), with different equations for upward vs downward motion (deceleration g + cv²/m going up). Questions give c or v_t and ask for terminal speed, velocity profiles, or maximum height with drag.","sort":152,"aliases":[]},{"id":154,"code":"03.08","level":2,"parent_id":114,"name":"Variable-Mass Systems and Rocket Thrust","description":"Parent node on variable-mass dynamics: F_ext = m(dv/dt) − u(dm/dt) with thrust = u|dm/dt| for ejected mass at relative speed u. Covers rockets, conveyor belts gaining mass, and falling chains.","sort":153,"aliases":["momentum-flux systems","rocket equation"]},{"id":155,"code":"03.08.01","level":3,"parent_id":154,"name":"Variable-mass and momentum-flux systems","description":"Momentum-flux problems: force from mass entering/leaving a system at rate dm/dt with relative velocity, e.g., sand dropped onto a conveyor belt (extra force = v dm/dt), sand leaking from a cart, water jet hitting a surface. Apply F = d(mv)/dt carefully distinguishing the flux term.","sort":154,"aliases":[]},{"id":156,"code":"03.08.02","level":3,"parent_id":154,"name":"Variable-mass systems (rocket/thrust equation)","description":"Rocket equation problems: v = v0 + u ln(m0/m), acceleration a = (u/m)(dm/dt) − g, and thrust T = u(dm/dt). Questions give burn rate, exhaust speed, and mass ratio, asking for velocity, acceleration, or fuel fraction needed for a target Δv.","sort":155,"aliases":[]},{"id":157,"code":"04","level":1,"parent_id":null,"name":"Work, Power, Energy & Collisions","description":"JEE chapter 04: Work, Power, Energy & Collisions. Concepts: Work and work-energy theorem; Potential energy and conservation of mechanical energy; Power; Collisions (elastic and inelastic).","sort":156,"aliases":["Work, Energy and Power"]},{"id":158,"code":"04.01","level":2,"parent_id":157,"name":"Work and work-energy theorem","description":"Parent node for work, kinetic energy, and the work–energy theorem; tag here only for general work–energy questions not matching a subconcept.","sort":157,"aliases":["work done by a force","work done by a variable force","kinetic energy"]},{"id":159,"code":"04.01.01","level":3,"parent_id":158,"name":"Kinetic energy and the work-energy theorem","description":"Problems using W_net = ΔK = ½mv² − ½mu²: find final speed from net work, stopping distance from initial speed and friction force, or work needed to accelerate a body; includes the definitions K = ½mv² and K = p²/2m.","sort":158,"aliases":[]},{"id":160,"code":"04.01.02","level":3,"parent_id":158,"name":"Work done by a force","description":"Work by a constant force: W = F s cosθ as a dot product, with questions on positive/negative/zero work, work by gravity, normal, friction, or an applied force on blocks on inclines, and work when force and displacement are at an angle.","sort":159,"aliases":[]},{"id":161,"code":"04.01.03","level":3,"parent_id":158,"name":"Work done by a variable force (F-x graph)","description":"Work by a position-dependent force found as the area under an F–x graph or W = ∫F dx; classic cases are spring work ½kx², force varying linearly with x, and comparing areas for two paths between the same endpoints.","sort":160,"aliases":[]},{"id":162,"code":"04.02","level":2,"parent_id":157,"name":"Potential energy and conservation of mechanical energy","description":"Parent node for potential energy and conservation of mechanical energy; tag here only if the question mixes these ideas without fitting a subconcept.","sort":161,"aliases":["potential energy (gravitational and spring)","conservation of energy"]},{"id":163,"code":"04.02.01","level":3,"parent_id":162,"name":"Conservation of mechanical energy","description":"Problems where K + U is constant (no friction): speed at the bottom of a swing or slide, height reached by a projectile or pendulum, motion over tracks and hills, and comparing speeds/energies at two points via ½mv₁² + U₁ = ½mv₂² + U₂.","sort":162,"aliases":[]},{"id":164,"code":"04.02.02","level":3,"parent_id":162,"name":"Potential energy (gravitational and spring)","description":"Computing stored potential energy: U = mgh for gravity (with a chosen reference level) and U = ½kx² for springs, including energy stored on compressing/stretching a spring and changes in gravitational PE between heights.","sort":163,"aliases":[]},{"id":165,"code":"04.02.03","level":3,"parent_id":162,"name":"Equilibrium from an interaction potential","description":"Given a potential U(x) (e.g., U = −a/x + b/x² or a polynomial), find equilibrium points from F = −dU/dx = 0, classify stable/unstable via the second derivative, and find small-oscillation frequency or minimum energy for bound motion.","sort":164,"aliases":[]},{"id":166,"code":"04.02.04","level":3,"parent_id":162,"name":"Conservative forces and potential energy","description":"Conceptual/derivation questions on conservative forces: path independence of work, zero work over a closed loop, F = −dU/dx recovering force from potential (e.g., gravity, spring), and identifying whether a given force is conservative.","sort":165,"aliases":[]},{"id":167,"code":"04.02.05","level":3,"parent_id":162,"name":"Work done by friction and loss of mechanical energy","description":"Friction dissipating mechanical energy: ΔE = μ m g d (or f·d) equals heat generated; questions on a block sliding down a rough incline, stopping distance on a rough floor, or energy lost looping a track, using K_i + U_i − f d = K_f + U_f.","sort":166,"aliases":[]},{"id":168,"code":"04.03","level":2,"parent_id":157,"name":"Power","description":"Parent node for power; tag here only for general power questions not matching a subconcept.","sort":167,"aliases":[]},{"id":169,"code":"04.03.01","level":3,"parent_id":168,"name":"Average and instantaneous power","description":"Average power P = W/t and instantaneous power P = dW/dt = F·v cosθ; questions ask power of a force on a moving block, power at a given speed, or average power over an interval (e.g., engine accelerating a car).","sort":168,"aliases":[]},{"id":170,"code":"04.03.02","level":3,"parent_id":168,"name":"Power of engines and pumps","description":"Engine/pump problems: power needed to lift mass m through height h in time t (P = mgh/t), water pumped per second, output power with efficiency η, or power delivered against gravity/friction at constant speed.","sort":169,"aliases":[]},{"id":171,"code":"04.03.03","level":3,"parent_id":168,"name":"Motion under constant power","description":"Motion of a body (vehicle) whose engine delivers constant power P: use P = Fv = m a v, giving v² = v₀² + 2Pt/m, v–t and x–t relations, or limiting (terminal) speed when resistance is present.","sort":170,"aliases":[]},{"id":172,"code":"04.04","level":2,"parent_id":157,"name":"Collisions (elastic and inelastic)","description":"Parent node for collisions between bodies; tag here only if the question is generic and doesn't match the specific collision subconcept.","sort":171,"aliases":["collisions","elastic collisions","inelastic collisions","coefficient of restitution"]},{"id":173,"code":"04.04.01","level":3,"parent_id":172,"name":"Collisions","description":"Standard collision computations: 1D head-on collisions using momentum conservation plus coefficient of restitution e = (v₂−v₁)/(u₁−u₂); elastic (e = 1, KE conserved), perfectly inelastic (bodies stick, maximum KE loss), and finding final velocities or fraction of KE lost.","sort":172,"aliases":[]},{"id":174,"code":"04.04.02","level":3,"parent_id":172,"name":"Elastic collisions","description":"Head-on elastic collisions where kinetic energy is conserved: use v1' = ((m1-m2)u1 + 2m2u2)/(m1+m2) and v2' = ((m2-m1)u2 + 2m1u1)/(m1+m2). Typical questions: equal masses exchanging velocities, a ball rebounding off a massive wall, spring-compressed collision between blocks, or pendulum bobs colliding and finding post-impact speeds.","sort":173,"aliases":[]},{"id":175,"code":"04.04.03","level":3,"parent_id":172,"name":"Inelastic collisions","description":"Collisions with kinetic energy loss: perfectly inelastic case where bodies stick and move with common velocity v = (m1u1 + m2u2)/(m1+m2), plus partially inelastic cases. Typical questions: bullet embedding in a block, ballistic pendulum rise height, fraction/amount of KE lost in a crash, or maximum compression of a spring during impact.","sort":174,"aliases":[]},{"id":176,"code":"04.04.04","level":3,"parent_id":172,"name":"Coefficient of restitution","description":"Coefficient of restitution e = (relative velocity of separation)/(relative velocity of approach) = (v2-v1)/(u1-u2), with e=1 elastic and e=0 perfectly inelastic. Typical questions: ball dropped from height h rebounding to e²h, height after the nth bounce, total distance/time before a ball stops bouncing, or finding e from given pre- and post-collision velocities.","sort":175,"aliases":[]},{"id":177,"code":"04.04.05","level":3,"parent_id":172,"name":"Collisions in two dimensions (oblique collisions)","description":"Two-dimensional (oblique/glancing) collisions: conserve momentum along x and y separately, or along and perpendicular to the line of impact. Typical questions: billiard-ball collisions where identical masses scatter at 90° after elastic impact, a ball striking a wall at an angle and rebounding, or finding final velocity vectors after a glancing collision.","sort":176,"aliases":[]},{"id":178,"code":"05","level":1,"parent_id":null,"name":"System of Particles & Rotational Dynamics","description":"JEE chapter 05: System of Particles & Rotational Dynamics. Concepts: Centre of mass; Moment of inertia; Angular momentum; Rigid body dynamics; Torque; Rolling motion.","sort":177,"aliases":["System of Particles and Rotational Motion"]},{"id":179,"code":"05.01","level":2,"parent_id":178,"name":"Centre of mass","description":"Questions on locating the centre of mass of discrete particle systems or continuous bodies (rods, discs, composite shapes) using Σmr/Σm or integration.","sort":178,"aliases":[]},{"id":180,"code":"05.01.01","level":3,"parent_id":179,"name":"Centre of mass calculation for discrete and continuous systems","description":"Questions on calculating centre of mass coordinates for discrete point masses, uniform rods, semicircular rings/discs, or composite bodies with removed portions.","sort":179,"aliases":[]},{"id":181,"code":"05.01.02","level":3,"parent_id":179,"name":"Motion of centre of mass and momentum conservation","description":"Questions using V_cm = Σm_i v_i/M and F_ext = M a_cm, e.g., a man walking on a plank/boat, a person climbing a ladder on a trolley, or recoil of a cart, where zero external force means the centre of mass stays fixed or moves uniformly and momentum is conserved.","sort":180,"aliases":[]},{"id":182,"code":"05.01.03","level":3,"parent_id":179,"name":"Centre of mass of composite bodies and cut-out shapes","description":"Finding CM coordinates of composite objects (two rods joined in an L, disc+ring, sphere on a block) or shapes with holes using x_cm = Σm_i x_i/Σm_i and the negative-mass trick for cut-outs.","sort":181,"aliases":[]},{"id":183,"code":"05.01.04","level":3,"parent_id":179,"name":"Centre of mass shift on mass removal or addition","description":"Numericals on how far the CM shifts when mass is removed, added, or relocated, using Δx = (m/M)·d — e.g., a circular piece cut from a disc, a boy moving from one end of a boat to the other, or sand leaking from a cart.","sort":182,"aliases":[]},{"id":184,"code":"05.01.05","level":3,"parent_id":179,"name":"Centre of mass of two-body systems and reduced mass","description":"Two-body setups where the CM divides the line joining the masses (x_cm = (m1x1+m2x2)/(m1+m2), r1/r2 = m2/m1) and reduced mass μ = m1m2/(m1+m2) is used to split motion into CM plus relative motion, e.g., in gravitation or two-block spring problems.","sort":183,"aliases":[]},{"id":185,"code":"05.01.06","level":3,"parent_id":179,"name":"Centre of mass in exploding and fragmenting systems","description":"Explosion/fragmentation problems where internal forces cannot change the CM: a shell exploding at the top of its trajectory with fragments landing at given ranges, or velocities of pieces found from momentum conservation with the CM continuing on its original path.","sort":184,"aliases":[]},{"id":186,"code":"05.02","level":2,"parent_id":178,"name":"Moment of inertia","description":"General moment-of-inertia questions: computing I = Σmr² or ∫r²dm for bodies, recalling standard results, applying axis theorems, and using I in τ = Iα and rotational energy — the umbrella node for any MI-based calculation.","sort":185,"aliases":["inertia tensor","principal axes"]},{"id":187,"code":"05.02.01","level":3,"parent_id":186,"name":"Inertia tensor and nonparallel L and omega","description":"Advanced questions where L is not parallel to ω because the inertia tensor has off-diagonal terms: computing L components from L_i = ΣI_ij ω_j for a rotating asymmetric body or a rod rotating about a skewed axis.","sort":186,"aliases":[]},{"id":188,"code":"05.02.02","level":3,"parent_id":186,"name":"Inertia tensor and principal axes","description":"Questions on principal axes — directions where the inertia tensor diagonalizes and products of inertia vanish — e.g., identifying principal axes of a symmetric body (disc, rectangular block) and writing L = I₁ω₁ê₁ + I₂ω₂ê₂.","sort":187,"aliases":[]},{"id":189,"code":"05.02.03","level":3,"parent_id":186,"name":"Moment of inertia of standard bodies","description":"Recall-and-apply questions on standard MI results about stated axes: ring and hollow cylinder MR², disc/solid cylinder MR²/2, rod ML²/12 or ML²/3 about an end, solid sphere 2MR²/5, hollow sphere 2MR²/3, cone, rectangular plate.","sort":188,"aliases":[]},{"id":190,"code":"05.02.04","level":3,"parent_id":186,"name":"Parallel and perpendicular axis theorems","description":"Problems applying the parallel axis theorem I = I_cm + Md² (MI about an edge, a tangent, or an off-centre axis) and the perpendicular axis theorem I_z = I_x + I_y for planar laminas like discs, rings, and triangular plates.","sort":189,"aliases":[]},{"id":191,"code":"05.02.05","level":3,"parent_id":186,"name":"Moment of inertia of composite and cut-out bodies","description":"MI of built-up or punctured bodies: adding contributions of welded parts (rod+disc, two spheres on a rod) or subtracting the hole's MI via the parallel axis theorem, e.g., a disc with an off-centre circular hole.","sort":190,"aliases":[]},{"id":192,"code":"05.02.06","level":3,"parent_id":186,"name":"Radius of gyration","description":"Radius-of-gyration questions using I = Mk², k = √(I/M): finding k of standard bodies about given axes, comparing k values, or converting a given k back into I for use in dynamics.","sort":191,"aliases":[]},{"id":193,"code":"05.02.07","level":3,"parent_id":186,"name":"Moment of inertia by integration for non-standard shapes","description":"Deriving MI from scratch by integration for non-standard objects: a rod with density varying as x, a triangular or semicircular lamina, an annular disc, using dm = λ dx or σ dA and integrating r² dm.","sort":192,"aliases":[]},{"id":194,"code":"05.03","level":2,"parent_id":178,"name":"Angular momentum","description":"Umbrella node for angular momentum: L = r×p for particles, L = Iω for rigid bodies, torque–angular momentum relations, and conservation — any question whose central quantity is L.","sort":193,"aliases":["torque-angular momentum relation"]},{"id":195,"code":"05.03.01","level":3,"parent_id":194,"name":"Torque and angular momentum","description":"Questions connecting torque and angular momentum broadly: computing net torque about a point or axis from applied forces and relating it to the change in L, including angular impulse (τ dt = dL) problems.","sort":194,"aliases":[]},{"id":196,"code":"05.03.02","level":3,"parent_id":194,"name":"Torque-free precession of a symmetric top","description":"Advanced torque-free motion of a symmetric top where ω precesses about the fixed angular momentum L with precession rate Ω = (I₃ − I)/I · ω₃, e.g., a freely spinning symmetric body's wobble — beyond standard JEE numericals.","sort":195,"aliases":[]},{"id":197,"code":"05.03.03","level":3,"parent_id":194,"name":"Torque and angular momentum of a particle","description":"Single-particle angular momentum: L = m(r×v) = mvr sinθ about a point or axis, τ = r×F, e.g., a particle in straight-line or circular motion, central-force problems where L about the force centre is constant.","sort":196,"aliases":[]},{"id":198,"code":"05.03.04","level":3,"parent_id":194,"name":"Angular momentum of rigid bodies and systems of particles","description":"Angular momentum of many-particle systems and rigid bodies: L = Σr_i×m_i v_i, decomposition L = R_cm×MV_cm + L_about_cm, and L = Iω for rotation about a fixed axis, e.g., two masses on a rod or a rotating disc.","sort":197,"aliases":[]},{"id":199,"code":"05.03.05","level":3,"parent_id":194,"name":"Conservation of angular momentum and its applications","description":"Conservation of L (Iω = constant when net external torque is zero) in classic setups: a spinning skater pulling arms in, a person on a rotating stool/disc dropping masses, a collapsing star, a bead sliding on a rotating rod, or a planet's areal velocity.","sort":198,"aliases":[]},{"id":200,"code":"05.03.06","level":3,"parent_id":194,"name":"Relation between torque and rate of change of angular momentum","description":"Questions built specifically on dL/dt = τ_net: deriving it, finding angular acceleration from a known torque, or finding the rate of change of L for a rotating body whose orientation changes, e.g., a rod falling about a hinge.","sort":199,"aliases":[]},{"id":201,"code":"05.04","level":2,"parent_id":178,"name":"Rigid body dynamics","description":"Umbrella node for rigid body motion: rotation about a fixed axis, rolling, rotational equilibrium, and combined translation–rotation — any question treating an extended body as rigid.","sort":200,"aliases":["rotational kinematics and dynamics","Euler equations","rotation"]},{"id":202,"code":"05.04.01","level":3,"parent_id":201,"name":"Euler equations and the intermediate-axis theorem","description":"Advanced Euler equations I₁ω̇₁ = (I₂−I₃)ω₂ω₃ (and cyclic) and the intermediate-axis (tennis-racket) theorem on stability of rotation about principal axes — beyond standard JEE-level problems.","sort":201,"aliases":[]},{"id":203,"code":"05.04.02","level":3,"parent_id":201,"name":"Rotational kinematics and dynamics","description":"General rotational kinematics and dynamics: relating θ, ω, α, using τ = Iα for fixed-axis rotation, and rotational analogues of Newton's laws — the mid-level node for standard fixed-axis problems.","sort":202,"aliases":[]},{"id":204,"code":"05.04.03","level":3,"parent_id":201,"name":"Rotational kinematics with constant angular acceleration","description":"Constant-angular-acceleration kinematics using ω = ω₀ + αt, θ = ω₀t + ½αt², ω² = ω₀² + 2αθ, e.g., a wheel spun up from rest, a fan switching off, or number-of-revolutions-before-stopping problems.","sort":203,"aliases":[]},{"id":205,"code":"05.04.04","level":3,"parent_id":201,"name":"Combined translational and rotational motion","description":"Rolling and translation-plus-rotation problems: rolling without slipping (v = ωR), velocity/acceleration of the top and bottom points of a wheel, rolling down an incline with a = g sinθ/(1 + I/MR²), total KE = ½Mv² + ½Iω², and friction in rolling/slipping.","sort":204,"aliases":[]},{"id":206,"code":"05.04.05","level":3,"parent_id":201,"name":"Instantaneous axis of rotation","description":"Questions using the instantaneous centre of rotation: for a rolling wheel/cylinder the contact point is momentarily at rest, so velocity of any point is ω×r from that point (top of wheel moves at 2v). Tasks include finding velocities of points on a rolling disc, locating the IC for a rod sliding with ends on wall/floor, or computing v of a point given v_cm and ω.","sort":205,"aliases":[]},{"id":207,"code":"05.04.06","level":3,"parent_id":201,"name":"Toppling versus sliding of rigid bodies","description":"Questions deciding whether a block/box on a rough incline or pushed by a horizontal force slides or topples: compare the torque of applied force and weight about the lower edge against the maximum friction limit μN, or find the critical height/force/angle at which toppling begins. Typical setups include a cube pushed at height h and a block on an incline with given μ and base-to-height ratio.","sort":206,"aliases":[]},{"id":208,"code":"05.04.07","level":3,"parent_id":201,"name":"Kinetic energy of rotating and rolling bodies","description":"Questions computing kinetic energy as ½Iω² for pure rotation or ½mv² + ½Iω² for rolling, and the fraction of total KE that is rotational (e.g., ring ½, disc 1/3, solid sphere 2/7). Common tasks: same KE but different shapes compared by v or ω, or KE ratio of two rolling bodies.","sort":207,"aliases":[]},{"id":209,"code":"05.04.08","level":3,"parent_id":201,"name":"Rotation matrices and Euler's rotation theorem","description":"Advanced-math questions on representing finite rotations by 3×3 rotation matrices, composing rotations about different axes, Euler angles, and Euler's theorem that any displacement of a rigid body about a fixed point equals a single rotation about some axis through that point. Expect matrix multiplication of rotation operators and axis/angle extraction rather than standard JEE mechanics.","sort":208,"aliases":[]},{"id":210,"code":"05.05","level":2,"parent_id":178,"name":"Torque","description":"Parent node for torque: τ = r×F with magnitude rF sinθ, τ = Iα for a rigid body about a fixed axis, and sign/direction conventions (right-hand rule, clockwise vs anticlockwise). Tag here for generic torque calculation or Newton's-second-law-for-rotation questions that don't fit a more specific child.","sort":209,"aliases":["equilibrium of rigid bodies"]},{"id":211,"code":"05.05.01","level":3,"parent_id":210,"name":"Pivoted pipe equilibrium under fluid-jet torque","description":"Questions where a pipe or L-shaped tube is pivoted at a point and a water jet flowing through or striking it creates a torque; the pipe is held in equilibrium or rotates until jet thrust torque balances weight or spring torque. Use force on jet ρAv² (or ρA(v−u)²) at the nozzle lever arm and set net torque about the pivot to zero.","sort":210,"aliases":[]},{"id":212,"code":"05.05.02","level":3,"parent_id":210,"name":"Equilibrium of rigid bodies","description":"Parent node for static equilibrium of rigid bodies: conditions ΣF = 0 and Στ = 0 (about any point), finding unknown support reactions, tensions, or applied forces. Tag here for general equilibrium setups not specifically ladders/beams, coplanar-force systems, or couples.","sort":211,"aliases":[]},{"id":213,"code":"05.05.03","level":3,"parent_id":210,"name":"Torque of a force and moment of a couple","description":"Questions computing the moment of a single force about a point or axis (τ = rF sinθ, perpendicular-distance method) and the moment of a couple: two equal, antiparallel forces separated by distance d give a pure moment Fd independent of the reference point. Includes recognizing that a couple produces rotation without translation.","sort":212,"aliases":[]},{"id":214,"code":"05.05.04","level":3,"parent_id":210,"name":"Equilibrium of rigid bodies under coplanar forces","description":"Questions on bodies in equilibrium under several coplanar forces: resolve forces in x and y, take moments about a cleverly chosen point to eliminate unknowns, and use the three-force principle (lines of action concurrent) when applicable. Typical content: a rod held by two strings, a bar resting on two supports, forces at angles given via trigonometry.","sort":213,"aliases":[]},{"id":215,"code":"05.05.05","level":3,"parent_id":210,"name":"Ladder, beam and hinged-rod equilibrium problems","description":"Classic ladder/beam/hinged-rod problems: a ladder leaning on a frictionless or rough wall with friction at the floor (find minimum μ or maximum angle), a beam on two supports or with a hanging load, and a rod hinged at one end held by a string or cable (find hinge reaction and string tension via torque about the hinge).","sort":214,"aliases":[]},{"id":216,"code":"05.05.06","level":3,"parent_id":210,"name":"Torque due to fluid jets and continuous streams","description":"Questions where a continuous fluid stream or jet exerts force and hence torque: jet striking a rotating vane, paddle wheel, or hinged plate, using F = (dm/dt)(v − u) = ρAv(v − u) and torque = F × lever arm. Includes finding angular speed of a water wheel or torque about a pivot from a steady stream.","sort":215,"aliases":[]},{"id":217,"code":"05.05.07","level":3,"parent_id":210,"name":"Shear force and bending moment in an accelerating bar","description":"Questions on internal forces in a bar/rod undergoing linear acceleration: cut the bar at a section and find the shear force (needed to accelerate the outer part, F_s = ma of the outer segment) and bending moment distribution, often for a rod pulled or pushed along its length or hanging in an accelerating lift.","sort":216,"aliases":[]},{"id":218,"code":"05.06","level":2,"parent_id":178,"name":"Rolling motion","description":"Parent node for rolling motion: condition v_cm = ωR for rolling without slipping, combined translation plus rotation about the centre, and rolling as instantaneous rotation about the contact point. Tag here for general rolling questions that don't fit a specific child.","sort":217,"aliases":["rolling without slipping","rolling body problems"]},{"id":219,"code":"05.06.01","level":3,"parent_id":218,"name":"Rolling body with an internal sliding mass","description":"Questions where a rolling body (shell, sphere, cylinder) contains a mass that slides inside it, altering the motion: e.g., a particle slipping inside a rolling sphere changes the effective inertia and friction requirement, or the internal mass's position shifts the centre of mass. Apply Newton's laws and torque about the contact point or centre with the shifting internal mass included.","sort":218,"aliases":[]},{"id":220,"code":"05.06.02","level":3,"parent_id":218,"name":"Pure rolling on horizontal and inclined planes","description":"Standard pure-rolling questions: acceleration a = g sinθ/(1 + I/mR²) down an incline, friction force f = (I/R²)·a required (static, no energy loss), minimum μ_s = (tanθ)/(1 + mR²/I) to sustain rolling, and rolling on horizontal plane under an applied force at height h or via a string wound on the body.","sort":219,"aliases":[]},{"id":221,"code":"05.06.03","level":3,"parent_id":218,"name":"Rolling with slipping and friction analysis","description":"Questions where the rolling body slips: kinetic friction μ_kN acts, giving both linear acceleration and angular acceleration until v = ωR is achieved. Tasks: initial forward/backward spin given, time and distance to reach pure rolling, final velocity, and energy dissipated as heat (loss = μ_k mg × relative slip distance).","sort":220,"aliases":[]},{"id":222,"code":"05.06.04","level":3,"parent_id":218,"name":"Energy in rolling bodies and rolling races","description":"Energy-based rolling questions: conservation of energy mgh = ½mv² + ½Iω² down an incline, and 'rolling races' comparing which body (ring, disc, solid/hollow sphere) reaches the bottom first — smaller I/mR² wins regardless of mass and radius. Also fraction of KE in rotation and speed at the base.","sort":221,"aliases":[]},{"id":223,"code":"05.06.05","level":3,"parent_id":218,"name":"Rolling body with internal moving mass","description":"Questions where a mass moves relative to a rolling body, e.g., a person walking inside a rolling cylinder or a bead/trolley moving within a rolling shell: use conservation of linear momentum and energy with the internal mass's relative velocity, or find how internal motion changes v_cm and ω of the combined system.","sort":222,"aliases":[]},{"id":224,"code":"06","level":1,"parent_id":null,"name":"Gravitation","description":"JEE chapter 06: Gravitation. Concepts: Kepler's laws; Gravitational field and intensity; Gravitational potential and potential energy; Satellites, orbital velocity and escape velocity; Acceleration due to gravity and its variation; Newton's law of gravitation.","sort":223,"aliases":[]},{"id":225,"code":"06.01","level":2,"parent_id":224,"name":"Kepler's laws","description":"Parent node for Kepler's empirical laws of planetary and satellite motion. Tag here when the question invokes orbital ellipses, equal-area law, or the period–semi-major-axis relation rather than Newton's law of gravitation directly.","sort":224,"aliases":[]},{"id":226,"code":"06.01.01","level":3,"parent_id":225,"name":"Kepler's laws of planetary motion","description":"Questions applying Kepler's three laws: elliptical orbits with the Sun at a focus, equal areas in equal times (angular momentum conservation), and T²∝a³. Typical content: comparing periods of two planets from orbital radii, finding speed ratio at perihelion vs aphelion, or computing a satellite's period from its orbital radius.","sort":225,"aliases":[]},{"id":227,"code":"06.01.02","level":3,"parent_id":225,"name":"Kepler's second law and areal velocity","description":"Questions using dA/dt = L/2m = constant (equal areas in equal times) from angular momentum conservation; typical items ask for perihelion/aphelion speed ratios via v_p r_p = v_a r_a, or area swept / time spent in halves of an elliptical orbit.","sort":226,"aliases":[]},{"id":228,"code":"06.01.03","level":3,"parent_id":225,"name":"Kepler's third law and period-distance relation","description":"Questions applying T² = 4π²a³/GM to compare periods of planets or satellites, find the mass of the Sun/planet from T and a, or compute ratios like T₁/T₂ from orbital radii; includes binary stars orbiting their common centre of mass.","sort":227,"aliases":[]},{"id":229,"code":"06.01.04","level":3,"parent_id":225,"name":"Orbits as conic sections and energy of elliptical orbits","description":"Questions classifying trajectories (circle/ellipse/parabola/hyperbola) from total energy E = −GMm/2a and using the vis-viva relation v² = GM(2/r − 1/a); also eccentricity from energy and angular momentum, and speeds/energies at the apsides of an ellipse.","sort":228,"aliases":[]},{"id":230,"code":"06.01.05","level":3,"parent_id":225,"name":"Reduced mass and effective potential in central-force motion","description":"Questions reducing the two-body problem with reduced mass μ = m₁m₂/(m₁+m₂) and analysing motion via the effective potential U_eff(r) = −GMμ/r + L²/2μr²; typical items find turning points, stability of circular orbits, or radial motion in a central inverse-square force.","sort":229,"aliases":[]},{"id":231,"code":"06.01.06","level":3,"parent_id":225,"name":"Rutherford gravitational scattering","description":"Questions on gravitational (Rutherford-type) scattering of a light particle by a heavy mass M: hyperbolic trajectory, impact parameter vs scattering angle, and distance of closest approach from conservation of energy and angular momentum.","sort":230,"aliases":[]},{"id":232,"code":"06.02","level":2,"parent_id":224,"name":"Gravitational field and intensity","description":"Questions computing gravitational intensity E = F/m = GM/r² for point masses and superposing vectorially (e.g., field at a corner or centre of a triangle of masses); includes direction of the net field, null points, and units N/kg.","sort":231,"aliases":["gravitational field"]},{"id":233,"code":"06.02.01","level":3,"parent_id":232,"name":"Gravitational field of a rod on its perpendicular bisector","description":"Questions integrating to get the field at a point on the perpendicular bisector of a uniform rod, E = 2GM/(d√(L²+4d²)) directed toward the rod's midpoint; variants take limits d→∞ (point mass) or d→0 (near the rod).","sort":232,"aliases":[]},{"id":234,"code":"06.02.02","level":3,"parent_id":232,"name":"On-axis gravitational field of a rod","description":"Questions finding the on-axis field of a uniform rod by integration, E = Gλ(1/a − 1/b) for a point on the rod's line, e.g. Gλ(1/x − 1/(x+L)) outside an end; includes field at the end face of a rod and long-rod approximations.","sort":233,"aliases":[]},{"id":235,"code":"06.02.03","level":3,"parent_id":232,"name":"Gravitational field of spherical shells and solid spheres","description":"Questions using the shell theorem: E = 0 inside a shell and GM/r² outside; for a uniform solid sphere E = GMr/R³ inside and GM/r² outside; typical items plot or compare g vs r, or find the field at a depth inside the Earth.","sort":234,"aliases":[]},{"id":236,"code":"06.02.04","level":3,"parent_id":232,"name":"Gravitational field inside a spherical cavity","description":"Questions on the field inside a spherical cavity in a uniform sphere, found by superposing the full sphere and a negative-mass sphere; key result is a uniform field E = (4/3)πGρ·a (a = vector between centres), asked as magnitude/direction or as the motion of a particle released in the cavity.","sort":235,"aliases":[]},{"id":237,"code":"06.02.05","level":3,"parent_id":232,"name":"Field on the axis of a ring and disc","description":"Questions computing the axial field of a ring, E = GMx/(x²+R²)^{3/2} with maximum at x = R/√2, and of a uniform disc, E = 2Gσ(1 − x/√(x²+R²)); includes limits x→0 and x→∞ and ring-vs-disc comparisons.","sort":236,"aliases":[]},{"id":238,"code":"06.03","level":2,"parent_id":224,"name":"Gravitational potential and potential energy","description":"Umbrella for questions computing gravitational potential V = −GM/r and potential energy U = −GMm/r, superposing potentials of extended bodies, and using W_ext = mΔV or E = −dV/dr; children specialise to self-energy, ring/disc/rod potentials, and overlapping spheres.","sort":237,"aliases":["gravitational potential"]},{"id":239,"code":"06.03.01","level":3,"parent_id":238,"name":"Gravitational potential energy and potential","description":"Questions on potential of point-mass systems V = −ΣGmᵢ/rᵢ, potential energy of assemblies U = −GΣmᵢmⱼ/rᵢⱼ, and work done moving a mass between two points (W = mΔV); includes potential difference vs absolute potential with zero at infinity.","sort":238,"aliases":[]},{"id":240,"code":"06.03.02","level":3,"parent_id":238,"name":"Self-energy of uniform spheres and shells","description":"Questions computing self-energy: U = −GM²/2R for a uniform shell and U = −3GM²/5R for a uniform solid sphere; typical items ask the energy needed to disperse a planet to infinity, energy released when fragments coalesce, or work done in changing a sphere's radius/mass.","sort":239,"aliases":[]},{"id":241,"code":"06.03.03","level":3,"parent_id":238,"name":"Gravitational potential of a ring, disc and rod","description":"Questions finding potential of extended bodies by integration: ring V = −GM/√(x²+R²) on the axis, disc V = −2Gσ(√(x²+R²) − x), and rod potentials at axial or perpendicular-bisector points; often followed by deriving the field via E = −dV/dr.","sort":240,"aliases":[]},{"id":242,"code":"06.03.04","level":3,"parent_id":238,"name":"Potential and field due to overlapping spheres","description":"Questions on potential and field where two or more uniform spheres overlap, using superposition of solid-sphere results (inside: V = −GM(3R² − r²)/2R³); classic setups are a sphere with an off-centre cavity, locating zero-field points, and showing the overlap field is uniform.","sort":241,"aliases":[]},{"id":243,"code":"06.04","level":2,"parent_id":224,"name":"Satellites, orbital velocity and escape velocity","description":"Umbrella for satellite-orbit questions: circular orbital speed v = √(GM/r), escape speed, geostationary/polar orbits, orbital energy, and orbit-changing manoeuvres; children break these into specific computations.","sort":242,"aliases":["satellites","escape velocity","orbital motion of satellites","central force motion","effective potential and orbit energy diagrams","orbit energy, bound and unbound orbits"]},{"id":244,"code":"06.04.01","level":3,"parent_id":243,"name":"Satellites and orbital velocity","description":"Questions computing orbital velocity v_o = √(GM/r) = √(gR²/r) and period T = 2π√(r³/GM) for circular orbits, finding a satellite's height from its period, and comparing v_o across planets/altitudes; includes v_o = √(gR) for a surface-grazing orbit.","sort":243,"aliases":[]},{"id":245,"code":"06.04.02","level":3,"parent_id":243,"name":"Escape velocity","description":"Questions on escape velocity v_e = √(2GM/R) = √(2gR) ≈ 11.2 km/s for Earth, derived from energy conservation (just reaching infinity with zero speed); variants ask v_e for the Moon/other planets, launching with speed less or more than v_e, and v_e = √2·v_o.","sort":244,"aliases":[]},{"id":246,"code":"06.04.03","level":3,"parent_id":243,"name":"Geostationary and polar orbits","description":"Questions on geostationary satellites (T = 24 h, equatorial plane, h ≈ 35,800 km, r ≈ 42,000 km) and polar/sun-synchronous orbits with their uses; typical items compute the geostationary radius/height or decide which orbit can keep a satellite fixed over a point.","sort":245,"aliases":[]},{"id":247,"code":"06.04.04","level":3,"parent_id":243,"name":"Orbital transitions and Hohmann transfers","description":"Questions on moving a satellite between circular orbits: work/energy needed to change radius, and Hohmann transfer ellipses tangent to two circular orbits with perigee/apogee burns; typical items compute Δv, transfer time, or the transfer orbit's semi-major axis and energy.","sort":246,"aliases":[]},{"id":248,"code":"06.04.05","level":3,"parent_id":243,"name":"Total energy, binding energy and orbital decay","description":"Questions using E_total = −GMm/2r = −KE = PE/2 and binding energy GMm/2r for circular orbits; includes atmospheric drag causing orbital decay (radius shrinks while speed and KE increase, total energy decreases) and energy dissipated in decay.","sort":247,"aliases":[]},{"id":249,"code":"06.04.06","level":3,"parent_id":243,"name":"Lagrange points and restricted three-body problem","description":"Questions on the five Lagrange points of the restricted three-body problem, where a small mass stays fixed in the rotating frame; typical items locate L1 between the two primaries (Earth–Sun or Earth–Moon) via the net-force equation and discuss stability of L4/L5.","sort":248,"aliases":[]},{"id":250,"code":"06.05","level":2,"parent_id":224,"name":"Acceleration due to gravity and its variation","description":"Umbrella for questions on surface gravity g = GM/R² = (4/3)πGρR and how it changes with position; children cover variation with height, depth, and rotation/latitude.","sort":249,"aliases":[]},{"id":251,"code":"06.05.01","level":3,"parent_id":250,"name":"Variation of g with height, depth and latitude","description":"Questions applying g_h = g(1 − 2h/R) for h ≪ R, g_d = g(1 − d/R) for depth, and g' = g − ω²R cos²λ for latitude/rotation; typical items compare percentage changes, find the depth and height giving equal decrease, or contrast g at the poles vs equator.","sort":250,"aliases":[]},{"id":252,"code":"06.05.02","level":3,"parent_id":250,"name":"Effect of Earth's rotation on g","description":"Questions on how the measured acceleration due to gravity g decreases with latitude due to the centrifugal effect of Earth's rotation, using g_eff = g - ω²R cos²λ and related calculations.","sort":251,"aliases":[]},{"id":253,"code":"06.05.03","level":3,"parent_id":250,"name":"g inside and on the surface of shells and spheres","description":"Questions applying the shell theorem to find gravitational field or g at points inside, on, or outside uniform spherical shells and solid spheres, often using g ∝ r inside a solid sphere.","sort":252,"aliases":[]},{"id":254,"code":"06.06","level":2,"parent_id":224,"name":"Newton's law of gravitation","description":"Questions involving the inverse-square force law F = Gm₁m₂/r², gravitational field, potential energy U = -Gm₁m₂/r, and calculations of force between point masses or extended bodies.","sort":253,"aliases":["inertial mass vs gravitational mass","tidal forces"]},{"id":255,"code":"06.06.01","level":3,"parent_id":254,"name":"Inertial mass versus gravitational mass","description":"Questions distinguishing inertial mass (resistance to acceleration, F=ma) from gravitational mass (source/responder to gravity), including Eötvös-type experiments and equivalence of the two masses.","sort":254,"aliases":[]},{"id":256,"code":"06.06.02","level":3,"parent_id":254,"name":"Tidal forces and the equivalence principle","description":"Questions on differential gravitational forces causing tidal bulges, tidal acceleration, Roche limit concepts, and the equivalence principle relating uniform acceleration to a uniform gravitational field.","sort":255,"aliases":[]},{"id":257,"code":"06.06.03","level":3,"parent_id":254,"name":"Universal law of gravitation and superposition","description":"Questions using the universal law of gravitation with vector superposition to find net gravitational force, field, or potential at a point due to multiple masses or continuous mass distributions.","sort":256,"aliases":[]},{"id":258,"code":"07","level":1,"parent_id":null,"name":"Mechanical Properties of Solids (Elasticity)","description":"JEE chapter 07: Mechanical Properties of Solids (Elasticity). Concepts: Stress, strain and Hooke's law; Elastic behaviour and stress-strain curve; Elastic moduli (Young's, shear, bulk) and compressibility; Elastic potential energy of a stretched wire (strain energy density).","sort":257,"aliases":["Mechanical Properties of Solids"]},{"id":259,"code":"07.01","level":2,"parent_id":258,"name":"Stress, strain and Hooke's law","description":"Parent node for stress, strain, and Hooke's law: defining stress = F/A and strain = ΔL/L, and using stress ∝ strain within the elastic limit for wires, rods, and blocks under load.","sort":258,"aliases":["Hooke's law"]},{"id":260,"code":"07.01.01","level":3,"parent_id":259,"name":"Stress and strain","description":"Computing longitudinal stress (F/A in N/m²) and strain (ΔL/L, dimensionless) for wires, rods, and columns under tensile or compressive loads.","sort":259,"aliases":[]},{"id":261,"code":"07.01.02","level":3,"parent_id":259,"name":"Types of stress (longitudinal, volumetric and shearing stress)","description":"Identifying and calculating the three stress types: longitudinal (F/A along length), volumetric/hydrostatic (pressure change ΔP), and shearing (tangential force per unit area) in blocks and cuboids.","sort":260,"aliases":[]},{"id":262,"code":"07.01.03","level":3,"parent_id":259,"name":"Types of strain (longitudinal, volumetric and shear strain)","description":"Identifying and calculating the three strain types: longitudinal ΔL/L, volumetric ΔV/V, and shear strain (angle of shear, tanθ ≈ θ) for stretched wires, compressed bodies, and sheared cuboids.","sort":261,"aliases":[]},{"id":263,"code":"07.01.04","level":3,"parent_id":259,"name":"Hooke's law and constitutive relations","description":"Applying Hooke's law (stress ∝ strain up to elastic limit) and modulus = stress/strain as constitutive relations; verifying linearity from data tables and finding the force constant of a wire.","sort":262,"aliases":[]},{"id":264,"code":"07.01.05","level":3,"parent_id":259,"name":"Thermal stress and thermal strain","description":"Thermal stress when a rod is clamped between rigid supports and heated/cooled: thermal stress = YαΔθ, strain = αΔθ, and problems computing the gap or force needed to prevent expansion.","sort":263,"aliases":[]},{"id":265,"code":"07.02","level":2,"parent_id":258,"name":"Elastic behaviour and stress-strain curve","description":"Parent node for elastic behavior of materials and the stress-strain curve: proportional limit, elastic limit, yield point, UTS, fracture point, and distinguishing ductile, brittle, and elastomer behavior.","sort":264,"aliases":["elastic limit","yield point and fracture","ductile and brittle materials","elastomers"]},{"id":266,"code":"07.02.01","level":3,"parent_id":265,"name":"Stress-strain curve","description":"Reading and interpreting the stress-strain graph: identifying proportional limit, elastic limit, yield point, ultimate tensile strength, and fracture point; comparing ductile vs. brittle materials from curve shape.","sort":265,"aliases":[]},{"id":267,"code":"07.02.02","level":3,"parent_id":265,"name":"Elastic limit, yield point and plastic deformation","description":"Concepts of elastic limit, yield point, and plastic (permanent) deformation: permanent set after unloading beyond the elastic limit, and behavior of ductile vs. brittle materials past yielding.","sort":266,"aliases":[]},{"id":268,"code":"07.02.03","level":3,"parent_id":265,"name":"Breaking stress, ultimate tensile strength and factor of safety","description":"Numericals on breaking stress (breaking force/area), ultimate tensile strength, maximum load a wire can support, and factor of safety = UTS/working stress in wire and cable problems.","sort":267,"aliases":[]},{"id":269,"code":"07.02.04","level":3,"parent_id":265,"name":"Elastic hysteresis and elastomers","description":"Elastic hysteresis: loading and unloading curves not coinciding, area of the hysteresis loop as energy dissipated per cycle, and properties of elastomers like rubber (no linear region, low Young's modulus).","sort":268,"aliases":[]},{"id":270,"code":"07.02.05","level":3,"parent_id":265,"name":"Factors affecting elasticity (temperature, impurities, annealing, hammering)","description":"Conceptual questions on how elasticity changes with temperature (decreases on heating), impurity addition, annealing (increases elasticity), and hammering/rolling (decreases elasticity).","sort":269,"aliases":[]},{"id":271,"code":"07.03","level":2,"parent_id":258,"name":"Elastic moduli (Young's, shear, bulk) and compressibility","description":"Parent node for the three elastic moduli: Young's modulus Y, shear modulus G (η), bulk modulus K, and compressibility = 1/K, with their defining stress/strain ratios.","sort":270,"aliases":["Young's modulus","modulus of rigidity","compressibility","elasticity"]},{"id":272,"code":"07.03.01","level":3,"parent_id":271,"name":"Hooke's law and elastic moduli","description":"Using Hooke's law to compute Y, K, or G from given stress-strain data, and relations among elastic constants (Y, K, G, Poisson's ratio) in combined numericals.","sort":271,"aliases":[]},{"id":273,"code":"07.03.02","level":3,"parent_id":271,"name":"Young's modulus and elongation of a wire (ΔL = FL/AY)","description":"Standard wire-elongation problems using ΔL = FL/AY: finding extension, Young's modulus, or load; also wires in series/parallel and composite wires of two materials.","sort":272,"aliases":[]},{"id":274,"code":"07.03.03","level":3,"parent_id":271,"name":"Elongation of a bar under its own weight","description":"Extension of a vertically hanging bar/rod due to its own weight: ΔL = ρgL²/2Y = WL/2AY, with comparisons between bars of different lengths, areas, or hanging vs. lying configurations.","sort":273,"aliases":[]},{"id":275,"code":"07.03.04","level":3,"parent_id":271,"name":"Bulk modulus and compressibility","description":"Questions using B = -ΔP/(ΔV/V) and compressibility k = 1/B: fractional volume change of a solid or liquid under applied pressure, hydraulic compression of a sphere or block, bulk modulus of water/metals, or finding pressure needed for a given volume change.","sort":274,"aliases":[]},{"id":276,"code":"07.03.05","level":3,"parent_id":271,"name":"Shear modulus (modulus of rigidity)","description":"Questions using rigidity modulus G = (F/A)/θ or G = F·L/(A·Δx): a tangential force applied to the top face of a cube/block fixed at the bottom, lateral shift of the upper face, angle of shear, or comparing G with Y for the same material.","sort":275,"aliases":[]},{"id":277,"code":"07.03.06","level":3,"parent_id":271,"name":"Poisson's ratio and inter-relations between elastic constants","description":"Questions using σ = lateral strain/longitudinal strain and the identities Y = 3K(1-2σ), Y = 2G(1+σ), 9/Y = 3/G + 1/K: finding lateral contraction of a stretched wire, computing one elastic constant from the others, or allowed limits of Poisson's ratio (-1 to 0.5).","sort":276,"aliases":[]},{"id":278,"code":"07.03.07","level":3,"parent_id":271,"name":"Series and parallel combination of wires and rods","description":"Questions with two wires/rods of different Y, A, L sharing a load (series: same force, elongations add; parallel: same elongation, forces divide) or a rigid bar supported by two wires — finding individual elongations, stresses, or equivalent Young's modulus/effective spring constant.","sort":277,"aliases":[]},{"id":279,"code":"07.04","level":2,"parent_id":258,"name":"Elastic potential energy of a stretched wire (strain energy density)","description":"Parent node for work-energy questions on deforming elastic bodies: work done in stretching a wire, energy stored at a given extension or up to breaking, energy per unit volume, and comparing energy stored in two wires under equal force vs equal extension.","sort":278,"aliases":[]},{"id":280,"code":"07.04.01","level":3,"parent_id":279,"name":"Elastic potential energy","description":"Questions computing elastic PE of a stretched wire/spring-like body via U = ½FΔL = ½k(ΔL)² or area under force-extension graph, including work done by a gradually applied load and energy stored just before breaking (breaking stress based).","sort":279,"aliases":[]},{"id":281,"code":"07.04.02","level":3,"parent_id":279,"name":"Strain energy and energy density in a stretched wire (u = ½ × stress × strain)","description":"Questions on strain energy per unit volume u = ½ × stress × strain = ½Y(strain)²: finding energy density in a loaded wire, total energy via u × volume, or reading energy density as the area under a stress-strain curve.","sort":280,"aliases":[]},{"id":282,"code":"08","level":1,"parent_id":null,"name":"Fluid Mechanics","description":"JEE chapter 08: Fluid Mechanics. Concepts: Surface tension and capillarity; Fluid dynamics: continuity equation and Bernoulli's principle; Viscosity and Stokes' law; Fluid pressure and Pascal's law; Buoyancy and Archimedes' principle; Density and Specific Gravity.","sort":281,"aliases":["Mechanical Properties of Fluids"]},{"id":283,"code":"08.01","level":2,"parent_id":282,"name":"Surface tension and capillarity","description":"Parent node on surface tension: S = F/l, excess pressure inside drops/bubbles (2S/r for drop, 4S/r for soap bubble), capillary rise h = 2S cosθ/(rρg), and forces to lift rings/needles/plates from a liquid surface.","sort":282,"aliases":["surface tension","capillary rise","excess pressure in drops and bubbles","surface energy"]},{"id":284,"code":"08.01.01","level":3,"parent_id":283,"name":"Capillary force between liquid-bridged plates","description":"Two parallel plates separated by a thin liquid film (wetted area A, separation d): pressure difference 2S/d pulls the plates together, giving attraction force F = 2SA/d. Questions ask the force needed to pull the plates apart or the separation given the force.","sort":283,"aliases":[]},{"id":285,"code":"08.01.02","level":3,"parent_id":283,"name":"Surface energy, surface tension and angle of contact","description":"Surface energy and contact angle: work to create/increase surface area W = S·ΔA, energy change when a drop splits into n droplets (n^(1/3) radius scaling), and angle of contact determining meniscus shape and wetting vs non-wetting. Questions compute work, energy released, or contact angle effects on capillary rise.","sort":284,"aliases":[]},{"id":286,"code":"08.01.03","level":3,"parent_id":283,"name":"Excess pressure inside liquid drops and soap bubbles (Laplace's law)","description":"Questions applying Laplace's law ΔP = 2T/r for a liquid drop and ΔP = 4T/r for a soap bubble: finding excess pressure, comparing pressures of bubbles of different radii, behavior when two bubbles connect (small empties into large), and pressure just inside/outside a bubble surface.","sort":285,"aliases":[]},{"id":287,"code":"08.01.04","level":3,"parent_id":283,"name":"Capillary rise and Jurin's law","description":"Questions using Jurin's law h = 2Tcosθ/(rρg) for capillary rise: height in capillaries of different radii, capillaries of varying bore or tilted/inclined tubes, mercury depression, effect of contact angle, and splitting a capillary into sections.","sort":286,"aliases":[]},{"id":288,"code":"08.01.05","level":3,"parent_id":283,"name":"Surface tension force on rings, frames and floating needles","description":"Questions computing surface tension forces F = T×(total contact length): pulling a ring or needle off a liquid surface (factor 2 for two surfaces, 4πrT for a ring), soap film on a U-frame with sliding wire (F = 2TL), and equilibrium of a floating needle or blade.","sort":287,"aliases":[]},{"id":289,"code":"08.01.06","level":3,"parent_id":283,"name":"Energy changes in coalescence and splitting of drops","description":"Questions on surface energy change T×ΔA when n small drops coalesce into one big drop (R = n^(1/3)r, energy released) or a big drop splits into n droplets (energy absorbed), sometimes asking the accompanying temperature rise using specific heat.","sort":288,"aliases":[]},{"id":290,"code":"08.02","level":2,"parent_id":282,"name":"Fluid dynamics: continuity equation and Bernoulli's principle","description":"Parent node for ideal-fluid dynamics: questions on incompressible, non-viscous streamline flow using the continuity equation A₁v₁ = A₂v₂ and Bernoulli's equation P + ½ρv² + ρgh = constant in pipes, tanks, jets and flow-measuring devices.","sort":289,"aliases":["fluid dynamics","fluid mechanics","fluid flow","streamline flow","equation of continuity","Torricelli's theorem","venturi meter","fluid jet force"]},{"id":291,"code":"08.02.01","level":3,"parent_id":290,"name":"Fluid jet force on a moving plate","description":"Questions on force exerted by a fluid jet striking a plate: F = ρAv² for a stationary plate and F = ρAv(v−u) for a plate moving away at speed u, including jets pushing carts, hinged plates, or finding terminal speed of a plate held by a jet.","sort":290,"aliases":[]},{"id":292,"code":"08.02.02","level":3,"parent_id":290,"name":"Streamline flow and Bernoulli's principle","description":"Conceptual and numerical questions on streamline (steady) flow properties and Bernoulli's principle: streamlines never intersect, pressure is lower where speed is higher, and qualitative reasoning about pressure–velocity trade-offs in pipes of varying cross-section or height.","sort":291,"aliases":[]},{"id":293,"code":"08.02.03","level":3,"parent_id":290,"name":"Equation of continuity and volume flow rate","description":"Questions on the continuity equation A₁v₁ = A₂v₂ and volume flow rate Q = Av: speeds in tapering pipes, mass/volume of fluid crossing a section per second, and time to fill or empty tanks through pipes of given bore.","sort":292,"aliases":[]},{"id":294,"code":"08.02.04","level":3,"parent_id":290,"name":"Bernoulli's equation and its applications","description":"Questions applying Bernoulli's equation P + ½ρv² + ρgh = constant: pressure difference between two points in a pipe, speed of water in a main from pressure gauges, atomizers, spinning-ball deflection, and combined height-plus-velocity problems.","sort":293,"aliases":[]},{"id":295,"code":"08.02.05","level":3,"parent_id":290,"name":"Torricelli's law: velocity of efflux and jet dynamics","description":"Questions on efflux from a hole in a tank using Torricelli's law v = √(2gh): jet range and time of flight, horizontal distance where the jet lands, time to empty a tank (proportional to √h), holes at different depths, and efflux from an accelerating or moving tank.","sort":294,"aliases":[]},{"id":296,"code":"08.02.06","level":3,"parent_id":290,"name":"Venturimeter and flow speed measurement","description":"Questions on the venturimeter: relating manometer height difference h to flow speed via v = a₁a₂√(2ρgh/(a₁²−a₂²)) for throat and inlet areas a₁, a₂, and computing volume flow rate or pressure drop between wide and constricted sections.","sort":295,"aliases":[]},{"id":297,"code":"08.02.07","level":3,"parent_id":290,"name":"Dynamic lift: Magnus effect and aerofoil lift","description":"Questions on dynamic lift from Bernoulli's principle: Magnus effect on spinning balls (curving of tennis/football trajectories) and aerofoil lift from faster flow above the wing, usually asking direction of net force or pressure difference.","sort":296,"aliases":[]},{"id":298,"code":"08.03","level":2,"parent_id":282,"name":"Viscosity and Stokes' law","description":"Parent node for viscosity: questions using Newton's viscous force law F = −ηA(dv/dx), Stokes' drag 6πηrv, terminal velocity, Poiseuille flow Q = πPr⁴/(8ηl), and Reynolds number classification of flow.","sort":297,"aliases":["viscosity","terminal velocity in a viscous fluid","Poiseuille's law"]},{"id":299,"code":"08.03.01","level":3,"parent_id":298,"name":"Newton's law of viscosity and velocity gradient","description":"Questions on Newton's law of viscosity F = −ηA(dv/dx): tangential force needed to slide a plate over a liquid film, velocity gradient between parallel plates, units and dimensions of coefficient of viscosity, and force on a layer in Couette-type flow.","sort":298,"aliases":[]},{"id":300,"code":"08.03.02","level":3,"parent_id":298,"name":"Stokes' law and terminal velocity","description":"Questions using Stokes' law F = 6πηrv and terminal velocity v_t = 2r²(ρ−σ)g/(9η): sphere falling through oil or glycerine, time/distance to attain terminal velocity, apparent weight during fall, finding viscosity or radius from measured terminal speed, and two spheres of different radii.","sort":299,"aliases":[]},{"id":301,"code":"08.03.03","level":3,"parent_id":298,"name":"Poiseuille's equation for viscous flow through a tube","description":"Questions on Poiseuille's equation Q = πPr⁴/(8ηl): volume flow rate through a capillary under a pressure head, pressure drop along a tube, effect of halving radius or length, and tubes joined in series or parallel with combined flow resistance.","sort":300,"aliases":[]},{"id":302,"code":"08.03.04","level":3,"parent_id":298,"name":"Reynolds number and laminar vs turbulent flow","description":"Questions on Reynolds number Re = ρvd/η: computing Re for water/air flow in pipes, critical velocity v_c = Re·η/(ρd), and classifying flow as laminar, unsteady, or turbulent, including why large-diameter or high-speed flows become turbulent.","sort":301,"aliases":[]},{"id":303,"code":"08.04","level":2,"parent_id":282,"name":"Fluid pressure and Pascal's law","description":"Parent node for fluid statics: questions on pressure P = F/A, hydrostatic pressure P = P₀ + ρgh, Pascal's law transmission in hydraulic devices, barometers/manometers, forces on tank walls, and pressure in accelerating or rotating fluids.","sort":302,"aliases":["pressure due to a fluid column","hydraulic lift","manometer and barometer"]},{"id":304,"code":"08.04.01","level":3,"parent_id":303,"name":"Pressure and Pascal's law","description":"Questions on pressure definition P = F/A and Pascal's law: pressure exerted by solids on fluids, transmission of applied pressure to all points, equal pressure at the same level in connected vessels, and force multiplication in enclosed liquids.","sort":303,"aliases":[]},{"id":305,"code":"08.04.02","level":3,"parent_id":303,"name":"Hydrostatic pressure and variation with depth","description":"Questions on hydrostatic pressure P = P₀ + ρgh: pressure at the bottom of tanks containing layered immiscible liquids, gauge vs absolute pressure, pressure difference between two depths, U-tube balance problems, and pressure variation in accelerating containers.","sort":304,"aliases":[]},{"id":306,"code":"08.04.03","level":3,"parent_id":303,"name":"Pascal's law and hydraulic machines","description":"Questions on hydraulic machines via Pascal's law F₁/A₁ = F₂/A₂: hydraulic press, lift, and brakes—finding force/pressure ratios, mechanical advantage, distance moved by pistons, and work input vs output in a hydraulic system.","sort":305,"aliases":[]},{"id":307,"code":"08.04.04","level":3,"parent_id":303,"name":"Atmospheric pressure, barometers and manometers","description":"Questions on atmospheric pressure (1 atm ≈ 1.013×10⁵ Pa, 76 cm Hg): Torricelli's mercury barometer, effect of using water or a tilted/incomplete barometer, open-tube manometer readings for gas pressure, and pressure at depth using mercury or other liquids.","sort":306,"aliases":[]},{"id":308,"code":"08.04.05","level":3,"parent_id":303,"name":"Hydrostatic force on tank walls and bottoms","description":"Questions on total hydrostatic thrust on tank bottoms and vertical walls: F = P_avg × A with average pressure ρgh/2 on a rectangular wall, force on circular/rectangular gates and dams, point of application of thrust, and comparison of forces on wall vs bottom.","sort":307,"aliases":[]},{"id":309,"code":"08.04.06","level":3,"parent_id":303,"name":"Pressure distribution and free surface in a rotating fluid","description":"Questions on fluids in rigid rotation: parabolic free surface shape, height difference between rim and center h = ω²R²/(2g), pressure at a point in a rotating cylindrical vessel, and effective gravity angle tanθ = ω²r/g for surfaces or bubbles.","sort":308,"aliases":[]},{"id":310,"code":"08.05","level":2,"parent_id":282,"name":"Buoyancy and Archimedes' principle","description":"Questions on Archimedes' principle and buoyancy: buoyant force = weight of displaced fluid, fraction of a floating body submerged (ρ_body/ρ_liquid), apparent weight of immersed objects, blocks floating between two liquids, and density determination from immersion depth.","sort":309,"aliases":["buoyancy","floatation","apparent weight in a fluid"]},{"id":311,"code":"08.05.01","level":3,"parent_id":310,"name":"Pressure and buoyancy in accelerating fluids","description":"Questions where the container of fluid accelerates (lift going up/down, horizontally accelerating tank), so effective gravity changes and buoyant force becomes F_b = ρV(g±a); also free-surface tilt and pressure gradient dp/dz = -ρ(g+a) in non-inertial frames.","sort":310,"aliases":[]},{"id":312,"code":"08.05.02","level":3,"parent_id":310,"name":"Archimedes' principle and buoyant force","description":"Direct applications of Archimedes' principle: buoyant force F_b = ρ_fluid·V_displaced·g, loss of weight of a body immersed in a liquid, spring-balance readings of blocks hung in water, and tension in strings supporting submerged objects.","sort":311,"aliases":[]},{"id":313,"code":"08.05.03","level":3,"parent_id":310,"name":"Floatation, fraction submerged and apparent weight","description":"Floating-body problems using fraction submerged = ρ_body/ρ_liquid: percentage of volume above the liquid surface, apparent weight of a floating body (zero), and comparing how deep identical blocks sink in different liquids.","sort":312,"aliases":[]},{"id":314,"code":"08.05.04","level":3,"parent_id":310,"name":"Stability of floating bodies: center of buoyancy and metacenter","description":"Conceptual/numerical questions on equilibrium of tilted floating bodies: center of buoyancy shift, metacentric height GM, and conditions for stable (GM>0), unstable, or neutral equilibrium of ships, blocks, and cylinders.","sort":313,"aliases":[]},{"id":315,"code":"08.05.05","level":3,"parent_id":310,"name":"Ice melting and water level change problems","description":"Classic 'what happens to water level' problems: ice cube melting in water (level unchanged), ice containing a stone, air bubble, or lead shot, and ice floating in liquids denser or lighter than water — decide rise/fall/no change with reasoning.","sort":314,"aliases":[]},{"id":316,"code":"08.05.06","level":3,"parent_id":310,"name":"Buoyancy in immiscible liquid layers","description":"Objects floating at the interface of two immiscible liquids, where total buoyancy = ρ₁V₁g + ρ₂V₂g; find fraction of volume in each layer or the density of the floating body.","sort":315,"aliases":[]},{"id":317,"code":"08.06","level":2,"parent_id":282,"name":"Density and Specific Gravity","description":"Parent node for density ρ = m/V and specific gravity (relative density to water): basic mass-volume computations, unit conversions (g/cm³ to kg/m³), and density from weighing in air and water.","sort":316,"aliases":["relative density","density of fluids and mixtures"]},{"id":318,"code":"08.06.01","level":3,"parent_id":317,"name":"Density and specific gravity of solids and fluids","description":"Numericals computing density or specific gravity of solids and liquids from mass and volume data, or from buoyancy measurements (weight in air vs. weight when immersed).","sort":317,"aliases":[]},{"id":319,"code":"08.06.02","level":3,"parent_id":317,"name":"Solution density under dilution with overflow","description":"Problems where water is added to a brine/solution already filling a vessel, causing overflow; final density found by mass balance of added water and overflowed solution.","sort":318,"aliases":[]},{"id":320,"code":"09","level":1,"parent_id":null,"name":"Thermal Physics & Heat Transfer","description":"JEE chapter 09: Thermal Physics & Heat Transfer. Concepts: Thermal expansion; Temperature, specific heat and change of state; Calorimetry; Heat transfer.","sort":319,"aliases":["Thermal Properties of Matter"]},{"id":321,"code":"09.01","level":2,"parent_id":320,"name":"Thermal expansion","description":"Parent node for thermal expansion: solids and liquids expand on heating, with coefficients α (linear), β (area), γ (volume) and ΔL = LαΔT type relations. Tag here for general expansion questions that don't fit a specific child.","sort":320,"aliases":[]},{"id":322,"code":"09.01.01","level":3,"parent_id":321,"name":"Linear, area and volume expansion of solids","description":"Questions on ΔL = LαΔT, ΔA = AβΔT, ΔV = VγΔT with β = 2α, γ = 3α: finding change in length/area/volume/density, comparing expansion of two rods, reading a vernier/scale correction, or determining α from experimental length-vs-temperature data.","sort":321,"aliases":[]},{"id":323,"code":"09.01.02","level":3,"parent_id":321,"name":"Thermal stress in rigidly clamped rods","description":"Questions where a rod is rigidly clamped between walls (or two rods joined and fixed) and heated/cooled: thermal stress = YαΔθ, thermal force = YAαΔθ, comparing stresses in rods of different materials, and finding the force exerted on the supports or whether a composite rod buckles.","sort":322,"aliases":[]},{"id":324,"code":"09.01.03","level":3,"parent_id":321,"name":"Real and apparent expansion of liquids; anomalous expansion of water","description":"Questions on liquid expansion measured in a container: γ_real = γ_apparent + γ_vessel, finding real/apparent coefficients from overflow volume, and the anomalous expansion of water (density maximum at 4 °C) explaining why lakes freeze top-down and survival of aquatic life.","sort":323,"aliases":[]},{"id":325,"code":"09.01.04","level":3,"parent_id":321,"name":"Bimetallic strip and differential expansion","description":"Questions on a bimetallic strip of two metals with different α that bends on heating (which side becomes convex), radius of curvature of bending, thermostat applications, and differential expansion of two rods joined end-to-end or riveted, including finding the temperature at which a gap closes or stress develops.","sort":324,"aliases":[]},{"id":326,"code":"09.01.05","level":3,"parent_id":321,"name":"Thermal expansion of pendulum clocks and timekeeping","description":"Questions on a pendulum clock whose rod expands with temperature: T = 2π√(L/g) increases, so the clock runs slow; fractional change ΔT/T = ½αΔθ, time lost/gained per day, and temperature at which a compensated pendulum keeps correct time.","sort":325,"aliases":[]},{"id":327,"code":"09.01.06","level":3,"parent_id":321,"name":"Density and buoyancy changes with temperature","description":"Questions on how heating changes density (ρ = ρ₀/(1 + γΔθ)) and hence buoyancy: fraction of a floating body submerged increases with temperature, apparent weight of a submerged solid changes, level of liquid in a vessel with a floating ice block as temperature varies, and hydrometer-style problems.","sort":326,"aliases":[]},{"id":328,"code":"09.02","level":2,"parent_id":320,"name":"Temperature, specific heat and change of state","description":"Questions on temperature scales, heat Q = msΔT, latent heats, and phase changes of substances (melting, boiling, sublimation). Includes thermometer behavior, humidity, and numerical heat-accounting for heating/cooling/phase-transition sequences.","sort":327,"aliases":["heat","temperature and heat","temperature scales and thermometry","specific heat","latent heat"]},{"id":329,"code":"09.02.01","level":3,"parent_id":328,"name":"Specific heat and latent heat","description":"Direct use of Q = msΔθ and Q = mL for melting (L_f ≈ 80 cal/g for ice) and vaporization (L_v ≈ 540 cal/g for water). Typical tasks: heat needed to convert ice to steam, comparing specific heats from temperature rise data, or finding latent heat from mixture results.","sort":328,"aliases":[]},{"id":330,"code":"09.02.02","level":3,"parent_id":328,"name":"Clausius-Clapeyron relation","description":"Uses dP/dT = L/(T·ΔV) to find how boiling or melting point shifts with pressure, or to compute latent heat from a vapour-pressure curve. Archetypes: why ice melts under pressure, pressure-cooker boiling temperature, slope of the fusion/vaporisation curve.","sort":329,"aliases":[]},{"id":331,"code":"09.02.03","level":3,"parent_id":328,"name":"Temperature and heat","description":"Conceptual and numerical questions distinguishing heat (energy in transit) from temperature, and converting between Celsius, Fahrenheit and Kelvin scales. Includes identifying which body 'contains more heat' and temperature of mixtures at equilibrium.","sort":330,"aliases":[]},{"id":332,"code":"09.02.04","level":3,"parent_id":328,"name":"Thermometer calibration and resolution","description":"Calibration of thermometers using fixed points (ice and steam), linear interpolation of a thermometric property (mercury length, resistance, emf), and least-count/resolution errors. Archetypes: a thermometer reads X at ice point and Y at steam point—find true temperature of a reading Z.","sort":331,"aliases":[]},{"id":333,"code":"09.02.05","level":3,"parent_id":328,"name":"Vapour pressure, humidity and condensation","description":"Saturated vapour pressure, relative humidity = (vapour pressure / SVP at that T) × 100%, and dew point. Questions on when dew/fog/condensation forms, humidity changes on cooling or compressing air, and reading SVP tables.","sort":332,"aliases":[]},{"id":334,"code":"09.03","level":2,"parent_id":320,"name":"Calorimetry","description":"Calorimetry problems: energy balance using heat lost = heat gained, with calorimeter heat capacity and phase changes included. Covers method of mixtures, ice-water-steam bookkeeping, and variable specific heat integrals.","sort":333,"aliases":[]},{"id":335,"code":"09.03.01","level":3,"parent_id":334,"name":"Principle of calorimetry and method of mixtures","description":"Method of mixtures: mixing hot and cold water/solids and solving m₁s₁(T₁−T) = m₂s₂(T−T₂) for the final temperature. Archetypes: hot metal dropped into water, finding specific heat of a solid, temperature after mixing two liquids.","sort":334,"aliases":[]},{"id":336,"code":"09.03.02","level":3,"parent_id":334,"name":"Water equivalent and heat capacity of a calorimeter","description":"Includes the calorimeter in the energy balance via its water equivalent w (mass of water absorbing the same heat) or heat capacity ms. Questions give calorimeter mass and specific heat and ask for corrected final temperatures or the calorimeter constant.","sort":335,"aliases":[]},{"id":337,"code":"09.03.03","level":3,"parent_id":334,"name":"Ice-water-steam mixture problems (latent heat of phase change)","description":"Multi-phase bookkeeping: adding ice to water, injecting steam into water, or mixing ice and steam, deciding the final state by checking whether all ice melts or all steam condenses. Uses L_f = 80 cal/g and L_v = 540 cal/g with staged heat calculations.","sort":336,"aliases":[]},{"id":338,"code":"09.03.04","level":3,"parent_id":334,"name":"Temperature-dependent specific heat (calorimetry by integration)","description":"Calorimetry when s depends on temperature, s = a + bT, so Q = m∫s(T)dT between limits. Tasks: heat required to raise temperature from T₁ to T₂, mean specific heat over a range, or final temperature of mixtures with temperature-dependent c.","sort":337,"aliases":[]},{"id":339,"code":"09.04","level":2,"parent_id":320,"name":"Heat transfer","description":"Heat transfer by conduction, convection and radiation: thermal resistance networks, cooling laws, and Stefan–Boltzmann/Wien radiation problems. Includes identifying the dominant mode and computing rates of heat flow or cooling times.","sort":338,"aliases":["conduction, convection and radiation","Newton's law of cooling","Stefan-Boltzmann law","Wien's displacement law","black-body radiation"]},{"id":340,"code":"09.04.01","level":3,"parent_id":339,"name":"Conduction","description":"Steady-state conduction: dQ/dt = kA dT/dx, thermal resistance R = L/kA, and series/parallel composite slabs or rods. Archetypes: temperature at the junction of two joined rods, equivalent conductivity of slabs, heat current through an insulated wall or a rod with variable area.","sort":339,"aliases":[]},{"id":341,"code":"09.04.02","level":3,"parent_id":339,"name":"Convection","description":"Heat carried by bulk fluid motion (natural and forced convection), treated mostly qualitatively. Questions on why sea breezes occur, how hot water/radiator heats a room, and estimating convective heat loss with a heat-transfer coefficient h.","sort":340,"aliases":[]},{"id":342,"code":"09.04.03","level":3,"parent_id":339,"name":"Radiation","description":"Radiation questions using Stefan's law P = eσA(T⁴ − T₀⁴), emissivity, and black-body behaviour. Includes net radiative exchange between a body and surroundings and cooling of a hot body by emission.","sort":341,"aliases":[]},{"id":343,"code":"09.04.04","level":3,"parent_id":339,"name":"Newton's law of cooling","description":"Newton's law of cooling, dT/dt = −k(T − T_s), valid for small temperature excess. Archetypes: time for a body to cool from T₁ to T₂ in a room at T_s, reading cooling curves (log(T−T_s) vs t), and comparing cooling rates at different excess temperatures.","sort":342,"aliases":[]},{"id":344,"code":"09.04.05","level":3,"parent_id":339,"name":"Stefan-Boltzmann cooling in vacuum","description":"Cooling purely by radiation in vacuum: mc(dT/dt) = −eσA(T⁴ − T₀⁴), often integrated or approximated. Tasks: initial cooling rate of a sphere/sphere in vacuum, time to cool when T ≫ T₀, and comparing cooling of bodies of different emissivity or size.","sort":343,"aliases":[]},{"id":345,"code":"09.04.06","level":3,"parent_id":339,"name":"Wien black-body scaling law","description":"Scaling of the black-body spectrum with temperature: the spectral distribution has the form λ⁻⁵f(λT), so the whole curve shifts and its peak height grows as T⁵. Questions ask how the emitted spectrum or peak intensity changes when a body's temperature is doubled.","sort":344,"aliases":[]},{"id":346,"code":"09.04.07","level":3,"parent_id":339,"name":"Wien's displacement law","description":"Wien's displacement law, λ_m T = b = 2.9×10⁻³ m·K, applied to find a star's or filament's temperature from its peak wavelength, or the peak wavelength from a known temperature. Includes comparing surface temperatures of the Sun, stars, and red-hot vs white-hot bodies.","sort":345,"aliases":[]},{"id":347,"code":"09.04.08","level":3,"parent_id":339,"name":"Black-body radiative equilibrium of a planet","description":"Radiative equilibrium of a planet: absorbed solar power (1−a)SπR² equals emitted σ(4πR²)T⁴, giving T = [S(1−a)/(4σ)]^{1/4}. Tasks: estimate Earth's (or another planet's) temperature from the solar constant, albedo, and distance via the inverse-square law.","sort":346,"aliases":[]},{"id":348,"code":"09.04.09","level":3,"parent_id":339,"name":"Kirchhoff's law of radiation and emissivity","description":"Kirchhoff's law: at thermal equilibrium the ratio of emissive to absorptive power equals that of a black body, so good absorbers are good emitters (E/a = E_black, e = a). Questions compare emissive powers of polished vs rough surfaces, or use emissivity e to correct Stefan-law powers.","sort":347,"aliases":[]},{"id":349,"code":"10","level":1,"parent_id":null,"name":"Thermodynamics & Kinetic Theory of Gases","description":"JEE chapter 10: Thermodynamics & Kinetic Theory of Gases. Concepts: First law of thermodynamics; Thermodynamic processes; Second law of thermodynamics and entropy; Carnot engine; Ideal gas equation and gas laws; Kinetic model of an ideal gas: pressure, molecular speeds and Maxwell distribution; Degrees of freedom and specific heats of gases; Mean free path; Real gases and Van der Waals equation.","sort":348,"aliases":["Thermodynamics","Kinetic Theory"]},{"id":350,"code":"10.01","level":2,"parent_id":349,"name":"First law of thermodynamics","description":"Questions on the first law of thermodynamics ΔQ = ΔU + W, sign conventions, and applying energy conservation to heat, internal energy, and work in closed systems.","sort":349,"aliases":["zeroth law of thermodynamics","thermal equilibrium","laws of thermodynamics","internal energy","heat and work"]},{"id":351,"code":"10.01.01","level":3,"parent_id":350,"name":"Ideal-gas process with heat capacity C(T)","description":"Problems where an ideal gas undergoes a process with a specified heat capacity C(T) or C(V), requiring integration of dQ = nC dT and use of the first law to find work or internal energy change.","sort":350,"aliases":[]},{"id":352,"code":"10.01.02","level":3,"parent_id":350,"name":"Thermal equilibrium and the zeroth law","description":"Questions about thermal equilibrium, temperature measurement, and the zeroth law as the basis for thermometers and comparing temperatures of bodies in contact.","sort":351,"aliases":[]},{"id":353,"code":"10.01.03","level":3,"parent_id":350,"name":"First law of thermodynamics: internal energy, work and heat","description":"Direct application of ΔQ = ΔU + W: calculating internal energy change, work done by/on a gas, and heat supplied/removed in simple processes, including sign conventions.","sort":352,"aliases":[]},{"id":354,"code":"10.01.04","level":3,"parent_id":350,"name":"Degrees of freedom and equipartition of energy","description":"Problems using degrees of freedom f and equipartition to find internal energy U = (f/2)nRT, molar specific heats, and energy per molecule for monatomic, diatomic, or polyatomic gases.","sort":353,"aliases":[]},{"id":355,"code":"10.01.05","level":3,"parent_id":350,"name":"Molar specific heats and Mayer's relation","description":"Questions on molar specific heats Cp and Cv, their ratio γ, and Mayer's relation Cp − Cv = R, including numerical values for different gas types.","sort":354,"aliases":[]},{"id":356,"code":"10.01.06","level":3,"parent_id":350,"name":"Mean free path of gas molecules","description":"Calculations of mean free path λ = 1/(√2 π d² n) using molecular diameter and number density, and related collision frequency or average distance between collisions.","sort":355,"aliases":[]},{"id":357,"code":"10.01.07","level":3,"parent_id":350,"name":"First law applied to thermodynamic processes","description":"Applying the first law to specific thermodynamic processes: isothermal, adiabatic, isobaric, isochoric, and cyclic processes, finding Q, W, and ΔU for each step.","sort":356,"aliases":[]},{"id":358,"code":"10.02","level":2,"parent_id":349,"name":"Thermodynamic processes","description":"General questions on thermodynamic processes, their P-V, P-T, or V-T representations, and identifying process types from graphs or equations.","sort":357,"aliases":["isothermal process","adiabatic process","isobaric process","isochoric process","cyclic process","P-V diagram","work done by gas"]},{"id":359,"code":"10.02.01","level":3,"parent_id":358,"name":"Ideal-gas thermodynamic processes","description":"Problems on ideal-gas processes described by equations like PV = constant, P/T = constant, or V/T = constant, and computing state variables or work/heat for such processes.","sort":358,"aliases":[]},{"id":360,"code":"10.02.02","level":3,"parent_id":358,"name":"Pressure-temperature graphs with a nonzero intercept","description":"Questions involving P-T graphs where the line has a nonzero intercept (e.g., P = aT + b), requiring analysis of whether volume increases, decreases, or stays constant along the process.","sort":359,"aliases":[]},{"id":361,"code":"10.02.03","level":3,"parent_id":358,"name":"Maxwell equal-area construction","description":"Problems on the Maxwell equal-area construction for van der Waals isotherms, locating the horizontal tie-line for liquid-vapor coexistence using equal areas above and below the line.","sort":360,"aliases":[]},{"id":362,"code":"10.02.04","level":3,"parent_id":358,"name":"Polytropic process","description":"Questions on polytropic processes PVⁿ = constant, finding molar specific heat C = Cv + R/(1−n), work done, and relating n to isothermal, adiabatic, isobaric, and isochoric limits.","sort":361,"aliases":[]},{"id":363,"code":"10.02.05","level":3,"parent_id":358,"name":"Ideal gas postulates and equation of state","description":"Questions on ideal gas postulates, the equation of state PV = nRT, and using it to relate pressure, volume, temperature, and number of moles in various states.","sort":362,"aliases":[]},{"id":364,"code":"10.02.06","level":3,"parent_id":358,"name":"Isothermal, adiabatic, isobaric and isochoric processes","description":"Problems specifically on isothermal (PV = constant), adiabatic (PV^γ = constant), isobaric (P = constant), and isochoric (V = constant) processes, including work, heat, and internal energy changes.","sort":363,"aliases":[]},{"id":365,"code":"10.02.07","level":3,"parent_id":358,"name":"Work done on P-V diagram","description":"Calculating work done by or on a gas as the area under a P-V curve, including cyclic processes where net work is the enclosed area, with sign determined by traversal direction.","sort":364,"aliases":[]},{"id":366,"code":"10.03","level":2,"parent_id":349,"name":"Second law of thermodynamics and entropy","description":"Questions on the second law of thermodynamics, entropy, and its implications for spontaneity, irreversibility, and the direction of natural processes.","sort":365,"aliases":["entropy","reversible and irreversible processes","Kelvin-Planck statement","Clausius statement","entropy change"]},{"id":367,"code":"10.03.01","level":3,"parent_id":366,"name":"Second law and entropy","description":"Problems on entropy as a state function, calculating ΔS = ∫dQ_rev/T for various processes, and using entropy change to determine spontaneity or reversibility.","sort":366,"aliases":[]},{"id":368,"code":"10.03.02","level":3,"parent_id":366,"name":"Gas microstates and Boltzmann entropy","description":"Questions relating entropy to the number of microstates Ω via S = k_B ln Ω, counting microstates for gas particles in compartments, and computing entropy changes during free expansion or mixing.","sort":367,"aliases":[]},{"id":369,"code":"10.03.03","level":3,"parent_id":366,"name":"Clausius and Kelvin-Planck statements of second law","description":"Questions on the Clausius statement (heat cannot flow spontaneously from cold to hot) and Kelvin-Planck statement (no perfect engine with 100% efficiency), and their equivalence.","sort":368,"aliases":[]},{"id":370,"code":"10.03.04","level":3,"parent_id":366,"name":"Entropy change in reversible and irreversible processes","description":"Calculating entropy change for reversible and irreversible processes between the same initial and final states, using reversible paths, and showing ΔS_universe ≥ 0 for irreversible processes.","sort":369,"aliases":[]},{"id":371,"code":"10.04","level":2,"parent_id":349,"name":"Carnot engine","description":"Questions on the Carnot engine, its cycle, efficiency, and coefficient of performance for Carnot refrigerators or heat pumps.","sort":370,"aliases":["heat engines","refrigerators","Carnot cycle"]},{"id":372,"code":"10.04.01","level":3,"parent_id":371,"name":"Heat engines, refrigerators and the Carnot cycle","description":"Problems on heat engines, refrigerators, and heat pumps: work output/input, heat absorbed/rejected, efficiency η = W/Q_h, and coefficient of performance for refrigerators and heat pumps.","sort":371,"aliases":[]},{"id":373,"code":"10.04.02","level":3,"parent_id":371,"name":"Carnot engine efficiency and coefficient of performance","description":"Calculating Carnot engine efficiency η = 1 − T_c/T_h and coefficient of performance for Carnot refrigerators/heat pumps using reservoir temperatures, and comparing with real engines.","sort":372,"aliases":[]},{"id":374,"code":"10.04.03","level":3,"parent_id":371,"name":"Carnot cycle and its P-V diagram","description":"Questions on the Carnot cycle's four steps (isothermal expansion, adiabatic expansion, isothermal compression, adiabatic compression), its P-V diagram, and computing work, heat, and efficiency for the cycle.","sort":373,"aliases":[]},{"id":375,"code":"10.04.04","level":3,"parent_id":371,"name":"Carnot theorem and reversible heat engines","description":"Questions on Carnot theorem: no engine between two reservoirs can exceed Carnot efficiency, efficiency of reversible engines, and comparisons of reversible heat engine performance.","sort":374,"aliases":[]},{"id":376,"code":"10.05","level":2,"parent_id":349,"name":"Ideal gas equation and gas laws","description":"Questions applying Boyle's law, Charles's law, Gay-Lussac's law, and the combined gas law PV/T = constant to ideal gas processes and state changes.","sort":375,"aliases":["ideal gas equation","Boyle's law","Charles's law","Gay-Lussac's law","Avogadro's law"]},{"id":377,"code":"10.05.01","level":3,"parent_id":376,"name":"Ideal gas equation","description":"Questions using the ideal gas equation PV = nRT or PV = NkT to relate pressure, volume, temperature, and amount of gas, including molar mass and density forms.","sort":376,"aliases":[]},{"id":378,"code":"10.05.02","level":3,"parent_id":376,"name":"Isothermal gas stratification and center of mass","description":"Questions on the density and pressure distribution of an isothermal gas column in a gravitational field, and finding the center of mass of such a stratified gas column.","sort":377,"aliases":[]},{"id":379,"code":"10.05.03","level":3,"parent_id":376,"name":"Gas pressure under gravity or centrifugal force","description":"Questions on gas pressure variation in a container under gravity or in a rotating frame, where pressure depends on height or radial position due to body forces.","sort":378,"aliases":[]},{"id":380,"code":"10.05.04","level":3,"parent_id":376,"name":"Barometric formula with variable temperature","description":"Questions deriving or applying the barometric formula P = P₀ exp(-Mgh/RT) when temperature varies with altitude, requiring integration over the temperature profile.","sort":379,"aliases":[]},{"id":381,"code":"10.05.05","level":3,"parent_id":376,"name":"Barometric formula (isothermal atmosphere)","description":"Questions on the exponential decrease of pressure with height in an isothermal atmosphere, P = P₀ exp(-Mgh/RT), and related density or number density calculations.","sort":380,"aliases":[]},{"id":382,"code":"10.05.06","level":3,"parent_id":376,"name":"Barometric formula with height-varying T","description":"Questions on atmospheric pressure variation with altitude when temperature is a specified function of height, requiring integration of dP/dh = -ρg with variable T.","sort":381,"aliases":[]},{"id":383,"code":"10.05.07","level":3,"parent_id":376,"name":"Van der Waals equation of state","description":"Questions using the Van der Waals equation (P + a/V²)(V - b) = RT to account for finite molecular size and intermolecular attraction in real gases.","sort":382,"aliases":[]},{"id":384,"code":"10.05.08","level":3,"parent_id":376,"name":"Van der Waals real-gas equation of state","description":"Questions on real-gas behavior via the Van der Waals equation of state, including critical constants, corrections to ideal gas law, and comparison with ideal gas predictions.","sort":383,"aliases":[]},{"id":385,"code":"10.05.09","level":3,"parent_id":376,"name":"Molecule distribution between connected vessel subvolumes","description":"Questions on how gas molecules distribute between connected vessels or subvolumes at equilibrium, using equal pressure or chemical potential conditions and the ideal gas law.","sort":384,"aliases":[]},{"id":386,"code":"10.06","level":2,"parent_id":349,"name":"Kinetic model of an ideal gas: pressure, molecular speeds and Maxwell distribution","description":"Questions on the kinetic model of an ideal gas, including pressure derivation P = (1/3)ρv²_rms, molecular speeds, and the Maxwell speed distribution.","sort":385,"aliases":["Maxwell speed distribution","distribution of molecular speeds","rms speed","root mean square speed","average speed","mean speed","most probable speed","Maxwell-Boltzmann distribution","kinetic interpretation of temperature","pressure of an ideal gas"]},{"id":387,"code":"10.06.01","level":3,"parent_id":386,"name":"Molecular model of an ideal gas","description":"Questions on the molecular model of an ideal gas: assumptions of point-like non-interacting molecules, elastic collisions with walls, and derivation of pressure from molecular impacts.","sort":386,"aliases":[]},{"id":388,"code":"10.06.02","level":3,"parent_id":386,"name":"Kinetic interpretation of temperature","description":"Questions relating average molecular kinetic energy to absolute temperature, (1/2)mv²_rms = (3/2)kT, and calculating rms speed from temperature and molar mass.","sort":387,"aliases":[]},{"id":389,"code":"10.06.03","level":3,"parent_id":386,"name":"Maxwell speed distribution","description":"Questions on the Maxwell speed distribution function, most probable speed, average speed, and rms speed, and their relationships for an ideal gas at temperature T.","sort":388,"aliases":[]},{"id":390,"code":"10.06.04","level":3,"parent_id":386,"name":"Maxwell-Boltzmann speed distribution","description":"Questions on the Maxwell-Boltzmann speed distribution, including the distribution function f(v), peak shifts with temperature or mass, and calculating fractions of molecules in speed ranges.","sort":389,"aliases":[]},{"id":391,"code":"10.07","level":2,"parent_id":349,"name":"Degrees of freedom and specific heats of gases","description":"Questions on degrees of freedom of gas molecules (translational, rotational, vibrational) and their effect on specific heats and internal energy of gases.","sort":390,"aliases":["law of equipartition of energy","specific heat capacities of gases","Mayer's relation"]},{"id":392,"code":"10.07.01","level":3,"parent_id":391,"name":"Degrees of freedom and equipartition of energy","description":"Questions applying the equipartition theorem: each degree of freedom contributes (1/2)kT per molecule, leading to internal energy U = (f/2)nRT and degrees of freedom counting.","sort":391,"aliases":[]},{"id":393,"code":"10.07.02","level":3,"parent_id":391,"name":"Specific heats of gases","description":"Questions on molar specific heats C_p and C_v of gases, their ratio γ = C_p/C_v, relation C_p - C_v = R, and dependence on degrees of freedom.","sort":392,"aliases":[]},{"id":394,"code":"10.08","level":2,"parent_id":349,"name":"Mean free path","description":"Questions on mean free path λ = 1/(√2 π d² n) of gas molecules, its dependence on pressure, temperature, and molecular diameter, and related collision frequency calculations.","sort":393,"aliases":["collision frequency"]},{"id":395,"code":"10.08.01","level":3,"parent_id":394,"name":"Knudsen effusion versus viscous orifice flow","description":"Questions contrasting effusion rate (∝ P/√m, valid when mean free path exceeds hole diameter, Knudsen/molecular regime) with viscous or continuum flow through larger holes where rate scales differently with pressure (roughly ∝ P²). Typical items ask how leak rate changes with pressure, hole size, or gas molar mass in each regime.","sort":394,"aliases":[]},{"id":396,"code":"10.08.02","level":3,"parent_id":394,"name":"Gas transport coefficients from kinetic theory","description":"Computing transport coefficients from kinetic theory: viscosity η = (1/3)ρλ⟨v⟩, thermal conductivity κ = (1/3)ρC_v λ⟨v⟩, diffusion coefficient D = (1/3)λ⟨v⟩. Common questions: show η is independent of pressure, find η ∝ √T, compare coefficients across gases.","sort":395,"aliases":[]},{"id":397,"code":"10.08.03","level":3,"parent_id":394,"name":"Mean free path","description":"Mean free path λ = 1/(√2 π d² n) = kT/(√2 π d² P) and collision frequency ⟨v⟩/λ. Questions compute λ for air at given P, T, molecular diameter, or ask how λ scales when pressure or temperature changes, and when continuum assumptions break down.","sort":396,"aliases":[]},{"id":398,"code":"10.09","level":2,"parent_id":349,"name":"Real gases and Van der Waals equation","description":"Parent node for deviations from ideal-gas behavior: compressibility factor Z = PV/nRT, why real gases deviate at high P and low T, intermolecular forces and finite molecular size, liquefaction, Boyle temperature, and critical phenomena. Any question invoking non-ideal gas corrections belongs here.","sort":397,"aliases":["Van der Waals real-gas equation"]},{"id":399,"code":"10.09.01","level":3,"parent_id":398,"name":"Van der Waals real-gas equation","description":"Van der Waals equation (P + a n²/V²)(V − nb) = nRT: physical meaning of attraction constant a and co-volume b, units of a and b, critical constants (P_c = a/27b², V_c = 3b, T_c = 8a/27Rb), and Boyle temperature T_B = a/Rb. Questions give a, b and ask for critical temperature/pressure or compare two gases.","sort":398,"aliases":[]},{"id":400,"code":"11","level":1,"parent_id":null,"name":"Oscillations (Simple Harmonic Motion)","description":"JEE chapter 11: Oscillations (Simple Harmonic Motion). Concepts: Damped and forced oscillations, resonance; Simple harmonic motion; Pendulums; Spring-mass systems; Energy in simple harmonic motion; Coupled oscillations and normal modes; Small oscillations near a potential-energy minimum.","sort":399,"aliases":["Oscillations"]},{"id":401,"code":"11.01","level":2,"parent_id":400,"name":"Damped and forced oscillations, resonance","description":"Umbrella node for any question involving oscillations with resistive forces or an external periodic drive: damping equations, amplitude/energy decay, critical damping, resonance conditions, and Q factor.","sort":400,"aliases":["damped oscillations","forced oscillations","mechanical resonance","driven oscillator"]},{"id":402,"code":"11.01.01","level":3,"parent_id":401,"name":"Damped oscillations: equation, amplitude and energy decay","description":"Questions on the damped oscillator equation m x'' + b x' + k x = 0: amplitude decay A = A₀e^(−bt/2m), energy decaying as e^(−bt/m), time for amplitude or energy to fall to a given fraction, and comparing underdamped/critically damped/overdamped behavior.","sort":401,"aliases":[]},{"id":403,"code":"11.01.02","level":3,"parent_id":401,"name":"Critical damping and its applications","description":"Questions on critical damping (b = 2√(mk)): no oscillation with fastest return to equilibrium, and applications such as dead-beat galvanometers, vehicle shock absorbers, and door-closing mechanisms; may ask to compute the damping constant or identify which system is critically damped.","sort":402,"aliases":[]},{"id":404,"code":"11.01.03","level":3,"parent_id":401,"name":"Forced oscillations and resonance","description":"Questions with an external periodic force F₀ sin ωt on a damped oscillator: steady-state amplitude A = F₀/√((k − mω²)² + (bω)²), phase lag of displacement, amplitude maximum near ω₀ = √(k/m), and resonance phenomena like a soldier breaking step on a bridge or a tuning-fork box sounding loudly.","sort":403,"aliases":[]},{"id":405,"code":"11.01.04","level":3,"parent_id":401,"name":"Sharpness of resonance and quality factor","description":"Questions on sharpness of resonance and quality factor: Q = ω₀m/b (or 2π × energy stored/energy lost per cycle), bandwidth Δω = b/m, and how smaller damping makes the resonance peak taller and narrower; often asks to rank oscillators by Q or compute Q from decay data.","sort":404,"aliases":[]},{"id":406,"code":"11.02","level":2,"parent_id":400,"name":"Simple harmonic motion","description":"Umbrella node for all simple harmonic motion questions: defining a = −ω²x, displacement/velocity/acceleration relations, energy of SHM, standard oscillators, and superposition of SHMs.","sort":405,"aliases":["SHM kinematics"]},{"id":407,"code":"11.02.01","level":3,"parent_id":406,"name":"Periodic motion and simple harmonic motion","description":"Questions testing the definition of SHM versus general periodic motion: identifying which of several motions (uniform circular motion projection, F = −kx systems, a pendulum at large angle) are SHM, using the condition a ∝ −x, and finding ω from the force law.","sort":406,"aliases":[]},{"id":408,"code":"11.02.02","level":3,"parent_id":406,"name":"Displacement, velocity and acceleration in SHM","description":"Questions using x = A sin(ωt + φ) with v = Aω cos(ωt + φ) and a = −Aω² sin(ωt + φ): finding v = ω√(A² − x²) at a given position, maximum speed/acceleration, phase differences between x, v and a, and time to go from one displacement to another.","sort":407,"aliases":[]},{"id":409,"code":"11.02.03","level":3,"parent_id":406,"name":"Linear SHM","description":"Questions on one-dimensional linear SHM with restoring force F = −kx: spring–mass oscillators, T = 2π√(m/k), energy ½kA² split into kinetic and potential at a given x, and amplitude/phase from initial conditions.","sort":408,"aliases":[]},{"id":410,"code":"11.02.04","level":3,"parent_id":406,"name":"Standard oscillators and their time periods","description":"Questions computing time periods of standard oscillators: spring–mass (including series/parallel springs and cut springs), simple pendulum, physical pendulum, torsional pendulum, liquid column in a U-tube, floating cylinder bobbing, and a piston oscillating over gas in a cylinder.","sort":409,"aliases":[]},{"id":411,"code":"11.02.05","level":3,"parent_id":406,"name":"Superposition of collinear SHMs (beats and phasor addition)","description":"Questions adding two collinear SHMs: same-frequency phasor addition giving R = √(A₁² + A₂² + 2A₁A₂ cos δ) with special cases of constructive/destructive interference, and different-frequency superposition producing beats x = 2A cos(Δω t/2) cos(ω̄ t) with beat period 2π/Δω.","sort":410,"aliases":[]},{"id":412,"code":"11.02.06","level":3,"parent_id":406,"name":"Superposition of perpendicular SHMs (Lissajous figures)","description":"Questions superposing perpendicular SHMs, x = A sin ωt and y = B sin(ωt + φ): identifying the resulting path as a straight line (φ = 0 or π), ellipse, or circle (equal amplitudes, φ = π/2), and sketching Lissajous figures for frequency ratios like 1:2.","sort":411,"aliases":[]},{"id":413,"code":"11.03","level":2,"parent_id":400,"name":"Pendulums","description":"Umbrella node for pendulum-based oscillation questions: simple, compound, torsional and bifilar pendulums, and pendulums modified by accelerating frames, fields, collisions or constraints.","sort":412,"aliases":["simple pendulum","physical pendulum","compound pendulum","torsional pendulum"]},{"id":414,"code":"11.03.01","level":3,"parent_id":413,"name":"Kapitza pendulum","description":"Questions on a pendulum whose suspension point vibrates vertically at high frequency (Kapitza pendulum): using the effective-potential/rapid-vibration approach to show the normally unstable inverted position becomes stable and finding the small-oscillation frequency about it.","sort":413,"aliases":[]},{"id":415,"code":"11.03.02","level":3,"parent_id":413,"name":"Physical (compound) pendulum","description":"Questions on a rigid body pivoted at a fixed point oscillating as a compound pendulum: T = 2π√(I/(mgd)), equivalent simple-pendulum length, center of oscillation and interchangeability of pivot points, minimum-period condition d = k (radius of gyration), e.g., a rod pivoted at one end or a disc pivoted at its rim.","sort":414,"aliases":[]},{"id":416,"code":"11.03.03","level":3,"parent_id":413,"name":"Simple pendulum and spring-mass systems","description":"Questions on the ideal simple pendulum T = 2π√(L/g) with small-angle approximation and the spring–mass system T = 2π√(m/k): effect of changing length, mass, g (different planets, mines, mountains), spring cutting or combining, and mixed spring–pendulum setups.","sort":415,"aliases":[]},{"id":417,"code":"11.03.04","level":3,"parent_id":413,"name":"Charged pendulum in a uniform field (effective gravity)","description":"Questions on a pendulum bob carrying charge q in a uniform electric field E: replacing gravity by an effective acceleration g_eff = √(g² + (qE/m)²) (or |g − qE/m| for a vertical field), finding the tilted equilibrium direction and the modified period T = 2π√(L/g_eff).","sort":416,"aliases":[]},{"id":418,"code":"11.03.05","level":3,"parent_id":413,"name":"Rolling body on a spring: effective-mass SHM","description":"Questions on a rolling cylinder or sphere attached to a spring and rolling without slipping: using energy conservation with rotational KE to get effective mass m(1 + I/mR²) and period T = 2π√(m(1 + I/mR²)/k), comparing solid sphere, hollow sphere, disc and ring.","sort":417,"aliases":[]},{"id":419,"code":"11.03.06","level":3,"parent_id":413,"name":"Pendulum with an elastic wall collision","description":"Questions where a pendulum's motion is interrupted by an elastic collision with a wall or by the string catching on a peg mid-swing: the period is the sum of two half-oscillations with different effective lengths, e.g., T = π√(L₁/g) + π√(L₂/g), possibly with amplitude unchanged in elastic bounce.","sort":418,"aliases":[]},{"id":420,"code":"11.03.07","level":3,"parent_id":413,"name":"Pendulum with an accelerating suspension","description":"Questions on a pendulum in a non-inertial frame: lift accelerating up/down giving g_eff = g + a or g − a, a cart accelerating horizontally giving g_eff = √(g² + a²) with tilted equilibrium, or a pendulum on an inclined accelerating wedge; find the new period or equilibrium angle.","sort":419,"aliases":[]},{"id":421,"code":"11.03.08","level":3,"parent_id":413,"name":"Cubic-restoring-force oscillator period","description":"Questions on an oscillator with restoring force F = −kx³ (potential ∝ x⁴), which is nonlinear so the period depends on amplitude: computing T via the energy integral T ∝ 1/A (e.g., T = (constant/A)∫dx/√(A⁴ − x⁴)) or comparing periods at different amplitudes.","sort":420,"aliases":[]},{"id":422,"code":"11.03.09","level":3,"parent_id":413,"name":"Torsional pendulum","description":"Questions on the torsional pendulum: a disc or body suspended by a wire executing I θ'' = −κθ oscillations with T = 2π√(I/κ), torsional rigidity κ = πηr⁴/(2L), and tasks like finding rigidity modulus, moment of inertia, or the new period when the wire's length/radius or the disc is changed.","sort":421,"aliases":[]},{"id":423,"code":"11.03.10","level":3,"parent_id":413,"name":"Bifilar torsional oscillation","description":"Questions on bifilar suspension, where a bar or disc hangs from two parallel wires and twists about the vertical axis: period T = 2π√(I l/(mg r²)) with l the wire length and r half the wire separation, used to measure the moment of inertia; typical tasks compute T or I when separation, length or mass changes.","sort":422,"aliases":[]},{"id":424,"code":"11.03.11","level":3,"parent_id":413,"name":"Swing suspended by perpendicular ropes","description":"Questions about a pendulum bob suspended by two perpendicular ropes, finding effective length, tension, or oscillation period from the geometry of the rope arrangement.","sort":423,"aliases":[]},{"id":425,"code":"11.04","level":2,"parent_id":400,"name":"Spring-mass systems","description":"Questions on block-spring oscillators: time period T=2π√(m/k), frequency, spring combinations (series/parallel), and force/energy relations for horizontal or vertical spring-mass setups.","sort":424,"aliases":["spring oscillator","block on spring","springs in series and parallel","effective spring constant","vertical spring oscillations","two-block spring systems"]},{"id":426,"code":"11.04.01","level":3,"parent_id":425,"name":"Simple pendulum and spring-mass systems","description":"Questions comparing or combining a simple pendulum and a spring-mass system, such as matching periods, finding equivalent lengths, or coupled pendulum-spring oscillations.","sort":425,"aliases":[]},{"id":427,"code":"11.04.02","level":3,"parent_id":425,"name":"Charged pendulum in a uniform field (effective gravity)","description":"Questions where a charged pendulum bob oscillates in a uniform electric or gravitational field, using effective gravity g_eff = g ± qE/m to find the modified time period.","sort":426,"aliases":[]},{"id":428,"code":"11.04.03","level":3,"parent_id":425,"name":"Rolling body on a spring: effective-mass SHM","description":"Questions on a rolling body (disc, sphere, cylinder) attached to a spring, where rotational kinetic energy modifies the SHM period via effective mass or reduced acceleration.","sort":427,"aliases":[]},{"id":429,"code":"11.04.04","level":3,"parent_id":425,"name":"Pendulum with an elastic wall collision","description":"Questions where a pendulum bob collides elastically with a wall or barrier during oscillation, altering the effective length or period of the resulting piecewise motion.","sort":428,"aliases":[]},{"id":430,"code":"11.04.05","level":3,"parent_id":425,"name":"Pendulum with an accelerating suspension","description":"Questions on a pendulum whose point of suspension accelerates (e.g., in a lift or cart), requiring effective gravity g_eff = g ± a to compute the oscillation period.","sort":429,"aliases":[]},{"id":431,"code":"11.04.06","level":3,"parent_id":425,"name":"Cubic-restoring-force oscillator period","description":"Questions on oscillators with a cubic restoring force F = -kx^3 or potential U ∝ x^4, asking for period dependence on amplitude using dimensional analysis or energy integrals.","sort":430,"aliases":[]},{"id":432,"code":"11.04.07","level":3,"parent_id":425,"name":"Coulomb-restoring SHM of a rolling charged sphere","description":"Questions on a charged sphere rolling under Coulomb or electrostatic restoring force, combining rolling constraints with SHM to find period or frequency.","sort":431,"aliases":[]},{"id":433,"code":"11.04.08","level":3,"parent_id":425,"name":"Gas as a spring: piston oscillations in a cylinder","description":"Questions on a piston oscillating in a gas-filled cylinder, treating adiabatic or isothermal gas compression as a spring to derive the SHM period from pressure-volume relations.","sort":432,"aliases":[]},{"id":434,"code":"11.04.09","level":3,"parent_id":425,"name":"Piecewise SHM with alternately slack elastic strings","description":"Questions where a mass is attached to two elastic strings that become alternately slack during motion, producing piecewise SHM with different periods in each half-cycle.","sort":433,"aliases":[]},{"id":435,"code":"11.05","level":2,"parent_id":400,"name":"Energy in simple harmonic motion","description":"Questions on kinetic, potential, and total energy in SHM: expressions ½kA², ½mω²(A²-x²), energy fraction at a given displacement, and average energy over a cycle.","sort":434,"aliases":["kinetic and potential energy in SHM"]},{"id":436,"code":"11.05.01","level":3,"parent_id":435,"name":"Energy in SHM","description":"Questions asking for potential energy, kinetic energy, or total energy of a simple harmonic oscillator at specific displacements, velocities, or phases, often using E = ½mω²A².","sort":435,"aliases":[]},{"id":437,"code":"11.06","level":2,"parent_id":400,"name":"Coupled oscillations and normal modes","description":"Questions on two or more coupled oscillators (masses connected by springs, coupled pendulums), finding normal mode frequencies, mode shapes, and beat phenomena.","sort":436,"aliases":["Coupled oscillators","Normal modes of vibration"]},{"id":438,"code":"11.06.01","level":3,"parent_id":437,"name":"Coupled oscillators and normal modes","description":"Questions on normal modes of coupled oscillators: symmetric and antisymmetric mode frequencies, energy exchange between coupled pendulums or spring-connected masses.","sort":437,"aliases":[]},{"id":439,"code":"11.07","level":2,"parent_id":400,"name":"Small oscillations near a potential-energy minimum","description":"Questions on small oscillations about a stable equilibrium of an arbitrary potential U(x), using U''(x₀) to find the angular frequency ω = √(U''/m) or period.","sort":438,"aliases":[]},{"id":440,"code":"11.07.01","level":3,"parent_id":439,"name":"Rocking oscillations on a curved support","description":"Questions on a body rocking without slipping on a curved surface or support, using the local curvature and geometry to find the small-oscillation period.","sort":439,"aliases":[]},{"id":441,"code":"12","level":1,"parent_id":null,"name":"Waves & Acoustics","description":"JEE chapter 12: Waves & Acoustics. Concepts: Wave basics and the wave equation; Standing waves; Sound waves; Superposition and beats; Doppler effect (sound and light); Reflection and transmission of waves at boundaries.","sort":440,"aliases":["Waves"]},{"id":442,"code":"12.01","level":2,"parent_id":441,"name":"Wave basics and the wave equation","description":"Parent node for basic travelling-wave questions: y(x,t) = A sin(kx − ωt + φ), v = fλ = ω/k, wave number, period, phase, direction of travel, and distinguishing wave speed from particle speed.","sort":441,"aliases":["wave speed and the wave equation","transverse and longitudinal waves","types of waves"]},{"id":443,"code":"12.01.01","level":3,"parent_id":442,"name":"Wave speed and the wave equation","description":"Direct use of v = fλ and v = ω/k: given any two of speed, frequency, wavelength (or period), find the third; converting between wave-function parameters and wave speed; how v, f, λ change when a wave enters a new medium.","sort":442,"aliases":[]},{"id":444,"code":"12.01.02","level":3,"parent_id":442,"name":"Power carried by a transverse string wave","description":"Average power transported by a sinusoidal wave on a string, P = 2π²μA²f²v (equivalently ½μω²A²v), scaling of power with amplitude and frequency, and related energy density/energy-flux questions.","sort":443,"aliases":[]},{"id":445,"code":"12.01.03","level":3,"parent_id":442,"name":"Sound speed in a gas with atmospheric height","description":"Problems where sound speed varies with altitude because atmospheric temperature T(h) varies (v = √(γRT/M)): computing travel times or turning heights by integrating dt = dh/v(h) through layered air.","sort":444,"aliases":[]},{"id":446,"code":"12.01.04","level":3,"parent_id":442,"name":"Wave-pulse reflection and transmission at a string junction","description":"A pulse on a string hitting a junction with a second string of different linear density (or a fixed/free end): partial reflection and transmission, phase inversion on reflection from a denser string or rigid wall, and relative speeds/amplitudes of reflected and transmitted pulses via v = √(T/μ).","sort":445,"aliases":[]},{"id":447,"code":"12.01.05","level":3,"parent_id":442,"name":"Sound intensity level (decibel scale)","description":"Decibel calculations with β = 10 log₁₀(I/I₀), I₀ = 10⁻¹² W/m²: converting intensity to dB and back, combining equal or unequal sources, and the +3 dB (doubling) and +10 dB (10×) rules.","sort":446,"aliases":[]},{"id":448,"code":"12.01.06","level":3,"parent_id":442,"name":"Wave speed on a rope with position-dependent tension","description":"Wave speed on a hanging rope/chain where tension comes from the weight below: T(y) = μgy, v(y) = √(gy), and questions asking for the time for a pulse to climb the rope (t = 2√(L/g)) or how v varies with position.","sort":447,"aliases":[]},{"id":449,"code":"12.01.07","level":3,"parent_id":442,"name":"Wave speed on a rotating string loop","description":"A string or loop rotating about an axis, where centripetal-force balance sets the tension (e.g. T = μω²r² for a spinning loop), and the transverse wave speed is v = √(T/μ); questions ask for wave speed or lap time in terms of angular speed.","sort":448,"aliases":[]},{"id":450,"code":"12.01.08","level":3,"parent_id":442,"name":"Wave speed on a wire with variable linear density","description":"Composite wires made of segments with different linear densities μ under the same tension: computing v = √(T/μ) in each segment, total travel time Σ Lᵢ/√(T/μᵢ), and comparing speeds across junctions.","sort":449,"aliases":[]},{"id":451,"code":"12.01.09","level":3,"parent_id":442,"name":"Amplitude falloff of a spherical wave","description":"Point-source/spherical-wave geometry: amplitude A ∝ 1/r and intensity I ∝ 1/r², so ratios of amplitudes or intensities at two distances from the source; contrast with plane or cylindrical waves.","sort":450,"aliases":[]},{"id":452,"code":"12.01.10","level":3,"parent_id":442,"name":"Speed of sound in humid air","description":"Comparing sound speed in humid vs dry air at the same T and P: water vapour lowers the mean molar mass M, so density drops and v = √(γP/ρ) = √(γRT/M) increases; questions ask which air carries sound faster and why.","sort":451,"aliases":[]},{"id":453,"code":"12.01.11","level":3,"parent_id":442,"name":"Sound-wave pressure amplitude, intensity and power","description":"Relating sound pressure amplitude to displacement amplitude and intensity: Δp_max = ρvωs_max, I = Δp²_max/(2ρv) = 2π²ρvf²s²_max, and questions converting between pressure amplitude, intensity, and total radiated power of a source.","sort":452,"aliases":[]},{"id":454,"code":"12.01.12","level":3,"parent_id":442,"name":"Frequency invariance across a wave boundary","description":"When a wave crosses into a medium with different wave speed, the frequency (set by the source) is unchanged while v and λ change (λ₂/λ₁ = v₂/v₁); questions test which of f, v, λ is conserved at a boundary.","sort":453,"aliases":[]},{"id":455,"code":"12.01.13","level":3,"parent_id":442,"name":"A single wave-generating event heard/detected via two media of different propagation speed arrives as two temporally separated signals (e.g. an underwater explosion heard first through water then again through air): the transit-time difference dt=d/v_slow-d/v_fast, not a Doppler effect, causes the same event to be perceived twice, and only an observer who samples both media in turn (e.g. submerged, then surfaced) can detect both arrivals","description":"One event (e.g. an underwater explosion) detected twice because sound reaches the observer through two media: Δt = d(1/v_slow − 1/v_fast), e.g. arrival first through water then through air; questions stress this is a transit-speed effect, not Doppler, and that both arrivals are sensed only by an observer who samples both media in turn.","sort":454,"aliases":[]},{"id":456,"code":"12.01.14","level":3,"parent_id":442,"name":"Free fall combined with the finite travel time of a returning sound signal (total elapsed time = fall time + distance/v_sound): successive-approximation estimate versus the exact quadratic-in-time solution","description":"Dropping an object into a well/shaft and hearing the impact: total time t = √(2h/g) + h/v_sound, solved exactly as a quadratic in h or by successive approximation; questions contrast the naive free-fall-only estimate with the corrected two-stage time.","sort":455,"aliases":[]},{"id":457,"code":"12.01.15","level":3,"parent_id":442,"name":"Sound speed from bulk modulus and density","description":"Computing v = √(B/ρ) from bulk modulus and density for solids/liquids, and for gases v = √(γP/ρ) = √(γRT/M) (Newton–Laplace correction); questions give B, ρ, P or T and ask for sound speed or its dependence on temperature and pressure.","sort":456,"aliases":[]},{"id":458,"code":"12.01.16","level":3,"parent_id":442,"name":"2D event localization by lateration","description":"Locating an event (explosion, lightning strike, epicentre) from arrival-time data at several detectors: time differences give distance differences (lateration, hyperbola intersection) or, with known origin time, circles whose intersection fixes the 2D position.","sort":457,"aliases":[]},{"id":459,"code":"12.01.17","level":3,"parent_id":442,"name":"Longitudinal (compression) wave speed along a stretched coiled spring (Slinky), derived by modeling it as segments-in-series with a scaled local force constant and taking the discrete-to-continuum limit (v=L*sqrt(k/m) for a spring of force constant k, mass m, stretched length L); shown to equal the transverse wave speed sqrt(T/mu) once the natural length is negligible compared to the stretched length","description":"Longitudinal pulse speed on a stretched Slinky modelled as segments in series: v = L√(k/m) for a spring of stiffness k, mass m, stretched length L, derived in the discrete-to-continuum limit and shown to equal the transverse speed √(T/μ) when natural length is negligible.","sort":458,"aliases":[]},{"id":460,"code":"12.01.18","level":3,"parent_id":442,"name":"Parameters from a harmonic travelling wave function","description":"Given y(x,t) = A sin(kx − ωt + φ) (or cosine), extract amplitude, wave number, wavelength, angular frequency, period, frequency, phase constant, speed and direction of travel; also find particle velocity ∂y/∂t, particle acceleration, and phase differences between two points or times.","sort":459,"aliases":[]},{"id":461,"code":"12.01.19","level":3,"parent_id":442,"name":"Transverse and longitudinal waves","description":"Classifying waves as transverse (particle motion ⊥ propagation: string, EM waves) or longitudinal (parallel: sound, compressions–rarefactions on a spring); questions ask to identify wave type from the disturbance description or from whether the medium can support shear.","sort":460,"aliases":[]},{"id":462,"code":"12.01.20","level":3,"parent_id":442,"name":"Transverse particle velocity and travelling-wave slope","description":"For a travelling wave y = A sin(kx − ωt), questions ask for transverse particle velocity ∂y/∂t = −Aω cos(kx−ωt) or acceleration at a given (x, t), the string slope ∂y/∂x = Ak cos(kx−ωt), and the relation v_particle = −v_wave·(slope). Typical items: compare particle speed with wave speed, find when particle velocity is maximum/zero, or determine direction of particle motion from the slope sign.","sort":461,"aliases":[]},{"id":463,"code":"12.01.21","level":3,"parent_id":442,"name":"Energy density and power of a travelling wave","description":"Questions on energy carried by a wave on a string or in a medium: average energy density u = ½ρω²A² (equal kinetic and potential parts), power transmitted P = ½μω²A²v across a point, and intensity I = 2π²ρvf²A² for sound. Typical items: compute power delivered by a vibrating source to a string, find how power/intensity changes when amplitude or frequency is doubled, or energy per cycle in a segment.","sort":462,"aliases":[]},{"id":464,"code":"12.02","level":2,"parent_id":441,"name":"Standing waves","description":"Parent node for stationary (standing) waves: questions use the standing-wave form y = 2A sin kx cos ωt, locate nodes and antinodes spaced λ/2 apart, and test that energy is not transported while particles between nodes vibrate in phase. Conceptual items contrast standing vs progressive waves (no net energy flow, fixed amplitude pattern, all particles same frequency).","sort":463,"aliases":["stationary waves","organ pipes (open and closed)","resonance tube and end correction","harmonics and overtones","normal modes of a string"]},{"id":465,"code":"12.02.01","level":3,"parent_id":464,"name":"Standing waves in strings and pipes","description":"Mid-level node for questions that combine strings and pipes or use general standing-wave conditions: counting nodes/antinodes for a given harmonic, matching fundamental frequencies of a string and a pipe, or deciding which harmonics are present from boundary conditions (fixed end = node, open end = antinode).","sort":464,"aliases":[]},{"id":466,"code":"12.02.02","level":3,"parent_id":464,"name":"Formation of standing waves by superposition of two identical waves","description":"Questions deriving the standing wave from superposition of two identical waves travelling oppositely, e.g. y = A sin(kx−ωt) + A sin(kx+ωt) = 2A sin kx cos ωt. Typical items: find the resultant equation from two given waves, amplitude at a specified x, node/antinode positions, or the phase difference between two points of the standing wave.","sort":465,"aliases":[]},{"id":467,"code":"12.02.03","level":3,"parent_id":464,"name":"Standing waves on a stretched string: normal modes and harmonics","description":"Normal modes of a string fixed at both ends: f_n = (n/2L)√(T/μ), all integer harmonics present, with p loops meaning nth harmonic. Typical items: find tension, length or linear density from a given harmonic frequency, predict frequency shift when T or L is changed, count nodes/antinodes, or compare two wires (sonometer/Melde-type setups).","sort":466,"aliases":[]},{"id":468,"code":"12.02.04","level":3,"parent_id":464,"name":"Organ pipes: open and closed pipe harmonics and end correction","description":"Organ pipe harmonics: open pipe f_n = nv/2L (all harmonics), closed pipe f_n = (2n−1)v/4L (odd harmonics only), with end correction e ≈ 0.6r so effective length is L + e. Typical items: fundamental and overtone frequencies of open/closed pipes, ratio questions (e.g. second overtone of closed vs open pipe), and effect of end correction on measured frequency.","sort":467,"aliases":[]},{"id":469,"code":"12.02.05","level":3,"parent_id":464,"name":"Resonance tube and sonometer experiments","description":"Lab-based questions: resonance tube gives first and second resonance lengths l₁, l₂ with l₂ − l₁ = λ/2, speed of sound v = 2f(l₂ − l₁) and end correction e = (l₂ − 3l₁)/2; sonometer relates a stretched wire's resonance length to a tuning-fork frequency via f ∝ √T/L. Typical items: compute v, unknown fork frequency, or end correction from resonance data.","sort":468,"aliases":[]},{"id":470,"code":"12.03","level":2,"parent_id":441,"name":"Sound waves","description":"Parent node for longitudinal sound waves: speed of sound v = √(γP/ρ) = √(γRT/M) (Newton–Laplace correction), dependence on temperature (v ∝ √T, ≈ 0.61 m/s per °C), humidity and pressure, plus identifying displacement/pressure waveforms of a longitudinal wave. Typical items: compare sound speeds in gases/solids/liquids, effect of temperature rise, and conceptual questions on why sound needs a medium.","sort":469,"aliases":["sound intensity and decibel level","speed of sound in gases, liquids and solids (Newton-Laplace)","dependence of speed of sound on temperature, pressure and humidity"]},{"id":471,"code":"12.03.01","level":3,"parent_id":470,"name":"Decibel sound level and intensity ratios","description":"Sound intensity level β = 10 log₁₀(I/I₀) with I₀ = 10⁻¹² W/m². Typical items: dB change when intensity changes by a factor (e.g. +10 dB for ×10, +3 dB for ×2), ratio of intensities from a given dB difference, combined level of two identical sources, or finding distance where level drops by a set amount using I ∝ 1/r².","sort":470,"aliases":[]},{"id":472,"code":"12.03.02","level":3,"parent_id":470,"name":"Displacement and pressure amplitude in a sound wave","description":"Relation between the displacement wave and pressure wave of sound: pressure amplitude Δp_max = Bks_max = ρvωs_max, intensity I = Δp²_max/(2ρv), and the 90° phase shift (pressure node at displacement antinode). Typical items: convert between displacement and pressure amplitudes, compute intensity from pressure amplitude, or identify maxima/minima positions of each wave.","sort":471,"aliases":[]},{"id":473,"code":"12.03.03","level":3,"parent_id":470,"name":"Characteristics of sound: loudness, pitch and quality","description":"Conceptual mapping of sound characteristics: loudness ↔ intensity/amplitude (measured in dB, logarithmic), pitch ↔ frequency, quality/timbre ↔ waveform and harmonic content. Typical items: identify which characteristic changes in a described situation, or explain why two instruments at the same pitch and loudness sound different.","sort":472,"aliases":[]},{"id":474,"code":"12.03.04","level":3,"parent_id":470,"name":"Audible, ultrasonic and infrasonic sound ranges","description":"Frequency ranges: audible 20 Hz–20 kHz, infrasonic below 20 Hz (earthquakes, some animals), ultrasonic above 20 kHz (bats, dolphins, SONAR, medical imaging, cleaning). Typical items: classify a given frequency or wavelength (using v = fλ in air) into a range, or pick the correct application of ultrasonics.","sort":473,"aliases":[]},{"id":475,"code":"12.03.05","level":3,"parent_id":470,"name":"Echo and reverberation of sound","description":"Echo problems: minimum distance for a distinct echo from the 0.1 s persistence of hearing (≈17.2 m in air), counting echoes with multiple reflectors, reverberation in halls, and echo combined with Doppler (moving source or moving cliff/wall changing the reflected frequency). Typical items: find distance of a cliff from echo time, or beat frequency between emitted and reflected sound.","sort":474,"aliases":[]},{"id":476,"code":"12.04","level":2,"parent_id":441,"name":"Superposition and beats","description":"Parent node for superposition of waves: resultant displacement is the algebraic/vector sum y = y₁ + y₂, applied to interference of two waves and to beats. Typical items: resultant amplitude for two waves with a phase difference, conditions for constructive/destructive interference, and beat-frequency calculations.","sort":475,"aliases":["principle of superposition","beats"]},{"id":477,"code":"12.04.01","level":3,"parent_id":476,"name":"Principle of superposition","description":"Principle of superposition questions: resultant amplitude A = √(A₁² + A₂² + 2A₁A₂cosφ) for phase difference φ, with φ = 2π·(path difference)/λ; constructive when φ = 2nπ, destructive when φ = (2n+1)π. Typical items: find resultant intensity/amplitude at a point given path difference, ratio I_max/I_min for two coherent sources, or number of maxima/minima along a line.","sort":476,"aliases":[]},{"id":478,"code":"12.04.02","level":3,"parent_id":476,"name":"Quincke tube interference","description":"Quincke's tube: moving the sliding tube by distance d changes the path difference by 2d, giving intensity I = 4I₀cos²(πd/λ) between maxima and silence. Typical items: distance the tube must be moved to go from a maximum to the next minimum (λ/4) or between successive silences (λ/2), and I_max/I_min from amplitudes of the two paths.","sort":477,"aliases":[]},{"id":479,"code":"12.04.03","level":3,"parent_id":476,"name":"Beats","description":"Beats: beat frequency = |f₁ − f₂| from two sources of nearby frequencies, heard as periodic waxing and waning. Typical items: unknown tuning-fork frequency from beat count (loading with wax lowers frequency, filing raises it), beats produced by two wires/forks under changed tension, and problems where two unknowns are found from two beat observations.","sort":478,"aliases":[]},{"id":480,"code":"12.04.04","level":3,"parent_id":476,"name":"Pulse duration and frequency bandwidth","description":"Relation between a pulse's time duration Δt and its spread of frequencies (bandwidth Δν ≈ 1/Δt): shorter pulses contain a wider range of frequencies. Typical items are conceptual/qualitative — e.g. which pulse is more 'pure' in frequency, or estimating the frequency spread of a short signal — rather than standard numerical wave problems.","sort":479,"aliases":[]},{"id":481,"code":"12.05","level":2,"parent_id":441,"name":"Doppler effect (sound and light)","description":"Parent node for the Doppler effect: apparent frequency (or wavelength) changes due to relative motion of source, observer or medium, for both sound and light. Typical items: sign-convention setup of the general formula, identifying whether frequency increases or decreases in a described approach/recede scenario.","sort":480,"aliases":[]},{"id":482,"code":"12.05.01","level":3,"parent_id":481,"name":"Doppler effect","description":"Standard sound Doppler problems using f' = f(v ± v_L)/(v ∓ v_S) with source and/or listener moving along the line joining them, including wind/medium velocity added to v. Typical items: frequency heard by an observer as a car/train approaches and then recedes, both moving simultaneously, source crossing the listener, and reflected sound from a moving wall (beats between direct and reflected waves).","sort":481,"aliases":[]},{"id":483,"code":"12.05.02","level":3,"parent_id":481,"name":"Doppler effect for a source in circular motion","description":"Doppler effect for a source moving in a circle (whistle on a rotating wheel/turntable) with the listener on the axis or in the plane: the radial velocity component v_s cosθ varies, so f' = fv/(v − v_s cosθ) oscillates between f·v/(v − v_s) and f·v/(v + v_s). Typical items: maximum and minimum apparent frequencies, frequency at a given instant/angle, or sketching f' versus time.","sort":482,"aliases":[]},{"id":484,"code":"12.05.03","level":3,"parent_id":481,"name":"Doppler effect in a uniformly moving medium","description":"Doppler effect when the medium itself moves uniformly (wind): the medium velocity adds to wave speed but only the component along the source–observer line matters, and a source drifting with the wind still shows a shift relative to a ground observer. Typical items: frequency heard with wind blowing from source to listener, whether wind alone (source and observer at rest in the medium frame) produces a shift, and combined source/observer/wind calculations.","sort":483,"aliases":[]},{"id":485,"code":"12.05.04","level":3,"parent_id":481,"name":"Spectroscopic-binary Doppler shift and unseen companion","description":"Astronomical Doppler shift of light: radial velocity v_r = c·Δλ/λ, with redshift for recession and blueshift for approach; in a spectroscopic binary the lines periodically split/shift as the stars orbit. Typical items: find orbital speed from the observed wavelength shift, detect an unseen companion from periodic line displacement, or compute the shift for a star moving at a given fraction of c.","sort":484,"aliases":[]},{"id":486,"code":"12.05.05","level":3,"parent_id":481,"name":"Mach cone angle and supersonic speed","description":"Shock waves from a supersonic source: Mach number M = v_source/v_sound and Mach cone half-angle sinθ = v/v_s = 1/M. Typical items: find the cone angle or Mach number for a jet at a given speed, the speed needed for a specified angle, or the time after the jet passes overhead that the sonic boom is heard at a ground point.","sort":485,"aliases":[]},{"id":487,"code":"12.05.06","level":3,"parent_id":481,"name":"Doppler shift of a source passing an off-line observer","description":"Source moving in a straight line past a stationary observer at finite closest-approach distance, so the line-of-sight velocity component v_s cosθ matters: f' = f v/(v + v_s cosθ) for sound. Questions give f vs t or f vs θ curves, ask the frequency at a given angle, the instant when f' = f (closest approach, cosθ = 0), or the high-to-low frequency transition as the source passes.","sort":486,"aliases":[]},{"id":488,"code":"12.05.07","level":3,"parent_id":481,"name":"Radar Doppler velocimetry","description":"Double Doppler shift when waves emitted by a stationary source reflect off a moving target (radar gun, moving car, reflected sound/EM wave): Δf/f₀ ≈ 2v/c for light, exact beat frequency between transmitted and reflected signals for sound. Questions ask target speed from the measured beat/shift frequency, or the shift for a given speed.","sort":487,"aliases":[]},{"id":489,"code":"12.05.08","level":3,"parent_id":481,"name":"Relativistic Doppler shift of light at arbitrary angle","description":"Relativistic Doppler formula for light at viewing angle θ: f' = f₀√(1−β²)/(1−β cosθ), including the purely transverse Doppler redshift at θ = 90° and longitudinal redshift/blueshift for receding/approaching sources. Questions involve stellar redshifts, recession speeds from wavelength ratios, or the transverse effect distinguishing relativistic from classical predictions.","sort":488,"aliases":[]},{"id":490,"code":"12.06","level":2,"parent_id":441,"name":"Reflection and transmission of waves at boundaries","description":"Parent node for what happens to a wave incident on the interface between two media: splitting into reflected and transmitted parts, phase changes on reflection, and partition of energy. Tag here only for general boundary questions not specifically about refraction/TIR (d129) or string/media reflection-transmission coefficients (d130).","sort":489,"aliases":["wave behaviour at media boundaries","Reflection, refraction and transmission of waves"]},{"id":491,"code":"12.06.01","level":3,"parent_id":490,"name":"Mechanical-wave refraction and total internal reflection","description":"Mechanical waves (sound, water waves, waves on strings) changing direction at a boundary because wave speed changes, with Snell's law sinθ₁/sinθ₂ = v₁/v₂, and total internal reflection when passing from slower to faster medium beyond the critical angle sinθ_c = v₁/v₂. Questions compute critical angles, refraction of sound in temperature/wind gradients, or shallow-to-deep water wave bending.","sort":490,"aliases":[]},{"id":492,"code":"12.06.02","level":3,"parent_id":490,"name":"Wave reflection and transmission at a media boundary","description":"Reflection/transmission of a pulse or sinusoidal wave at a junction of two strings/media: amplitude coefficients r = (v₁−v₂)/(v₁+v₂) = (√μ₁−√μ₂)/(√μ₁+√μ₂), t = 2v₁/(v₁+v₂), phase inversion at a fixed end or denser (slower) medium, no inversion at a free end or rarer medium. Questions ask fraction of amplitude/power reflected and transmitted, phase of the reflected pulse, or behavior at fixed/free ends.","sort":491,"aliases":[]},{"id":493,"code":"13","level":1,"parent_id":null,"name":"Electrostatics","description":"JEE chapter 13: Electrostatics. Concepts: Electric charge: properties, quantization and conservation; Coulomb's law; Electric field; Electric dipole; Gauss's law; Method of images and induced charges; Electric potential; Electrostatic potential energy; Electrostatics of conductors.","sort":492,"aliases":["Electric Charges and Fields","Electrostatic Potential and Capacitance"]},{"id":494,"code":"13.01","level":2,"parent_id":493,"name":"Electric charge: properties, quantization and conservation","description":"Basic electrostatics of charge itself: quantization q=ne, conservation, additivity, and charging processes. Parent tag for conceptual questions about the nature of charge rather than fields or forces.","sort":493,"aliases":["Electric charge and conservation","quantization of charge","conservation of charge","basic properties of electric charge","additivity of charge","charging by friction, conduction and induction","electrostatic induction","conductors and insulators"]},{"id":495,"code":"13.01.01","level":3,"parent_id":494,"name":"Electric charge and conservation","description":"Conservation of charge: total charge of an isolated system stays constant — pair production/annihilation (e⁺e⁻→2γ), nuclear reactions, rubbing two bodies so they acquire equal and opposite charges, or charge sharing between conductors where the sum before equals the sum after.","sort":494,"aliases":[]},{"id":496,"code":"13.01.02","level":3,"parent_id":494,"name":"Quantization of charge (q = ne)","description":"Quantization q=ne with e=1.6×10⁻¹⁹ C: find the number of electrons gained/lost for a body with charge Q (n=Q/e), check whether a stated charge is physically possible, or compute charge transferred when N electrons are removed. Millikan oil-drop style reasoning that all observed charges are integral multiples of e.","sort":495,"aliases":[]},{"id":497,"code":"13.01.03","level":3,"parent_id":494,"name":"Additivity, scalar nature and invariance of electric charge","description":"Charge is a scalar that adds algebraically (additivity), is invariant of the body's speed or frame, and does not depend on state of motion. Conceptual true/false style questions: does charge change with velocity, how do charges add when bodies are combined, why charge is not a vector.","sort":496,"aliases":[]},{"id":498,"code":"13.01.04","level":3,"parent_id":494,"name":"Methods of charging: friction, conduction and induction","description":"Charging methods: friction (triboelectric — both bodies get equal and opposite charges, e.g. glass/silk, ebonite/flannel), conduction (contact gives the same sign), and induction (charged rod brought near induces opposite charge on the near side; permanent charging with earthing). Electroscope and which-body-gets-which-sign questions.","sort":497,"aliases":[]},{"id":499,"code":"13.02","level":2,"parent_id":493,"name":"Coulomb's law","description":"Coulomb's law F=kq1q2/r² with k=1/4πε0=9×10⁹ N·m²/C², force acting along the line joining point charges. Parent tag for any question computing electrostatic force between charges, including superposition, equilibrium, and medium effects.","sort":498,"aliases":[]},{"id":500,"code":"13.02.01","level":3,"parent_id":499,"name":"Electrostatic hoop stress in a charged ring","description":"Tension (hoop stress) developed in a uniformly charged ring of radius R and total charge Q as Coulomb repulsion from the rest of the ring pulls each element outward: T=Q²/(16π²ε0R²)=kQ²/(4πR²). Questions find T, or the maximum charge the ring can hold before breaking given its tensile strength.","sort":499,"aliases":[]},{"id":501,"code":"13.02.02","level":3,"parent_id":499,"name":"Coulomb's law and superposition of forces","description":"Direct application of Coulomb's law with superposition: net force on a charge as the vector sum of pairwise forces, forces among charges at triangle/square vertices, force between charged spheres (point approximation when r≫size), and ratio problems like the force after two spheres touch and are reseparated.","sort":500,"aliases":[]},{"id":502,"code":"13.02.03","level":3,"parent_id":499,"name":"Equilibrium of point-charge systems and null points","description":"Equilibrium and null points in point-charge systems: where the net force on a third/test charge is zero (between two like charges, outside for unlike), the charge q that keeps a collinear three-charge system in equilibrium (q=−Q1Q2/(√Q1+√Q2)² type), and stability of such equilibria.","sort":501,"aliases":[]},{"id":503,"code":"13.02.04","level":3,"parent_id":499,"name":"Dependence of Coulomb force on the dielectric medium","description":"Coulomb force in a dielectric medium: F=F0/K with K=εr (k_medium=k_air/K); force between charges immersed in a liquid of dielectric constant K, or change in force when a dielectric slab is inserted between them. Questions compare forces in air vs medium or find K from the force ratio.","sort":502,"aliases":[]},{"id":504,"code":"13.02.05","level":3,"parent_id":499,"name":"Electrostatic vs gravitational force comparison","description":"Comparing electrostatic and gravitational forces between the same pair: Fe/Fg=Gm1m2/(kq1q2), e.g. ≈2.3×10³⁹ for an electron–proton pair and ≈10³⁶ for two protons. Questions compute the ratio or use it to argue gravity is negligible in atomic physics.","sort":503,"aliases":[]},{"id":505,"code":"13.02.06","level":3,"parent_id":499,"name":"Charge sharing between identical conductors and resulting force change","description":"Two identical conducting spheres/balls carrying charges q1 and q2 are touched together (or joined by a wire) and re-separated, each taking (q1+q2)/2; questions ask the new Coulomb force at separation r — when it vanishes, reverses to attraction, or changes by a given factor. Also covers charge sharing between non-identical conductors in proportion to radius/capacitance.","sort":504,"aliases":[]},{"id":506,"code":"13.02.07","level":3,"parent_id":499,"name":"Potential energy of a system of point charges","description":"U = Σ k q_i q_j / r_ij over charge pairs; archetypes are work done to assemble a configuration (e.g., charges at the corners of a triangle or square), work to bring a charge from infinity to a point, and energy change when charges are rearranged or released.","sort":505,"aliases":[]},{"id":507,"code":"13.03","level":2,"parent_id":493,"name":"Electric field","description":"Parent node for E = F/q and superposition of point-charge fields E = kq/r²; questions ask the resultant field vector at a point from several discrete charges, or the force on a test charge placed there. Use for generic point-charge field problems not covered by the specialized subnodes.","sort":506,"aliases":["charged particle motion in an electric field","field due to continuous charge distribution","linear, surface and volume charge density","charged ring on axis","charged arc","finite line charge","uniformly charged disc","superposition of electric fields","electric field due to point charges","electric field lines"]},{"id":508,"code":"13.03.01","level":3,"parent_id":507,"name":"Charged-particle motion in a uniform electric field","description":"Electron/proton/ion entering a uniform field between parallel plates: acceleration a = qE/m, parabolic deflection y = qEL²/(2mv₀ₓ²), time inside the field, whether it strikes a plate, and exit velocity/direction; also a charge released from rest undergoing uniform acceleration qE/m.","sort":507,"aliases":[]},{"id":509,"code":"13.03.02","level":3,"parent_id":507,"name":"Electric field and field lines","description":"Qualitative field-line questions: properties (lines never cross, run + to −, tangent gives direction, density ∝ |E|), picking the correct field-line diagram for a charge configuration, and inferring sign/magnitude of charges or relative field strength from line patterns.","sort":508,"aliases":[]},{"id":510,"code":"13.03.03","level":3,"parent_id":507,"name":"cos(phi)/cos(theta) surface-charge trick: modelling a sinusoidally-varying surface charge on a sphere or infinite cylinder as two overlapping, oppositely and uniformly volume-charged bodies displaced by an infinitesimal vector, yielding a uniform interior field","description":"The trick where σ = σ₀cosθ on a sphere (or σ₀cosφ on a cylinder) is modelled as two overlapping uniformly volume-charged bodies displaced by an infinitesimal d, giving a uniform interior field E = ρd/3ε₀ along the shift axis; questions ask the field at the centre or anywhere inside such a body.","sort":509,"aliases":[]},{"id":511,"code":"13.03.04","level":3,"parent_id":507,"name":"Charged-particle motion in a line charge's field","description":"Motion of a charged particle in an infinite line charge's field E = λ/(2πε₀r): released from rest and finding speed at distance r via energy conservation with V ∝ −ln r, radial infall/escape, or closest-approach problems where acceleration varies as 1/r.","sort":510,"aliases":[]},{"id":512,"code":"13.03.05","level":3,"parent_id":507,"name":"Field of continuous charge distributions by integration (arc, finite line, ring, disc)","description":"Setting up dE = k·dq/r² integrals for continuous distributions: field at the centre of an arc (2kλ sin(θ/2)/R), finite line charge E = kλ/d(sinθ₁+sinθ₂), ring on its axis, disc E = (σ/2ε₀)(1 − x/√(x²+R²)), semicircular ring; includes symmetry-based cancellation of components.","sort":511,"aliases":[]},{"id":513,"code":"13.04","level":2,"parent_id":493,"name":"Electric dipole","description":"Parent node for electric dipoles: dipole moment p = q(2a) directed from −q to +q and general dipole behaviour. Route here only if a dipole question fits none of the specific subnodes (field, non-uniform-field force, torque/oscillation, rigid assemblies).","sort":512,"aliases":["dipole field","dipole in a uniform field"]},{"id":514,"code":"13.04.01","level":3,"parent_id":513,"name":"Electric dipole and dipole field","description":"Field and potential of a dipole: axial E = 2kp/r³, equatorial E = kp/r³, resultant at a general point, V = kpcosθ/r², and identifying dipole moment from charge arrangements; includes axial-vs-equatorial comparisons and short-dipole (r ≫ a) limits.","sort":513,"aliases":[]},{"id":515,"code":"13.04.02","level":3,"parent_id":513,"name":"Force on an electric dipole in a non-uniform field","description":"Net force on a dipole in a non-uniform field, F = (p·∇)E, usually computed as q(E₊ − E₋) on the two charges; archetypes: dipole on the axis of a point charge or charged ring, force between two dipoles, and attraction of a dipole toward stronger-field regions.","sort":514,"aliases":[]},{"id":516,"code":"13.04.03","level":3,"parent_id":513,"name":"Rigid charge assembly equilibrium in a uniform field","description":"Statics of rigid charge assemblies (two charges on a rod, charged dumbbell, suspended charged bob, square of charges) in a uniform field: net force Q_total·E, torque about the centre of mass, tension/compression in rods or strings, hinge/pivot reactions, and equilibrium orientations.","sort":515,"aliases":[]},{"id":517,"code":"13.04.04","level":3,"parent_id":513,"name":"Torque, potential energy and oscillations of a dipole in a uniform field","description":"Torque τ = pE sinθ, potential energy U = −pE cosθ, work required to rotate a dipole between two angles, stable (θ=0) vs unstable (θ=π) equilibrium, and small-angle SHM of a dipole or compass needle with period T = 2π√(I/pE).","sort":516,"aliases":[]},{"id":518,"code":"13.05","level":2,"parent_id":493,"name":"Gauss's law","description":"Parent node for Gauss's law, Φ_closed = q_enc/ε₀, and symmetry-based reasoning about it. Route to subnodes for flux computations or the standard line/sheet/sphere field results.","sort":517,"aliases":["electric flux","electric flux and Gauss's law"]},{"id":519,"code":"13.05.01","level":3,"parent_id":518,"name":"Gauss's law and applications","description":"General applications of Gauss's law: choosing Gaussian surfaces, deriving E for symmetric distributions, flux through a closed surface being independent of its shape or of where the enclosed charge sits, flux through a cube with a charge at centre/corner/edge, and how flux changes when charges are added, moved, or the surface is deformed.","sort":518,"aliases":[]},{"id":520,"code":"13.05.02","level":3,"parent_id":518,"name":"On-axis field of a uniformly charged ring","description":"On-axis field of a uniformly charged ring, E = kQx/(x²+R²)^(3/2): zero at the centre, maximum at x = R/√2; archetypes include locating the field maximum and a charge oscillating along the axis through the ring's centre.","sort":519,"aliases":[]},{"id":521,"code":"13.05.03","level":3,"parent_id":518,"name":"Field inside an off-axis cavity in a charged body","description":"Cavity problems by superposition: a uniformly charged sphere/cylinder with an off-centre spherical/cylindrical cavity equals the full body plus a negative charge density filling the cavity, giving a uniform field inside the cavity E = ρd/3ε₀ directed from body centre to cavity centre; questions ask magnitude and direction at points inside the cavity.","sort":520,"aliases":[]},{"id":522,"code":"13.05.04","level":3,"parent_id":518,"name":"Line of charge","description":"Fields of line charges: infinite line E = λ/(2πε₀r) via a coaxial cylindrical Gaussian surface, finite line at perpendicular distance d, semi-infinite line, and field at the centre/end of a charged rod; includes the 1/r falloff and superposition of several parallel lines.","sort":521,"aliases":[]},{"id":523,"code":"13.05.05","level":3,"parent_id":518,"name":"Charged sheet","description":"Infinite charged sheets and slabs: E = σ/2ε₀ for a single sheet, σ/ε₀ just outside a conductor, field between and outside two parallel sheets (same or opposite charge), and combined configurations of sheets with point charges or conducting plates.","sort":522,"aliases":[]},{"id":524,"code":"13.05.06","level":3,"parent_id":518,"name":"Charged sphere","description":"Spherically symmetric charge: conducting sphere (charge on surface, E = 0 inside), uniformly charged solid sphere (E = ρr/3ε₀ inside, kQ/r² outside), thin shell, and E-vs-r graphs; also non-uniform densities like ρ ∝ r and superposition of nested shells.","sort":523,"aliases":[]},{"id":525,"code":"13.05.07","level":3,"parent_id":518,"name":"Electric flux","description":"Direct flux computations: Φ = EA cosθ for a flat surface in a uniform field, flux through cube faces from an enclosed charge (q/6ε₀ per face for a central charge), flux through a disc or hemisphere from a point charge via solid angle q/2ε₀(1 − cosθ), and closed-surface flux from q_enc/ε₀.","sort":524,"aliases":[]},{"id":526,"code":"13.06","level":2,"parent_id":493,"name":"Method of images and induced charges","description":"Parent node for the method of images and induced charges: replacing an induced conductor by fictitious image charges so that boundary conditions (constant or zero potential) are satisfied. Route to subnodes for the grounded-plane and grounded-sphere cases or conductor properties.","sort":525,"aliases":["image charge method","grounded conductors and induced charge"]},{"id":527,"code":"13.06.01","level":3,"parent_id":526,"name":"Grounded sphere: image charge and induced charge","description":"Point charge q at distance d from a grounded conducting sphere of radius R: image charge q′ = −qR/d located at R²/d from the centre, net induced charge equal to q′, force on q, and variants with an insulated neutral sphere (additional central image) or a sphere held at fixed potential.","sort":526,"aliases":[]},{"id":528,"code":"13.06.02","level":3,"parent_id":526,"name":"Point charge near a grounded conducting plane","description":"Point charge q a distance d from a grounded infinite conducting plane: image −q at the mirror position, attractive force kq²/(2d)², induced surface charge density σ(r) = −qd/(2π(r²+d²)^(3/2)), total induced charge −q, and work/energy to move the charge away from the plane.","sort":527,"aliases":[]},{"id":529,"code":"13.06.03","level":3,"parent_id":526,"name":"Properties of conductors: surface charge distribution, cavity shielding and earthing","description":"Conductor electrostatics: E = 0 inside, charge resides on the surface with E = σ/ε₀ just outside, larger σ at sharper points, a charge-free cavity is shielded while a charge inside a cavity induces equal charge on cavity walls and the outer surface, effect of earthing, and electrostatic pressure σ²/2ε₀.","sort":528,"aliases":[]},{"id":530,"code":"13.07","level":2,"parent_id":493,"name":"Electric potential","description":"Questions on electric potential and potential difference, equipotential surfaces, potential due to point charges/dipoles, and relation E = -dV/dr.","sort":529,"aliases":["potential due to a point charge","potential due to a dipole","equipotential surfaces","relation between field and potential"]},{"id":531,"code":"13.07.01","level":3,"parent_id":530,"name":"Electric potential and potential difference","description":"Definition of electric potential, potential difference, work done in moving charge, V = W/q, potential due to a point charge V = kq/r, and potential difference in uniform fields.","sort":530,"aliases":[]},{"id":532,"code":"13.07.02","level":3,"parent_id":530,"name":"Equipotential surfaces","description":"Properties of equipotential surfaces, relation to electric field lines, work done along equipotential, and identifying equipotential surfaces for point charges, dipoles, and uniform fields.","sort":531,"aliases":[]},{"id":533,"code":"13.07.03","level":3,"parent_id":530,"name":"Potential due to charges and a dipole","description":"Potential due to a dipole V = k p cosθ/r^2, potential due to multiple charges, axial/equatorial points, and potential of continuous charge distributions.","sort":532,"aliases":[]},{"id":534,"code":"13.07.04","level":3,"parent_id":530,"name":"Potential from field: E = -grad(phi)","description":"Finding potential from electric field using V = -∫ E·dr, E = -dV/dr, potential gradient, and relation between field and potential in uniform and radial fields.","sort":533,"aliases":[]},{"id":535,"code":"13.07.05","level":3,"parent_id":530,"name":"Potential due to point charges and systems of charges","description":"Superposition principle for potential due to point charges and systems of charges, V = Σ kq_i/r_i, potential at center of rings, corners of polygons, etc.","sort":534,"aliases":[]},{"id":536,"code":"13.07.06","level":3,"parent_id":530,"name":"Potential due to continuous charge distributions","description":"Questions asking for electric potential due to continuous charge distributions such as uniformly charged rods, rings, disks, or spheres using V = k∫dq/r; often involves on-axis, center, or general points with linear/surface/volume charge densities.","sort":535,"aliases":[]},{"id":537,"code":"13.07.07","level":3,"parent_id":530,"name":"Electric dipole in uniform and non-uniform electric fields","description":"Questions on the potential V = kp cosθ/r² due to a dipole, torque p×E, potential energy U = −p·E in uniform fields, and force/equilibrium behavior in non-uniform fields.","sort":536,"aliases":[]},{"id":538,"code":"13.08","level":2,"parent_id":493,"name":"Electrostatic potential energy","description":"General questions on electrostatic potential energy stored in charge configurations and capacitors, including work done to assemble charges or charge a capacitor.","sort":537,"aliases":["potential energy of a system of charges","self-energy of charge distributions","work done in assembling charges"]},{"id":539,"code":"13.08.01","level":3,"parent_id":538,"name":"Electrostatic potential energy of point charges","description":"Questions on potential energy of discrete point charges using U = (1/2)Σq_iV_i or pairwise kq_iq_j/r_ij; work to assemble or move point charges.","sort":538,"aliases":[]},{"id":540,"code":"13.08.02","level":3,"parent_id":538,"name":"Self-energy of a spherical charge distribution","description":"Questions on self-energy of spherical charge distributions, especially uniform sphere U = 3kQ²/(5R) and thin spherical shell U = kQ²/(2R); work to assemble the sphere.","sort":539,"aliases":[]},{"id":541,"code":"13.08.03","level":3,"parent_id":538,"name":"Potential energy of a dipole in an electric field","description":"Questions involve calculating U = -p·E = -pE cosθ, work done in rotating a dipole in a uniform electric field, and identifying stable/unstable equilibrium orientations.","sort":540,"aliases":[]},{"id":542,"code":"13.08.04","level":3,"parent_id":538,"name":"Work done in assembling charge distributions","description":"Questions ask for work required to bring charges from infinity to form a given configuration, using pairwise potential energy sums or integration for continuous distributions.","sort":541,"aliases":[]},{"id":543,"code":"13.09","level":2,"parent_id":493,"name":"Electrostatics of conductors","description":"General properties: conductors in electrostatic equilibrium have zero field inside, charge resides on surface, potential constant throughout, field just outside is σ/ε0 perpendicular to surface.","sort":542,"aliases":["properties of conductors","electrostatic pressure on a charged conductor","charge sharing between conductors"]},{"id":544,"code":"13.09.01","level":3,"parent_id":543,"name":"Electrostatic pressure on a charged conductor","description":"Questions compute outward pressure on a charged conductor surface using P = σ²/(2ε0) or energy density, often for spherical shells or soap bubbles.","sort":543,"aliases":[]},{"id":545,"code":"13.09.02","level":3,"parent_id":543,"name":"Sequential charge sharing between conducting spheres","description":"Problems where two conducting spheres of different radii are connected by a wire, charge redistributes until potentials equal, then disconnected; may involve repeated connections.","sort":544,"aliases":[]},{"id":546,"code":"13.09.03","level":3,"parent_id":543,"name":"Surface charge distribution and curvature of conductors","description":"Questions relate surface charge density to local curvature (σ ∝ 1/R for spheres, higher at sharp points), often comparing charge densities on different parts of a conductor.","sort":545,"aliases":[]},{"id":547,"code":"13.09.04","level":3,"parent_id":543,"name":"Electrostatic shielding and earthing","description":"Situations involving Faraday cages, grounding a conductor to fix its potential to zero, and shielding of electric fields inside cavities.","sort":546,"aliases":[]},{"id":548,"code":"13.09.05","level":3,"parent_id":543,"name":"Charge redistribution and common potential of connected conductors","description":"Problems where conductors of different capacitances and initial charges are connected, find final common potential V = (Q1+Q2)/(C1+C2) and final charges.","sort":547,"aliases":[]},{"id":549,"code":"13.09.06","level":3,"parent_id":543,"name":"Field inside a conductor and at its surface","description":"Questions test that E=0 inside a conductor in equilibrium, and just outside E = σ/ε0 perpendicular to surface, often with cavities or external fields.","sort":548,"aliases":[]},{"id":550,"code":"14","level":1,"parent_id":null,"name":"Capacitance & Dielectrics","description":"JEE chapter 14: Capacitance & Dielectrics. Concepts: Energy stored in capacitor; Capacitance; Dielectrics in capacitors; Combination of capacitors.","sort":549,"aliases":["Electrostatic Potential and Capacitance"]},{"id":551,"code":"14.01","level":2,"parent_id":550,"name":"Energy stored in capacitor","description":"Questions on energy stored in capacitors, work done in charging, energy density, and force between plates.","sort":550,"aliases":[]},{"id":552,"code":"14.01.01","level":3,"parent_id":551,"name":"Energy stored in a capacitor","description":"Using U = 1/2 CV^2 = Q^2/(2C) = 1/2 QV; energy stored in single and capacitor combinations, changes when connected/disconnected from a battery.","sort":551,"aliases":[]},{"id":553,"code":"14.01.02","level":3,"parent_id":551,"name":"Work done in charging a capacitor","description":"Work done by battery vs energy stored in charging a capacitor, energy loss in charging, and heat dissipated in connecting capacitors.","sort":552,"aliases":[]},{"id":554,"code":"14.01.03","level":3,"parent_id":551,"name":"Energy density of the electric field","description":"Energy density u = 1/2 ε0 E^2 in an electric field; energy stored in a charged capacitor from field, dielectric-filled capacitors.","sort":553,"aliases":[]},{"id":555,"code":"14.01.04","level":3,"parent_id":551,"name":"Force between capacitor plates at constant charge and constant potential","description":"Force between capacitor plates using F = Q^2/(2ε0 A) at constant charge or F = 1/2 ε0 A V^2/d^2 at constant potential; work done in changing plate separation.","sort":554,"aliases":[]},{"id":556,"code":"14.02","level":2,"parent_id":550,"name":"Capacitance","description":"General capacitor questions involving C = Q/V, stored charge and energy, geometric capacitance formulas, series/parallel combination, and the effect of dielectrics.","sort":555,"aliases":["parallel-plate capacitor","spherical and cylindrical capacitors","capacitance of an isolated conductor"]},{"id":557,"code":"14.02.01","level":3,"parent_id":556,"name":"Capacitance and definition of capacitance","description":"Questions asking for the definition C = Q/V, its units, interpretation from Q-V graphs, and basic capacitance of a conductor/configuration.","sort":556,"aliases":[]},{"id":558,"code":"14.02.02","level":3,"parent_id":556,"name":"Capacitance of an isolated spherical conductor","description":"Questions using C = 4πε₀R for an isolated spherical conductor, V = kQ/R, and related energy/charge problems for conducting spheres or shells.","sort":557,"aliases":[]},{"id":559,"code":"14.02.03","level":3,"parent_id":556,"name":"Parallel plate capacitor","description":"Questions on the parallel plate capacitor formula C = ε₀A/d, field E = σ/ε₀, potential difference V = Ed, and changes in A, d, or battery connection.","sort":558,"aliases":[]},{"id":560,"code":"14.02.04","level":3,"parent_id":556,"name":"Spherical capacitor","description":"Questions on concentric spherical capacitors using C = 4πε₀ab/(b−a), potential difference between shells, and related charge/energy calculations.","sort":559,"aliases":[]},{"id":561,"code":"14.02.05","level":3,"parent_id":556,"name":"Cylindrical capacitor","description":"Questions on coaxial cylindrical capacitors using C = 2πε₀L/ln(b/a) or capacitance per unit length, based on line-charge field integration.","sort":560,"aliases":[]},{"id":562,"code":"14.03","level":2,"parent_id":550,"name":"Dielectrics in capacitors","description":"General questions on inserting dielectrics into capacitors, including changes in capacitance, field, potential, charge, and stored energy with battery connected or disconnected.","sort":561,"aliases":["dielectrics","dielectric constant","capacitor with dielectric slab (partial and full)","polarisation and bound charge","force on a dielectric slab"]},{"id":563,"code":"14.03.01","level":3,"parent_id":562,"name":"Capacitors in series and parallel; dielectrics","description":"Questions combining dielectric-filled capacitors in series and parallel; compute equivalent capacitance and charge/voltage/energy distribution when different K values are present.","sort":562,"aliases":[]},{"id":564,"code":"14.03.02","level":3,"parent_id":562,"name":"Bound charge at a dielectric interface","description":"Questions on bound charge at dielectric interfaces using surface bound charge σ_b = P·n and volume bound charge −∇·P; relating polarization to induced charges and internal fields.","sort":563,"aliases":[]},{"id":565,"code":"14.03.03","level":3,"parent_id":562,"name":"Charge conservation at a floating capacitor junction","description":"Questions where a floating/isolated junction in a capacitor network conserves charge before and after dielectric insertion or switching; asks redistribution of charge and potential.","sort":564,"aliases":[]},{"id":566,"code":"14.03.04","level":3,"parent_id":562,"name":"Force on a partially inserted dielectric slab","description":"Questions on the force on a partially inserted dielectric slab between capacitor plates, using F = (1/2)V² dC/dx or constant-charge energy methods.","sort":565,"aliases":[]},{"id":567,"code":"14.03.05","level":3,"parent_id":562,"name":"Dielectric polarization","description":"Questions on dielectric polarization, polar vs non-polar dielectrics, polarization vector P, susceptibility χ, and relative permittivity K = 1+χ with E = E₀/K inside the dielectric.","sort":566,"aliases":[]},{"id":568,"code":"14.03.06","level":3,"parent_id":562,"name":"Capacitors with dielectrics in series and parallel","description":"Questions treating a capacitor partially filled with dielectric layers as series or parallel sub-capacitors; calculate combined capacitance for stacked or side-by-side dielectrics.","sort":567,"aliases":[]},{"id":569,"code":"14.03.07","level":3,"parent_id":562,"name":"Capacitance of a parallel plate capacitor with a dielectric slab","description":"Questions on a parallel plate capacitor with a dielectric slab of thickness t and constant K, using C = ε₀A/(d − t + t/K) with possible air gaps and field/potential changes.","sort":568,"aliases":[]},{"id":570,"code":"14.04","level":2,"parent_id":550,"name":"Combination of capacitors","description":"General questions on equivalent capacitance of series, parallel, mixed, bridge, and infinite capacitor networks, including charge and voltage division.","sort":569,"aliases":["series and parallel capacitors","equivalent capacitance","capacitor networks","charge redistribution on connecting capacitors","common potential after connection","energy loss on connecting capacitors","floating capacitor junction"]},{"id":571,"code":"14.04.01","level":3,"parent_id":570,"name":"Equivalent capacitance: series, parallel, bridge networks, and infinite ladders","description":"Questions asking for equivalent capacitance in series, parallel, Wheatstone bridge, symmetric, and infinite ladder networks; uses redrawing, equipotential points, or recurrences.","sort":570,"aliases":[]},{"id":572,"code":"14.04.02","level":3,"parent_id":570,"name":"Capacitors in series","description":"Questions on capacitors in series using 1/C_eq = Σ1/C_i, equal charge on each capacitor, and voltage division inversely proportional to capacitance.","sort":571,"aliases":[]},{"id":573,"code":"14.04.03","level":3,"parent_id":570,"name":"Capacitors in parallel","description":"Questions on capacitors in parallel using C_eq = ΣC_i, equal voltage across each capacitor, and charge division proportional to capacitance.","sort":572,"aliases":[]},{"id":574,"code":"14.04.04","level":3,"parent_id":570,"name":"Mixed series-parallel capacitor networks","description":"Questions on reducing mixed series-parallel capacitor networks stepwise to find equivalent capacitance and charge/voltage/energy on individual capacitors.","sort":573,"aliases":[]},{"id":575,"code":"14.04.05","level":3,"parent_id":570,"name":"Infinite ladder and symmetrical capacitor networks","description":"Questions on infinite ladder and symmetrical capacitor networks; uses recurrence C_eq = f(C_eq), symmetry/equipotential arguments, and balanced bridge reductions.","sort":574,"aliases":[]},{"id":576,"code":"15","level":1,"parent_id":null,"name":"Current Electricity","description":"JEE chapter 15: Current Electricity. Concepts: Ohm's law, electric current and drift velocity; Kirchhoff's laws; Resistors and series-parallel combinations; Cells, EMF, internal resistance and potentiometer; Wheatstone bridge and meter bridge; Heating effects of current (Joule heating); RC circuits: charging, discharging, and time constant.","sort":575,"aliases":[]},{"id":577,"code":"15.01","level":2,"parent_id":576,"name":"Ohm's law, electric current and drift velocity","description":"Questions on electric current, current density, drift velocity, mobility, and their relation to applied electric field and conductor properties.","sort":576,"aliases":["ohm's law","drift velocity","current density","resistivity","conductivity","temperature dependence of resistance","temperature coefficient of resistance","colour code of resistors"]},{"id":578,"code":"15.01.01","level":3,"parent_id":577,"name":"Ohm's law and resistivity","description":"Problems using Ohm's law, resistivity, conductivity, and their temperature dependence, including color codes of resistors and effect of temperature on resistance.","sort":577,"aliases":[]},{"id":579,"code":"15.01.02","level":3,"parent_id":577,"name":"Electric current and drift velocity","description":"Calculations of drift velocity, current from charge carrier density, relaxation time, and relation I = nAevd, including effect of length and area on current.","sort":578,"aliases":[]},{"id":580,"code":"15.01.03","level":3,"parent_id":577,"name":"Microscopic current: drift velocity, current density and temperature dependence of resistivity","description":"Microscopic derivation of Ohm's law, current density J = σE, temperature dependence of resistivity, and numerical problems on drift velocity and relaxation time.","sort":579,"aliases":[]},{"id":581,"code":"15.02","level":2,"parent_id":576,"name":"Kirchhoff's laws","description":"Questions applying Kirchhoff's current and voltage laws to solve for currents and voltages in multi-loop DC circuits with batteries and resistors.","sort":580,"aliases":[]},{"id":582,"code":"15.02.01","level":3,"parent_id":581,"name":"Kirchhoff's current law (KCL)","description":"Problems using KCL at junctions: sum of currents entering equals sum leaving, often combined with KVL to solve circuit networks.","sort":581,"aliases":[]},{"id":583,"code":"15.02.02","level":3,"parent_id":581,"name":"Kirchhoff's voltage law (KVL) and loop/mesh analysis","description":"Application of KVL around loops, sign conventions for voltage drops and rises, and solving simultaneous equations for loop currents in complex circuits.","sort":582,"aliases":[]},{"id":584,"code":"15.02.03","level":3,"parent_id":581,"name":"Symmetry circuits, node-voltage methods and star-delta transformations","description":"Circuit solving using symmetry, node-voltage analysis, and star-delta (Y-Δ) transformations to simplify resistor networks and find equivalent resistance.","sort":583,"aliases":[]},{"id":585,"code":"15.02.04","level":3,"parent_id":581,"name":"Multi-loop circuit solving using KCL and KVL","description":"Multi-loop circuit problems requiring systematic application of KCL and KVL, including circuits with multiple batteries, dependent sources, and unknown currents.","sort":584,"aliases":[]},{"id":586,"code":"15.03","level":2,"parent_id":576,"name":"Resistors and series-parallel combinations","description":"Questions on series and parallel combinations of resistors, equivalent resistance, current and voltage division, and mixed combination circuits.","sort":585,"aliases":["series and parallel resistances","equivalent resistance","equivalent resistance of networks","symmetry method","infinite ladder networks"]},{"id":587,"code":"15.03.01","level":3,"parent_id":586,"name":"Infinite ladder resistor network","description":"Questions on finding equivalent resistance of infinite ladder networks using self-similarity/recursion, setting R_eq = R + ... and solving a quadratic; includes symmetric infinite grids and ladders.","sort":586,"aliases":[]},{"id":588,"code":"15.03.02","level":3,"parent_id":586,"name":"Series and parallel resistances","description":"Basic identification and calculation of equivalent resistance for resistors in series and parallel, including mixed circuits, current/voltage division, and simplification of resistor networks.","sort":587,"aliases":[]},{"id":589,"code":"15.04","level":2,"parent_id":576,"name":"Cells, EMF, internal resistance and potentiometer","description":"Questions involving cells, EMF, internal resistance, terminal voltage, potentiometer principle and applications; includes combinations of cells and galvanometer-to-meter conversions.","sort":588,"aliases":["emf and internal resistance","potentiometer","potential gradient","comparison of emfs","terminal voltage","combination of cells","cells in series and parallel","grouping of cells"]},{"id":590,"code":"15.04.01","level":3,"parent_id":589,"name":"EMF, terminal voltage and internal resistance","description":"Relation V = E - Ir, terminal voltage vs current, internal resistance from V-I graphs, power delivered to a load, and effects of internal resistance in circuits.","sort":589,"aliases":[]},{"id":591,"code":"15.04.02","level":3,"parent_id":589,"name":"Series and parallel combination of cells","description":"Equivalent EMF and internal resistance for cells in series, parallel, and mixed grouping; condition for maximum current in mixed grouping of cells.","sort":590,"aliases":[]},{"id":592,"code":"15.04.03","level":3,"parent_id":589,"name":"Potentiometer principle and comparison of EMF","description":"Potentiometer principle based on uniform potential gradient; comparing EMFs of two cells using a potentiometer, balance length ratios, and sensitivity.","sort":591,"aliases":[]},{"id":593,"code":"15.04.04","level":3,"parent_id":589,"name":"Measurement of internal resistance and EMF using potentiometer","description":"Using a potentiometer to measure internal resistance of a cell and unknown EMF; balance length with/without shunt, formulas like r = R(l1/l2 - 1).","sort":592,"aliases":[]},{"id":594,"code":"15.04.05","level":3,"parent_id":589,"name":"Galvanometer, ammeter and voltmeter conversions","description":"Conversion of galvanometer to ammeter by a shunt and to voltmeter by a series high resistance; calculations of shunt resistance, multiplier resistance, and range extension.","sort":593,"aliases":[]},{"id":595,"code":"15.05","level":2,"parent_id":576,"name":"Wheatstone bridge and meter bridge","description":"Balanced Wheatstone bridge condition P/Q = R/S, meter bridge experiments, unknown resistance, end corrections, and sensitivity/unbalanced bridge.","sort":594,"aliases":["meter bridge","balanced bridge","slide wire bridge"]},{"id":596,"code":"15.06","level":2,"parent_id":576,"name":"Heating effects of current (Joule heating)","description":"Joule heating, power P = I^2 R = VI = V^2/R, heat generated in resistors, electric power in series/parallel, fuses, and efficiency.","sort":595,"aliases":[]},{"id":597,"code":"15.07","level":2,"parent_id":576,"name":"RC circuits: charging, discharging, and time constant","description":"Questions on transient behavior of RC circuits, time constant τ = RC, charging/discharging equations, and initial/steady-state analysis.","sort":596,"aliases":[]},{"id":598,"code":"15.07.01","level":3,"parent_id":597,"name":"RC circuit transients: charging, discharging and time constant","description":"Qualitative and quantitative transients in RC circuits: growth/decay of charge/current, time constant, half-life, and graphs of charging/discharging.","sort":597,"aliases":[]},{"id":599,"code":"15.07.02","level":3,"parent_id":597,"name":"Transient current and voltage equations for RC circuits","description":"Applying q = Q0(1 - e^{-t/RC}), i = (E/R)e^{-t/RC} for charging; q = Q0 e^{-t/RC}, i = -(Q0/RC)e^{-t/RC} for discharging; finding current/voltage at a given time.","sort":598,"aliases":[]},{"id":600,"code":"15.07.03","level":3,"parent_id":597,"name":"Initial, transient and steady-state circuit analysis for RC circuits","description":"Determining initial (t=0+) and steady-state (t→∞) capacitor voltage/charge and currents, using uncharged capacitor as short and charged capacitor as open at steady state; transient analysis.","sort":599,"aliases":[]},{"id":601,"code":"16","level":1,"parent_id":null,"name":"Magnetic Effects of Current","description":"JEE chapter 16: Magnetic Effects of Current. Concepts: Magnetic field; Ampere's law; Lorentz force and motion of charged particles in magnetic fields; Biot-Savart law; Current loop as magnetic dipole: torque, force and moving-coil galvanometer.","sort":600,"aliases":["Moving Charges and Magnetism"]},{"id":602,"code":"16.01","level":2,"parent_id":601,"name":"Magnetic field","description":"General concept of magnetic field produced by moving charges/currents, direction via right-hand rule, superposition of fields.","sort":601,"aliases":["magnetic force on current-carrying conductors","force between parallel currents"]},{"id":603,"code":"16.01.01","level":3,"parent_id":602,"name":"Force between parallel currents","description":"Questions compute force per unit length between two long parallel wires: F/L = μ0 I1 I2 / (2π d), attractive for same direction, repulsive for opposite.","sort":602,"aliases":[]},{"id":604,"code":"16.01.02","level":3,"parent_id":602,"name":"Magnetic flux density units (tesla and equivalents)","description":"Problems converting or identifying units of magnetic field: 1 T = 1 N/(A·m) = 1 Wb/m² = 10⁴ G, often in dimensional analysis.","sort":603,"aliases":[]},{"id":605,"code":"16.02","level":2,"parent_id":601,"name":"Ampere's law","description":"General application of Ampere's law to symmetric current distributions to find magnetic field.","sort":604,"aliases":["magnetic field of solenoid and toroid"]},{"id":606,"code":"16.02.01","level":3,"parent_id":605,"name":"Ampere's circuital law","description":"Questions use ∮ B·dl = μ0 I_enc for infinite wires, coaxial cables, thick cylinders, etc., choosing appropriate Amperian loops.","sort":605,"aliases":[]},{"id":607,"code":"16.02.02","level":3,"parent_id":605,"name":"Off-axis cavity field via current superposition","description":"Problems where a cylindrical conductor with a cavity carries current; field at a point in cavity found by superposing fields of full cylinder and opposite current in cavity region.","sort":606,"aliases":[]},{"id":608,"code":"16.02.03","level":3,"parent_id":605,"name":"Field of a solenoid and toroid","description":"Questions compute B = μ0 n I inside an ideal solenoid, B = μ0 N I / (2π r) inside a toroid, and zero outside.","sort":607,"aliases":[]},{"id":609,"code":"16.03","level":2,"parent_id":601,"name":"Lorentz force and motion of charged particles in magnetic fields","description":"General topic: force on moving charge in magnetic field, circular/helical motion, applications.","sort":608,"aliases":["Hall effect","cyclotron","velocity selector","helical motion","circular motion of a charge in a magnetic field","motion of charged particle in magnetic field","lorentz force"]},{"id":610,"code":"16.03.01","level":3,"parent_id":609,"name":"Lorentz force","description":"Questions apply F = q(v × B) to find force magnitude/direction, often combined with electric field (F = q(E + v×B)) for velocity selectors.","sort":609,"aliases":[]},{"id":611,"code":"16.03.02","level":3,"parent_id":609,"name":"Hall effect","description":"Problems involve Hall voltage V_H = B I / (n e t), determining carrier density, sign of charge carriers, or magnetic field from measured Hall voltage.","sort":610,"aliases":[]},{"id":612,"code":"16.03.03","level":3,"parent_id":609,"name":"Motion of charges in magnetic fields","description":"Questions on circular motion (r = mv/(qB), T = 2πm/(qB)), helical paths when velocity has component along B, and pitch calculations.","sort":611,"aliases":[]},{"id":613,"code":"16.04","level":2,"parent_id":601,"name":"Biot-Savart law","description":"General use of Biot-Savart law to compute magnetic field from current elements.","sort":612,"aliases":[]},{"id":614,"code":"16.04.01","level":3,"parent_id":613,"name":"Biot-Savart law","description":"Questions apply dB = (μ0/4π) (I dl × r̂)/r² to find field of straight wire segments, circular loops (on axis), arcs, etc., via integration.","sort":613,"aliases":[]},{"id":615,"code":"16.05","level":2,"parent_id":601,"name":"Current loop as magnetic dipole: torque, force and moving-coil galvanometer","description":"General topic: magnetic moment of current loop, torque in uniform field, force in non-uniform field, galvanometer principle.","sort":614,"aliases":["magnetic dipole moment of a current loop","current loop as magnetic dipole","galvanometer to ammeter and voltmeter conversion","force on a magnetic dipole in a non-uniform field"]},{"id":616,"code":"16.05.01","level":3,"parent_id":615,"name":"Current-loop magnetic moment and torque","description":"Questions compute magnetic moment m = NIA, torque τ = m × B, potential energy U = -m·B, and work to rotate loop in field.","sort":615,"aliases":[]},{"id":617,"code":"16.05.02","level":3,"parent_id":615,"name":"Moving-coil galvanometer","description":"Problems on galvanometer: current sensitivity, voltage sensitivity, conversion to ammeter (shunt) or voltmeter (series resistance), deflection angle proportional to current.","sort":616,"aliases":[]},{"id":618,"code":"16.05.03","level":3,"parent_id":615,"name":"Current-loop force in a non-uniform magnetic field","description":"Questions on net force on a current loop in a spatially varying magnetic field, using F = ∇(m·B) or integrating force on loop elements.","sort":617,"aliases":[]},{"id":619,"code":"16.05.04","level":3,"parent_id":615,"name":"Torque on a current loop and moving-coil galvanometer","description":"Questions on torque τ = m × B on a current loop and its application in moving-coil galvanometers, including deflection and current sensitivity.","sort":618,"aliases":[]},{"id":620,"code":"17","level":1,"parent_id":null,"name":"Magnetism & Matter","description":"JEE chapter 17: Magnetism & Matter. Concepts: Magnetic materials and hysteresis; Earth's magnetism and magnetic elements; Bar magnet and magnetic dipole: field, torque, potential energy and oscillations.","sort":619,"aliases":["Magnetism and Matter"]},{"id":621,"code":"17.01","level":2,"parent_id":620,"name":"Magnetic materials and hysteresis","description":"Questions classifying magnetic materials and computing material quantities: given χ, μᵣ, M, or H find B = μ₀(H + M), or identify material type from susceptibility/temperature behavior; parent node for material-magnetism questions.","sort":620,"aliases":["magnetic properties of matter","dia-, para- and ferromagnetism","magnetisation and magnetic intensity"]},{"id":622,"code":"17.01.01","level":3,"parent_id":621,"name":"Dia-, para- and ferromagnetism","description":"Questions distinguishing diamagnetic, paramagnetic, and ferromagnetic behavior: signs and typical ranges of χ and μᵣ, Curie law χ ∝ 1/T and Curie temperature, field-line behavior, and examples (bismuth, aluminium, iron).","sort":621,"aliases":[]},{"id":623,"code":"17.01.02","level":3,"parent_id":621,"name":"Hysteresis","description":"Questions on the B-H hysteresis loop: retentivity and coercivity values from a loop, energy dissipated per cycle proportional to loop area, and material selection for permanent magnets (high coercivity) versus transformer cores (narrow loop).","sort":622,"aliases":[]},{"id":624,"code":"17.01.03","level":3,"parent_id":621,"name":"Magnetisation and magnetic intensity","description":"Questions on the relations M, H, B = μ₀(H + M), χ = M/H, and μ = μ₀(1+χ): computing magnetization or intensity when a material is placed in a solenoid's field, or converting between B, H, and M.","sort":623,"aliases":[]},{"id":625,"code":"17.02","level":2,"parent_id":620,"name":"Earth's magnetism and magnetic elements","description":"Questions on Earth's magnetic field and its description: magnitude and direction at a location, elements of the field, neutral points with a bar magnet, and apparent changes of dip in non-magnetic-meridian planes; parent node for Earth's-field questions.","sort":624,"aliases":["declination","angle of dip","horizontal component of Earth's field"]},{"id":626,"code":"17.02.01","level":3,"parent_id":625,"name":"Earth's magnetism","description":"Questions on Earth's magnetism itself: origin/dipole picture, field strength and dip varying with latitude, horizontal/vertical component patterns, and neutral points formed when a bar magnet's field cancels Earth's field on a map.","sort":625,"aliases":[]},{"id":627,"code":"17.02.02","level":3,"parent_id":625,"name":"Magnetic elements: declination, dip and horizontal component","description":"Questions using the magnetic elements: B_H = B cosδ, B_V = B sinδ, tanδ = B_V/B_H, finding total field or dip from given components, dip at equator (0°) and poles (90°), and resolving a measured component into B_H and B_V.","sort":626,"aliases":[]},{"id":628,"code":"17.02.03","level":3,"parent_id":625,"name":"True dip and apparent dip relations","description":"Questions relating true dip δ to apparent dip δ' measured in a vertical plane at angle θ to the magnetic meridian: tanδ' = tanδ/cosθ, including finding true dip from two apparent dips in perpendicular or given planes.","sort":627,"aliases":[]},{"id":629,"code":"17.02.04","level":3,"parent_id":625,"name":"Neutral points of a bar magnet in Earth's field","description":"Questions locating neutral points where a bar magnet's field cancels Earth's horizontal component B_H: magnet with N-pole toward magnetic north gives neutral points on the equatorial line (μ0·2m/4πx³ = B_H), magnet perpendicular to meridian gives them on the axial line (μ0·m/4πx³ = B_H). Typical asks: find x given m and B_H, or infer B_H from neutral-point distance.","sort":628,"aliases":[]},{"id":630,"code":"17.02.05","level":3,"parent_id":625,"name":"Tangent law and tangent galvanometer","description":"Questions using the tangent law B = B_H tanθ and the tangent galvanometer I = K tanθ with reduction factor K = (2rB_H)/(μ0 N). Typical asks: compute current from deflection angle, find K or B_H, compare two currents via tanθ ratio, effect of reversing current or doubling turns, sensitivity conditions.","sort":629,"aliases":[]},{"id":631,"code":"17.02.06","level":3,"parent_id":625,"name":"Earth's field as a dipole: magnetic and geographic axes","description":"Questions treating Earth as a dipole with magnetic axis tilted from the geographic axis: declination, angle of dip δ, B_H = B cosδ, B_V = B sinδ, tanδ = B_V/B_H. Typical asks: find dip or total field from components, apparent dip in a vertical plane at angle θ from magnetic meridian (tanδ' = tanδ/cosθ), compare B_H at two latitudes.","sort":630,"aliases":[]},{"id":632,"code":"17.03","level":2,"parent_id":620,"name":"Bar magnet and magnetic dipole: field, torque, potential energy and oscillations","description":"Umbrella node for mixed bar-magnet/dipole questions that combine several sub-results: e.g., a magnet cut into pieces then placed in a field, torque plus oscillation data to find moment, or field at a point combined with force/energy of a small dipole placed there. Use when the question spans field, torque, energy, or oscillation aspects together.","sort":631,"aliases":["bar magnet","magnetic dipole moment","magnetic field lines of a bar magnet","field on axis and equator of a bar magnet","torque on a magnetic dipole","potential energy of a magnetic dipole","magnetic needle oscillations","vibration magnetometer","force between magnetic dipoles","Gauss's law for magnetism","solenoid as a bar magnet","Bar magnet and magnetic dipole"]},{"id":633,"code":"17.03.01","level":3,"parent_id":632,"name":"Bar magnet as a magnetic dipole","description":"Questions on the bar magnet as a dipole with pole strength q_m and moment M = q_m·L (directed S→N): cutting a magnet transversely or along its length (new M, q_m, force between poles F = μ0 q_m1 q_m2/4πr²), effective length, moment of a bent or reshaped magnet. Typical asks: M of each piece after n cuts, attraction/repulsion between pole pieces.","sort":632,"aliases":[]},{"id":634,"code":"17.03.02","level":3,"parent_id":632,"name":"Magnetic field lines","description":"Conceptual questions on magnetic field line properties: closed continuous loops (no monopoles), lines never intersect, tangent gives direction, crowding indicates stronger field, field line pattern of a bar magnet and its resemblance to a solenoid, behavior inside the magnet (S to N inside).","sort":633,"aliases":[]},{"id":635,"code":"17.03.03","level":3,"parent_id":632,"name":"Magnetic field due to a bar magnet: axial and equatorial points","description":"Numerical questions on B due to a bar magnet at a point on the axis (B = μ0·2M/4π(r²−l²)² ≈ μ0·2M/4πr³) or equatorial line (B = μ0·M/4π(r²+l²)^{3/2} ≈ μ0·M/4πr³), including the 2:1 axial-to-equatorial ratio for short magnets and vector combination of axial/equatorial fields at a general point.","sort":634,"aliases":[]},{"id":636,"code":"17.03.04","level":3,"parent_id":632,"name":"Torque on a magnetic dipole in a uniform field","description":"Questions on torque τ = M B sinθ on a magnet/current loop in a uniform field: stable equilibrium at θ=0, unstable at θ=π, zero net force in uniform field, torque at 45° or 90°, direction by right-hand rule, torque needed to hold a magnet at angle, behavior of a compass needle.","sort":635,"aliases":[]},{"id":637,"code":"17.03.05","level":3,"parent_id":632,"name":"Potential energy of a magnetic dipole in a field","description":"Questions on dipole potential energy U = −MB cosθ (zero at θ=90°): work done in rotating a magnet from θ₁ to θ₂ as W = MB(cosθ₁ − cosθ₂), energy difference between stable and unstable positions (2MB), minimum energy orientation, combining U with torque to find angle from energy ratio.","sort":636,"aliases":[]},{"id":638,"code":"17.03.06","level":3,"parent_id":632,"name":"Oscillations of a suspended magnet (vibration magnetometer)","description":"Questions on a suspended magnet oscillating in Earth's field with T = 2π√(I/MB_H): vibration magnetometer, comparing magnetic moments or B_H via T² ratios, combined magnets (sum/difference of moments, like/unlike poles together), effect of temperature or inertia changes on T.","sort":637,"aliases":[]},{"id":639,"code":"17.03.07","level":3,"parent_id":632,"name":"Gauss's law for magnetism","description":"Questions on Gauss's law for magnetism ∮B·dA = 0: net magnetic flux through any closed surface is zero, nonexistence of magnetic monopoles, flux entering equals flux leaving, conceptual checks on flux through surfaces enclosing a magnet or current loop.","sort":638,"aliases":[]},{"id":640,"code":"18","level":1,"parent_id":null,"name":"Electromagnetic Induction","description":"JEE chapter 18: Electromagnetic Induction. Concepts: Faraday's and Lenz's laws (magnetic flux and induced EMF); Motional EMF; Self inductance; LR circuit transients (growth and decay of current, time constant); Mutual inductance; Eddy currents.","sort":639,"aliases":[]},{"id":641,"code":"18.01","level":2,"parent_id":640,"name":"Faraday's and Lenz's laws (magnetic flux and induced EMF)","description":"Parent node for problems computing magnetic flux (Φ = BA cosθ) and induced EMF (ε = −N dΦ/dt) when flux changes via changing B, area, or orientation. Tag here only for general flux/EMF questions that don't fit a specific subcase like AC generator, motional EMF, or inductance.","sort":640,"aliases":["magnetic flux","AC generator","Faraday's and Lenz's laws","lenz's law","direction of induced current","Faraday's Law and Magnetic Flux"]},{"id":642,"code":"18.01.01","level":3,"parent_id":641,"name":"AC generator","description":"AC generator: a coil of N turns and area A rotating with angular frequency ω in a uniform field B, giving Φ = NBA cos ωt and ε = NBAω sin ωt. Questions ask peak EMF, RMS value, instantaneous EMF at a given angle/time, frequency dependence, or graph of ε vs t.","sort":641,"aliases":[]},{"id":643,"code":"18.01.02","level":3,"parent_id":641,"name":"Faraday's and Lenz's laws","description":"Direct application of ε = −N dΦ/dt: loops with time-varying B(t), shrinking/expanding loops, rotating or flipping loops, and flux-vs-time graphs where slope gives EMF. Includes finding magnitude, direction, and average EMF over an interval (ε_avg = NΔΦ/Δt).","sort":642,"aliases":[]},{"id":644,"code":"18.01.03","level":3,"parent_id":641,"name":"Magnetic flux","description":"Computing magnetic flux Φ = BA cosθ (or ∫B·dA) through planar loops, tilted coils, or loops in non-uniform fields; unit conversions to weber. Questions often give B, area, and angle and ask flux, or ask flux at a specific instant for a rotating coil.","sort":643,"aliases":[]},{"id":645,"code":"18.01.04","level":3,"parent_id":641,"name":"Induced EMF and induced charge (q = NΔΦ/R)","description":"Induced charge flowing when flux changes: q = NΔΦ/R, independent of how fast the flux changes. Typical questions give a coil, field change, and resistance (often with a galvanometer) and ask total charge passed, or reverse-engineer ΔB or N from measured charge.","sort":644,"aliases":[]},{"id":646,"code":"18.01.05","level":3,"parent_id":641,"name":"Lenz's law: direction of induced current and energy conservation","description":"Lenz's law direction questions: which way does the induced current flow (clockwise/anticlockwise) when a magnet approaches/recedes, B increases/decreases, or a loop enters a field region; also force opposing motion, ring repelling/attracting the magnet, and eddy-current braking justified by energy conservation.","sort":645,"aliases":[]},{"id":647,"code":"18.01.06","level":3,"parent_id":641,"name":"Induced electric field due to changing magnetic field (non-conservative E)","description":"Non-conservative induced electric field from a spatially uniform but time-varying B: ∮E·dl = −dΦ/dt, with E = (r/2)|dB/dt| inside and E = (R²/2r)|dB/dt| outside a cylindrical field region. Questions ask E at a radius, work done on a charge around a path, or why ∮E·dl ≠ 0.","sort":646,"aliases":[]},{"id":648,"code":"18.01.07","level":3,"parent_id":641,"name":"Betatron","description":"Betatron: electrons accelerated in a circular orbit by an induced E from a changing magnetic flux, with the orbit condition B_orbit = (½)B̄_avg (field at orbit equals half the average flux density inside). Questions test this 2:1 condition, orbit radius stability, or energy gained per turn.","sort":647,"aliases":[]},{"id":649,"code":"18.02","level":2,"parent_id":640,"name":"Motional EMF","description":"Parent node for EMF produced by conductors moving through a magnetic field (flux cutting), ε = Bvl for a rod, rotating rod/disc formulas, and rail problems. Use for general motional EMF questions not matching a specific subcase.","sort":648,"aliases":["rod on rails","rotating rod and disc EMF","emf = Blv","induced current, force and power in moving-conductor circuits"]},{"id":650,"code":"18.02.01","level":3,"parent_id":649,"name":"Motional EMF in a translating rod (EMF = Bvl)","description":"Straight rod of length l translating with velocity v in a uniform field B: ε = Bvl when v ⊥ B ⊥ l, zero if v ∥ B; direction from v × B or Fleming's right-hand rule. Questions give rod speed, B, length and ask EMF, polarity of ends, or current through a connected circuit.","sort":649,"aliases":[]},{"id":651,"code":"18.02.02","level":3,"parent_id":649,"name":"Rotating rod and rotating disc in a magnetic field","description":"Rotating conductors: rod of length l spinning with ω about one end in field B gives ε = Bωl²/2 between ends; a disc of radius R rotating gives ε = BωR²/2 between centre and rim. Questions ask EMF, polarity, current through an external resistance, or effect of doubling ω/R.","sort":650,"aliases":[]},{"id":652,"code":"18.02.03","level":3,"parent_id":649,"name":"Rod on rails: induced current, force and terminal velocity","description":"Rod sliding on conducting rails in field B: induced current i = Bvl/R, retarding force F = B²l²v/R, and terminal velocity problems (horizontal pull or on an incline under gravity), including v(t) = v₀e^{−B²l²t/mR} decay. Questions ask current, force at given speed, terminal velocity, or distance travelled.","sort":651,"aliases":[]},{"id":653,"code":"18.02.04","level":3,"parent_id":649,"name":"Mechanical power and energy balance in motional EMF","description":"Energy balance in motional EMF: mechanical power supplied Fv equals electrical power dissipated i²R = B²l²v²/R, with F = B²l²v/R. Questions ask power delivered, heat generated over a time, work done by external agent, or verifying energy conservation for a rod on rails.","sort":652,"aliases":[]},{"id":654,"code":"18.02.05","level":3,"parent_id":649,"name":"Motional EMF in moving loops and arbitrary shaped conductors","description":"Motional EMF in loops and curved/arbitrary conductors: ε = ∮(v × B)·dl, loops partially entering or leaving a field region, semicircular or L-shaped wires translating/rotating, and flux-cutting reasoning. Questions ask EMF between two points of a moving shaped wire or net EMF/current in a moving loop.","sort":653,"aliases":[]},{"id":655,"code":"18.03","level":2,"parent_id":640,"name":"Self inductance","description":"Parent node for self inductance: flux linkage NΦ = LI, back-EMF ε = −L dI/dt, energy ½LI², standard L formulas for solenoid/toroid/coax, and L–R combinations. Tag here for general self-inductance questions not matching a specific subcase.","sort":654,"aliases":["energy stored in an inductor","self inductance of solenoid and toroid"]},{"id":656,"code":"18.03.01","level":3,"parent_id":655,"name":"Self and mutual inductance","description":"Mutual inductance paired with self inductance: M = N₂Φ₂/I₁, EMF in secondary ε₂ = −M dI₁/dt, M = k√(L₁L₂) with coupling coefficient k, and M between coaxial solenoids. Questions ask M, induced secondary EMF for a given dI/dt, or k for given L₁, L₂, M.","sort":655,"aliases":[]},{"id":657,"code":"18.03.02","level":3,"parent_id":655,"name":"Self inductance: definition, flux linkage and L = NΦ/I","description":"Definition-level self inductance: L = NΦ/I (flux linkage per current), ε = −L dI/dt, SI unit henry, and the fact that L depends only on geometry and core material, not on current. Questions give N, Φ, I and ask L, or give L and dI/dt and ask back-EMF.","sort":656,"aliases":[]},{"id":658,"code":"18.03.03","level":3,"parent_id":655,"name":"Self inductance of solenoid, toroid and coaxial geometries","description":"Standard inductance formulas: solenoid L = μ₀μ_r n²Al, toroid L = μ₀N²h/(2π) ln(b/a), coaxial cable per length L/ℓ = (μ₀/2π) ln(b/a). Questions ask L for given geometry, change in L when an iron core is inserted or turns doubled, or L per unit length.","sort":657,"aliases":[]},{"id":659,"code":"18.03.04","level":3,"parent_id":655,"name":"Energy stored in an inductor and magnetic energy density","description":"Energy in inductors and fields: U = ½LI², magnetic energy density u = B²/2μ₀, and total energy in a solenoid/toroid volume. Questions ask stored energy at a given current, energy density at the centre of a solenoid, or ratio of energies at two currents.","sort":658,"aliases":[]},{"id":660,"code":"18.03.05","level":3,"parent_id":655,"name":"Series and parallel combination of inductors","description":"Combining inductors (no mutual coupling unless stated): series L_eq = L₁ + L₂ + …, parallel 1/L_eq = 1/L₁ + 1/L₂; with mutual coupling L_eq = L₁ + L₂ ± 2M. Questions ask equivalent L of a network or the effect of aiding/opposing series connection.","sort":659,"aliases":[]},{"id":661,"code":"18.04","level":2,"parent_id":640,"name":"LR circuit transients (growth and decay of current, time constant)","description":"Parent node for L–R circuit transients: current growth i = (E/R)(1 − e^{−t/τ}) and decay i = I₀e^{−t/τ} with τ = L/R, plus inductor behaviour at switch-on/off. Tag here for general transient questions not matching a specific subcase.","sort":660,"aliases":["LR circuit","growth and decay of current","time constant L/R"]},{"id":662,"code":"18.04.01","level":3,"parent_id":661,"name":"LR circuit transients and time constant","description":"General L–R transient analysis: finding i(t), di/dt, inductor EMF L di/dt, or charge/energy at a given time using τ = L/R, often with graphs of i vs t. Questions may combine a battery, resistor(s), and inductor with a switch thrown at t = 0.","sort":661,"aliases":[]},{"id":663,"code":"18.04.02","level":3,"parent_id":661,"name":"Growth and decay of current in LR circuit","description":"Explicit growth and decay equations: growth i = I₀(1 − e^{−Rt/L}) toward E/R and decay i = I₀e^{−Rt/L} after the source is removed, with total heat ½LI₀² dissipated during decay. Questions ask current after n time constants, time to reach half the final current, or energy dissipated.","sort":662,"aliases":[]},{"id":664,"code":"18.04.03","level":3,"parent_id":661,"name":"Time constant of LR circuit","description":"Time constant τ = L/R: current reaches 63.2% of final value (or decays to 36.8%) in one τ, dimensions of L/R are time, and τ sets how fast transients proceed. Questions ask τ for a given circuit, how τ changes when R or L is altered, or current at t = τ, 2τ, etc.","sort":663,"aliases":[]},{"id":665,"code":"18.04.04","level":3,"parent_id":661,"name":"Inductor behavior at switching instants (t = 0+ and steady state)","description":"Inductor behaviour at switching instants: i(0⁺) = i(0⁻) (current through L cannot jump), so at t = 0⁺ the inductor behaves as an open circuit (full EMF across it, di/dt = E/L), while at steady state it acts as a plain wire (ε = 0). Questions ask initial and final currents through each resistor in multi-branch L–R circuits just after and long after the switch closes/opens.","sort":664,"aliases":[]},{"id":666,"code":"18.04.05","level":3,"parent_id":661,"name":"LC oscillations","description":"Questions on LC oscillations, including charge and current as functions of time, energy exchange, and oscillation frequency.","sort":665,"aliases":[]},{"id":667,"code":"18.05","level":2,"parent_id":640,"name":"Mutual inductance","description":"Questions on mutual inductance, including definition, calculation, and applications in coupled circuits.","sort":666,"aliases":[]},{"id":668,"code":"18.05.01","level":3,"parent_id":667,"name":"Mutual inductance: definition, M = NΦ/I and reciprocity","description":"Questions on the definition of mutual inductance, the formula M = NΦ/I, and reciprocity of mutual inductance between two coils.","sort":667,"aliases":[]},{"id":669,"code":"18.05.02","level":3,"parent_id":667,"name":"Mutual inductance of coaxial solenoids","description":"Questions on mutual inductance M for two coaxial solenoids, using M = μ₀N₁N₂A/l and flux linkage through the inner solenoid.","sort":668,"aliases":[]},{"id":670,"code":"18.05.03","level":3,"parent_id":667,"name":"Mutual inductance of concentric coplanar coils","description":"Questions on mutual inductance between concentric coplanar circular coils, computing flux through one loop due to current in the other using the magnetic field at the center.","sort":669,"aliases":[]},{"id":671,"code":"18.05.04","level":3,"parent_id":667,"name":"EMF induced in secondary coil and coupled circuits","description":"Questions on induced EMF in a secondary coil due to changing current in a primary, including coupled LR/LC circuits and time-varying mutual induction.","sort":670,"aliases":[]},{"id":672,"code":"18.06","level":2,"parent_id":640,"name":"Eddy currents","description":"Questions on induced circulating currents in bulk conductors placed in changing magnetic flux, including their origin, direction, and energy dissipation.","sort":671,"aliases":[]},{"id":673,"code":"18.06.01","level":3,"parent_id":672,"name":"Eddy currents: origin and basic properties","description":"Questions on why eddy currents arise from Faraday's law in extended conductors, their dependence on conductivity, geometry, and rate of flux change.","sort":672,"aliases":[]},{"id":674,"code":"18.06.02","level":3,"parent_id":672,"name":"Applications of eddy currents (magnetic braking, induction furnace)","description":"Questions on practical uses of eddy currents such as magnetic braking, induction furnaces, metal detectors, and electromagnetic damping.","sort":673,"aliases":[]},{"id":675,"code":"18.06.03","level":3,"parent_id":672,"name":"Eddy current damping and core lamination","description":"Questions on reducing eddy current losses by laminating iron cores and on eddy current damping of moving conductors in magnetic fields.","sort":674,"aliases":[]},{"id":676,"code":"19","level":1,"parent_id":null,"name":"Alternating Current","description":"JEE chapter 19: Alternating Current. Concepts: AC Circuits: Voltage, Current and Phasor Analysis; LCR Circuit; Resonance; Power in AC Circuits and Power Factor; Transformers; LC Oscillations.","sort":675,"aliases":[]},{"id":677,"code":"19.01","level":2,"parent_id":676,"name":"AC Circuits: Voltage, Current and Phasor Analysis","description":"Questions on AC circuit analysis using phasors: phase relationships between voltage and current, phasor diagrams, and impedance calculations for series or parallel AC circuits.","sort":676,"aliases":["alternating current","AC fundamentals","RMS values","AC through R, L and C"]},{"id":678,"code":"19.01.01","level":3,"parent_id":677,"name":"AC through R, L and C","description":"Questions on AC applied separately to a resistor, inductor, or capacitor: phase differences (0, +π/2, -π/2), reactance formulas X_L=ωL, X_C=1/ωC, and current expressions.","sort":677,"aliases":[]},{"id":679,"code":"19.01.02","level":3,"parent_id":677,"name":"AC voltage and current; RMS values","description":"Questions on RMS and peak values of AC voltage/current, average power in resistive AC circuits, and relations like I_rms = I₀/√2, V_rms = V₀/√2.","sort":678,"aliases":[]},{"id":680,"code":"19.01.03","level":3,"parent_id":677,"name":"Parallel AC branches: phasor addition and impedance","description":"Questions on parallel AC branches containing R, L, or C, using phasor addition of branch currents, admittance, or impedance to find total current and phase angle.","sort":679,"aliases":[]},{"id":681,"code":"19.01.04","level":3,"parent_id":677,"name":"AC through resistor, inductor and capacitor","description":"Questions on AC through resistor, inductor, and capacitor individually or in series: reactance, impedance, phase angle, and voltage-current relations for each element.","sort":680,"aliases":[]},{"id":682,"code":"19.01.05","level":3,"parent_id":677,"name":"Phasor analysis of AC circuits","description":"Questions on representing sinusoidal AC voltages and currents as rotating phasors, drawing phasor diagrams, and using them to add or compare phase-shifted quantities in purely resistive, inductive, or capacitive circuits.","sort":681,"aliases":[]},{"id":683,"code":"19.01.06","level":3,"parent_id":677,"name":"RC low-pass and high-pass filters","description":"Questions about RC circuits used as frequency-dependent filters, including output taken across the resistor or capacitor, cutoff frequency, gain versus frequency, and phase shift in low-pass and high-pass configurations.","sort":682,"aliases":[]},{"id":684,"code":"19.02","level":2,"parent_id":676,"name":"LCR Circuit","description":"Questions involving a series or parallel combination of inductor, capacitor, and resistor driven by an AC source, including impedance, resonance, phase relationships, and voltage/current calculations.","sort":683,"aliases":["series LCR circuit","RLC series circuit"]},{"id":685,"code":"19.02.01","level":3,"parent_id":684,"name":"Series LCR circuit and resonance","description":"Questions on the series LCR circuit at resonance, where inductive and capacitive reactances cancel, current is maximum, and the circuit behaves resistively; includes resonance frequency and voltage magnification across L or C.","sort":684,"aliases":[]},{"id":686,"code":"19.02.02","level":3,"parent_id":684,"name":"Impedance, reactance and phase angle in series LCR","description":"Questions asking for impedance magnitude, inductive or capacitive reactance, net reactance, and phase angle between voltage and current in a series LCR circuit at a given frequency.","sort":685,"aliases":[]},{"id":687,"code":"19.02.03","level":3,"parent_id":684,"name":"Bandwidth and quality factor of series LCR","description":"Questions on the frequency range between half-power points of a series LCR resonance curve, and calculation of quality factor from bandwidth, resonance frequency, resistance, inductance, or capacitance.","sort":686,"aliases":[]},{"id":688,"code":"19.03","level":2,"parent_id":676,"name":"Resonance","description":"Questions on the general phenomenon of resonance in driven oscillatory systems, especially electrical resonance where applied frequency matches natural frequency and amplitude becomes maximum.","sort":687,"aliases":[]},{"id":689,"code":"19.03.01","level":3,"parent_id":688,"name":"Resonance frequency and condition","description":"Questions asking for the resonance frequency or condition in LCR circuits, including the formula f = 1/(2π√(LC)) and the condition XL = XC.","sort":688,"aliases":[]},{"id":690,"code":"19.03.02","level":3,"parent_id":688,"name":"Sharpness of resonance and Q factor","description":"Questions on how sharply a resonant circuit responds to frequencies near resonance, including sharpness, quality factor Q, relation to bandwidth, and effect of resistance on the resonance peak.","sort":689,"aliases":[]},{"id":691,"code":"19.04","level":2,"parent_id":676,"name":"Power in AC Circuits and Power Factor","description":"Questions on average power dissipated in AC circuits, apparent power, true power, and the power factor cos φ relating them.","sort":690,"aliases":["AC power","average power","wattless current"]},{"id":692,"code":"19.04.01","level":3,"parent_id":691,"name":"Power in AC circuits and power factor","description":"Questions asking for average power P = Vrms Irms cos φ, power factor, or phase angle in resistive, inductive, capacitive, or mixed AC circuits.","sort":691,"aliases":[]},{"id":693,"code":"19.04.02","level":3,"parent_id":691,"name":"Wattless current and reactive power","description":"Questions on the component of current that does no net work, wattless or idle current, reactive power, and its relation to power factor and phase angle in inductive or capacitive circuits.","sort":692,"aliases":[]},{"id":694,"code":"19.05","level":2,"parent_id":676,"name":"Transformers","description":"Questions on the working principle, construction, and operation of transformers, including mutual induction, core, primary and secondary coils, and ideal transformer assumptions.","sort":693,"aliases":[]},{"id":695,"code":"19.05.01","level":3,"parent_id":694,"name":"Step-up and step-down transformers","description":"Questions on step-up and step-down transformers, identifying which increases or decreases voltage, and relating turns ratio to voltage transformation.","sort":694,"aliases":[]},{"id":696,"code":"19.05.02","level":3,"parent_id":694,"name":"Transformer turn ratio, current and voltage relations","description":"Questions using the transformer equations Vp/Vs = Np/Ns and Ip/Is = Ns/Np to find unknown turns, voltages, or currents in ideal transformers.","sort":695,"aliases":[]},{"id":697,"code":"19.05.03","level":3,"parent_id":694,"name":"Transformer efficiency and energy losses","description":"Questions on transformer efficiency, output versus input power, and energy losses such as copper loss, eddy current loss, hysteresis loss, and flux leakage.","sort":696,"aliases":[]},{"id":698,"code":"19.06","level":2,"parent_id":676,"name":"LC Oscillations","description":"Questions on oscillatory exchange of energy between inductor and capacitor in an LC circuit, including natural frequency and analogy to mechanical oscillations.","sort":697,"aliases":["LC tank circuit","damped RLC oscillations"]},{"id":699,"code":"19.06.01","level":3,"parent_id":698,"name":"Free damped oscillations in a series RLC circuit","description":"Questions on free damped oscillations in a series RLC circuit without an external driving source, including damping factor, logarithmic decrement, and conditions for underdamped, critically damped, or overdamped behavior.","sort":698,"aliases":[]},{"id":700,"code":"19.06.02","level":3,"parent_id":698,"name":"LC oscillations: energy exchange and frequency","description":"Questions on the frequency of LC oscillations, f = 1/(2π√(LC)), and the periodic exchange of electric field energy in the capacitor with magnetic field energy in the inductor.","sort":699,"aliases":[]},{"id":701,"code":"20","level":1,"parent_id":null,"name":"Electromagnetic Waves","description":"JEE chapter 20: Electromagnetic Waves. Concepts: Nature and properties of electromagnetic waves; EM wave energy, intensity and radiation pressure; Electromagnetic spectrum: bands, sources and uses; Displacement Current and Maxwell's Equations.","sort":700,"aliases":[]},{"id":702,"code":"20.01","level":2,"parent_id":701,"name":"Nature and properties of electromagnetic waves","description":"Parent node for EM waves: transverse nature with E ⊥ B ⊥ propagation, E₀/B₀ = c, speed c = 1/√(μ₀ε₀), intensity I = ½ε₀E₀²c, radiation pressure, and questions identifying propagation direction from given E and B field vectors or computing field amplitudes from intensity.","sort":701,"aliases":["EM wave characteristics","EM waves as a consequence of Maxwell's equations","c = 1/sqrt(mu0 eps0)","ratio E0/B0 = c"]},{"id":703,"code":"20.01.01","level":3,"parent_id":702,"name":"Electric-dipole radiation from an accelerated charge","description":"Radiation from accelerated charges and oscillating electric dipoles: radiated power ∝ q²a² (Larmor-type scaling), dipole pattern ∝ sin²θ with no radiation along the dipole axis, and questions on why accelerating charges radiate or how antennas emit.","sort":702,"aliases":[]},{"id":704,"code":"20.01.02","level":3,"parent_id":702,"name":"Nature of electromagnetic waves","description":"Basic EM wave nature questions: transverse waves needing no medium, E and B in phase with E₀ = cB₀, direction of propagation from E × B, computing B₀ from E₀ or vice versa, energy density u = ε₀E², and Poynting vector/intensity of sunlight.","sort":703,"aliases":[]},{"id":705,"code":"20.01.03","level":3,"parent_id":702,"name":"Standing electromagnetic wave","description":"Standing EM waves from superposition of incident and reflected waves: E and B nodes displaced by λ/4, E-node/B-antinode at a conducting surface, nodal spacing λ/2, and energy oscillating between electric and magnetic forms; questions locate nodes/antinodes or write the standing-wave field expressions.","sort":704,"aliases":[]},{"id":706,"code":"20.01.04","level":3,"parent_id":702,"name":"Phase velocity, group velocity and dispersion of EM waves","description":"Phase velocity v_p = ω/k vs group velocity v_g = dω/dk, with v_p·v_g = c² for de Broglie/typical dispersive media; questions ask which velocity carries signal/energy, compute one from the other, or analyze dispersion from n(λ) data.","sort":705,"aliases":[]},{"id":707,"code":"20.01.05","level":3,"parent_id":702,"name":"Reflection and transmission of EM waves at a dielectric boundary","description":"Normal-incidence reflection/transmission of EM waves at a dielectric interface: amplitude coefficients r = (n₁−n₂)/(n₁+n₂), t = 2n₁/(n₁+n₂), reflectance R = ((n₁−n₂)/(n₁+n₂))², and the π phase flip on reflection from a denser medium; questions compute reflected/transmitted E amplitudes or fractions of intensity.","sort":706,"aliases":[]},{"id":708,"code":"20.01.06","level":3,"parent_id":702,"name":"EM wave propagation in conductors and plasmas (complex refractive index, skin depth)","description":"EM waves in conductors and plasmas: skin depth δ = √(2/(μσω)), complex/attenuated refractive index, plasma frequency ω_p with reflection of waves below ω_p (ionosphere, metals); questions compute penetration depth in copper or cutoff behavior in plasma.","sort":707,"aliases":[]},{"id":709,"code":"20.01.07","level":3,"parent_id":702,"name":"Doppler shift for light (relativistic Doppler effect)","description":"Relativistic Doppler effect for light: ν' = ν√((1−β)/(1+β)) for recession (redshift z = Δλ/λ), blueshift for approach, and transverse Doppler effect; typical questions find a galaxy's recession speed from redshift or a source speed from observed frequency shift.","sort":708,"aliases":[]},{"id":710,"code":"20.01.08","level":3,"parent_id":702,"name":"Aberration of light: stellar aberration and relativistic aberration","description":"Aberration of light: stellar aberration from Earth's orbital motion (~20.5″, rain-and-umbrella analogy, tanθ ≈ v/c), and relativistic aberration cosθ' = (cosθ − β)/(1 − β cosθ) including the relativistic headlight effect for beams from moving sources.","sort":709,"aliases":[]},{"id":711,"code":"20.01.09","level":3,"parent_id":702,"name":"Lorentz transformation of electric and magnetic fields and field invariants","description":"Lorentz transformation of E and B fields between inertial frames and the invariants E² − c²B² and E·B; questions transform the field of a moving charge, show a pure E field becomes E + B in another frame, or use invariants to classify fields.","sort":710,"aliases":[]},{"id":712,"code":"20.01.10","level":3,"parent_id":702,"name":"Fresnel drag of light in a moving medium","description":"Fresnel (Fizeau) drag: speed of light in a medium moving at v is c/n + v(1 − 1/n²), with drag coefficient f = 1 − 1/n²; questions compute the measured speed in flowing water and connect to the Fizeau experiment as evidence for relativistic velocity addition.","sort":711,"aliases":[]},{"id":713,"code":"20.01.11","level":3,"parent_id":702,"name":"Cherenkov (Vavilov-Cherenkov) radiation","description":"Cherenkov radiation emitted when a charged particle's speed exceeds c/n: threshold condition v > c/n, cone angle cosθ = c/(nv) = 1/(nβ), and the blue glow in reactor pools; questions find threshold energy or emission angle in water/glass.","sort":712,"aliases":[]},{"id":714,"code":"20.01.12","level":3,"parent_id":702,"name":"Smith-Purcell radiation","description":"Smith–Purcell radiation: an electron grazing a diffraction grating emits light with wavelength λ = (d/m)(c/v − cosθ), giving a tunable, grating-period-dependent spectrum; questions compute emitted wavelength, diffraction order, or electron speed from observed radiation.","sort":713,"aliases":[]},{"id":715,"code":"20.02","level":2,"parent_id":701,"name":"EM wave energy, intensity and radiation pressure","description":"Umbrella node for questions on energy carried by EM waves: computing intensity from field amplitudes, energy density, Poynting vector flux, and force/pressure exerted by light on surfaces.","sort":714,"aliases":["Poynting vector","intensity of electromagnetic waves","radiation pressure","photon momentum","energy density of EM waves"]},{"id":716,"code":"20.02.01","level":3,"parent_id":715,"name":"Poynting vector and EM-wave intensity","description":"Questions using S = (1/μ0) E×B, average intensity I = E0²/(2μ0c) = (1/2)ε0 E0²c, energy density u = ε0E², and I = uc. Typical tasks: given E0 or B0 find intensity, energy crossing an area per second, ratio E0/B0 = c, or instantaneous vs average Poynting flux.","sort":715,"aliases":[]},{"id":717,"code":"20.02.02","level":3,"parent_id":715,"name":"Radiation pressure and photon momentum","description":"Questions on radiation pressure P = I/c for absorption and P = 2I/c for perfect reflection, photon momentum p = E/c = h/λ, and force F = PA on sails/mirrors. Archetypes: laser beam pushing a mirror, solar sail acceleration, fraction of light absorbed vs reflected by a surface.","sort":716,"aliases":[]},{"id":718,"code":"20.02.03","level":3,"parent_id":715,"name":"Spin angular momentum of circularly polarized light","description":"Questions on angular momentum carried by circularly polarized light: each photon carries spin ±ħ, so a beam of intensity I delivers angular momentum flux I/ω per unit area. Typical task: torque exerted on a half-wave plate or absorbing target, number of photons needed to spin up a disk.","sort":717,"aliases":[]},{"id":719,"code":"20.03","level":2,"parent_id":701,"name":"Electromagnetic spectrum: bands, sources and uses","description":"Umbrella node for the electromagnetic spectrum: ordering bands by frequency/wavelength, identifying sources of each band, and matching bands to applications.","sort":718,"aliases":["EM spectrum bands and uses"]},{"id":720,"code":"20.03.01","level":3,"parent_id":719,"name":"Electromagnetic spectrum and uses","description":"Questions asking to arrange γ-rays, X-rays, UV, visible, IR, microwaves, radio waves in order of frequency or wavelength, identify a band from a given wavelength (e.g. 1 Å, 500 nm, 21 cm), or match uses like remote control, sterilization, radar, radiotherapy to the correct band.","sort":719,"aliases":[]},{"id":721,"code":"20.03.02","level":3,"parent_id":719,"name":"Radio waves and microwaves: production and applications","description":"Questions on radio waves and microwaves: production by oscillating LC circuits and accelerated charges, wavelength ranges (mm to km), and applications such as radio/TV communication, radar, microwave ovens (resonance with water molecules), and line-of-sight vs sky-wave propagation.","sort":720,"aliases":[]},{"id":722,"code":"20.03.03","level":3,"parent_id":719,"name":"Infrared radiation: production and applications","description":"Questions on infrared radiation: emitted by hot bodies and molecular vibrations, felt as heat, and applications like night-vision devices, remote controls, infrared photography, greenhouses, and physical therapy. Often paired with Wien's-type reasoning about warm objects radiating IR.","sort":721,"aliases":[]},{"id":723,"code":"20.03.04","level":3,"parent_id":719,"name":"Ultraviolet radiation: production and applications","description":"Questions on ultraviolet radiation: produced by very hot bodies (Sun, special lamps, gas discharges), ionizing enough to cause sunburn and fluorescence, absorbed by ozone, and used for sterilization, water purification, and detecting forged currency.","sort":722,"aliases":[]},{"id":724,"code":"20.03.05","level":3,"parent_id":719,"name":"X-rays: production and applications","description":"Questions on X-rays: production when fast electrons decelerate in a metal target (Coolidge tube), continuous spectrum with cutoff λmin = hc/eV, characteristic Kα/Kβ lines from electron transitions, Moseley-type reasoning, and medical/industrial imaging uses.","sort":723,"aliases":[]},{"id":725,"code":"20.03.06","level":3,"parent_id":719,"name":"Gamma rays: production and applications","description":"Questions on gamma rays: emitted by nuclear transitions and radioactive decay, shortest wavelengths and highest photon energies, greatest penetrating power, and uses in cancer radiotherapy, sterilizing medical equipment, and detecting nuclear reactions.","sort":724,"aliases":[]},{"id":726,"code":"20.03.07","level":3,"parent_id":719,"name":"Spectral line broadening: natural linewidth and Doppler broadening","description":"Questions on why spectral lines have finite width: natural broadening from the uncertainty principle ΔE·Δt ≈ ħ giving Δν ≈ 1/(2πτ) for excited-state lifetime τ, and Doppler broadening Δν/ν ≈ v/c from thermal motion of atoms. Typical tasks: compute linewidth from lifetime or from gas temperature.","sort":725,"aliases":[]},{"id":727,"code":"20.04","level":2,"parent_id":701,"name":"Displacement Current and Maxwell's Equations","description":"Umbrella node for the displacement current concept and Maxwell's four equations, including how Maxwell's addition of displacement current leads to EM waves.","sort":726,"aliases":["conduction vs displacement current","Ampere-Maxwell law"]},{"id":728,"code":"20.04.01","level":3,"parent_id":727,"name":"Displacement current","description":"Questions using displacement current Id = ε0 dΦE/dt and the Ampere–Maxwell law ∮B·dl = μ0(I + Id). Archetypes: parallel-plate capacitor being charged, find B between the plates at radius r (B ∝ r inside, B ∝ 1/r outside), show Id equals conduction current, and continuity of current around a circuit.","sort":727,"aliases":[]},{"id":729,"code":"20.04.02","level":3,"parent_id":727,"name":"Maxwell's equations (qualitative)","description":"Qualitative questions on Maxwell's equations: Gauss's law for E, ∮B·dl = 0 (no magnetic monopoles), Faraday's law of induction, and Ampere–Maxwell law. Tasks: identify which equation expresses absence of monopoles or symmetry of changing E and B, and derive/explain wave speed c = 1/√(μ0ε0).","sort":728,"aliases":[]},{"id":730,"code":"21","level":1,"parent_id":null,"name":"Ray Optics & Optical Instruments","description":"JEE chapter 21: Ray Optics & Optical Instruments. Concepts: Refraction and Total Internal Reflection; Lenses and Refraction at Spherical Surfaces; Prism and dispersion; Optical instruments (eye, microscope, telescope); Reflection and spherical mirrors; Rectilinear Propagation of Light.","sort":729,"aliases":["Ray Optics and Optical Instruments"]},{"id":731,"code":"21.01","level":2,"parent_id":730,"name":"Refraction and Total Internal Reflection","description":"Questions on Snell's law, critical angle, total internal reflection, and refraction at plane/spherical boundaries.","sort":730,"aliases":["TIR","Snell's law and total internal reflection","apparent depth and real depth","glass slab lateral shift","critical angle","optical fibre"]},{"id":732,"code":"21.01.01","level":3,"parent_id":731,"name":"Apparent depth with varying refractive index","description":"Questions on apparent depth when refractive index varies continuously with depth, using integration of n(y) to find image position.","sort":731,"aliases":[]},{"id":733,"code":"21.01.02","level":3,"parent_id":731,"name":"Atmospheric refraction and mirages","description":"Questions on atmospheric refraction, looming, mirage formation, and apparent position of celestial objects due to varying air refractive index.","sort":732,"aliases":[]},{"id":734,"code":"21.01.03","level":3,"parent_id":731,"name":"Optical-fibre bend radius for total internal reflection","description":"Questions on minimum bend radius of an optical fibre so that the angle of incidence at the core-cladding interface exceeds the critical angle.","sort":733,"aliases":[]},{"id":735,"code":"21.01.04","level":3,"parent_id":731,"name":"Ray bending in a spherical graded-index medium","description":"Questions on ray paths in a spherically symmetric graded-index medium, using Snell's law in spherical coordinates to find trajectories.","sort":734,"aliases":[]},{"id":736,"code":"21.01.05","level":3,"parent_id":731,"name":"Astronomical refraction","description":"Questions on astronomical refraction, including apparent elevation of stars, flattening of the sun's disc, and refraction angle near the horizon.","sort":735,"aliases":[]},{"id":737,"code":"21.01.06","level":3,"parent_id":731,"name":"Ray trajectories in a graded-index medium","description":"Questions on determining ray trajectories y(x) or r(θ) in media with refractive index varying with position, often via the ray equation or Snell's law.","sort":736,"aliases":[]},{"id":738,"code":"21.01.07","level":3,"parent_id":731,"name":"GRIN slab as a thin lens","description":"Questions on treating a GRIN slab as a thin lens, finding its focal length from the quadratic refractive index profile and slab thickness.","sort":737,"aliases":[]},{"id":739,"code":"21.01.08","level":3,"parent_id":731,"name":"Ray trajectory in a planar graded-index medium","description":"Questions on ray trajectory inside a planar graded-index medium, typically solving for sinusoidal or curved paths from n(y) profiles.","sort":738,"aliases":[]},{"id":740,"code":"21.01.09","level":3,"parent_id":731,"name":"Ray bending in a planar graded-index medium","description":"Questions on how a ray bends continuously in a planar medium with n varying perpendicular to the interface, including angle of deviation.","sort":739,"aliases":[]},{"id":741,"code":"21.01.10","level":3,"parent_id":731,"name":"Apparent depth and normal shift","description":"Questions on apparent depth, normal shift, and lateral shift for objects viewed through parallel or single refracting surfaces.","sort":740,"aliases":[]},{"id":742,"code":"21.01.11","level":3,"parent_id":731,"name":"Total internal reflection and optical fibres","description":"Questions on total internal reflection in optical fibres, including acceptance angle, numerical aperture, and critical angle at core-cladding boundary.","sort":741,"aliases":[]},{"id":743,"code":"21.01.12","level":3,"parent_id":731,"name":"Ray trajectories in graded-index media","description":"Questions on ray paths in graded-index media, including periodic focusing, sinusoidal trajectories, and comparison with step-index fibres.","sort":742,"aliases":[]},{"id":744,"code":"21.02","level":2,"parent_id":730,"name":"Lenses and Refraction at Spherical Surfaces","description":"Questions on refraction at spherical surfaces, thin lens formula, lens maker's formula, and image formation by lenses.","sort":743,"aliases":["Lens formula","Refraction at spherical surfaces","power of a lens","combination of thin lenses","silvered lens (lens-mirror combination)","displacement method","lens maker's formula"]},{"id":745,"code":"21.02.01","level":3,"parent_id":744,"name":"Refraction at spherical surfaces and lenses","description":"Questions combining refraction at spherical surfaces with thin lens equations, including sign conventions and image formation.","sort":744,"aliases":[]},{"id":746,"code":"21.02.02","level":3,"parent_id":744,"name":"Image velocity through a thin lens","description":"Questions on velocity of image when object or lens moves, using differentiation of the thin lens formula to relate image and object velocities.","sort":745,"aliases":[]},{"id":747,"code":"21.02.03","level":3,"parent_id":744,"name":"Refraction at spherical surfaces","description":"Questions on refraction at a single spherical surface using n₂/v - n₁/u = (n₂-n₁)/R, including real and virtual image cases.","sort":746,"aliases":[]},{"id":748,"code":"21.02.04","level":3,"parent_id":744,"name":"Thin lens formula and magnification","description":"Questions on 1/v - 1/u = 1/f, magnification m = v/u, and related numerical problems for thin lenses.","sort":747,"aliases":[]},{"id":749,"code":"21.02.05","level":3,"parent_id":744,"name":"Spherical and chromatic aberrations","description":"Questions on image defects in lenses, including spherical aberration from marginal rays and chromatic aberration due to wavelength-dependent focal length, often asking for causes, corrections, or comparisons.","sort":748,"aliases":[]},{"id":750,"code":"21.03","level":2,"parent_id":730,"name":"Prism and dispersion","description":"Questions involving refraction through a prism, angle of minimum deviation, dispersive power, and the splitting of white light into a spectrum.","sort":749,"aliases":["prisms","angle of minimum deviation","dispersion by a prism","angular dispersion and dispersive power"]},{"id":751,"code":"21.03.01","level":3,"parent_id":750,"name":"Primary and secondary rainbow formation","description":"Questions on how sunlight undergoes refraction, total internal reflection, and dispersion inside water droplets to form primary and secondary rainbows, including angle and color-order details.","sort":750,"aliases":[]},{"id":752,"code":"21.03.02","level":3,"parent_id":750,"name":"Prism refraction and dispersion relations","description":"Questions requiring prism formulas such as δ = i + e - A, minimum deviation relation n = sin((A+δm)/2)/sin(A/2), and calculations of angular dispersion or dispersive power.","sort":751,"aliases":[]},{"id":753,"code":"21.03.03","level":3,"parent_id":750,"name":"Minimum deviation and thin prism approximation","description":"Questions on prism minimum deviation condition, derivation of δ_min = 2i - A, and using the thin prism approximation δ = (μ - 1)A for small apex angles.","sort":752,"aliases":[]},{"id":754,"code":"21.03.04","level":3,"parent_id":750,"name":"Dispersion and angular dispersion","description":"Questions on splitting of white light by a prism, angular dispersion δ_v - δ_r, dispersive power, and combination of prisms for deviation without dispersion.","sort":753,"aliases":[]},{"id":755,"code":"21.04","level":2,"parent_id":730,"name":"Optical instruments (eye, microscope, telescope)","description":"Questions on image formation, magnification, and resolving power of the human eye, simple microscope, compound microscope, and astronomical telescope.","sort":754,"aliases":["optical instruments","simple microscope","compound microscope","astronomical telescope","magnifying power","defects of vision"]},{"id":756,"code":"21.04.01","level":3,"parent_id":755,"name":"Optical instruments (microscope, telescope, eye)","description":"Questions on ray diagrams, tube length, magnification formulas, and focal length combinations for microscopes and telescopes.","sort":755,"aliases":[]},{"id":757,"code":"21.04.02","level":3,"parent_id":755,"name":"Simple microscope and magnifying power","description":"Questions on a convex lens used as a simple microscope, magnifying power M = D/f or 1 + D/f, and image at near point or infinity.","sort":756,"aliases":[]},{"id":758,"code":"21.04.03","level":3,"parent_id":755,"name":"Compound microscope","description":"Questions on compound microscope construction, magnification M = (L/f_o)(D/f_e), tube length, and objective/eyepiece focal lengths.","sort":757,"aliases":[]},{"id":759,"code":"21.04.04","level":3,"parent_id":755,"name":"Astronomical and reflecting telescopes","description":"Questions on astronomical telescope magnification M = f_o/f_e, tube length f_o + f_e, and reflecting telescope advantages over refracting.","sort":758,"aliases":[]},{"id":760,"code":"21.04.05","level":3,"parent_id":755,"name":"Human eye and vision defects","description":"Questions on eye lens accommodation, near point, far point, myopia, hypermetropia, presbyopia, and corrective lens power calculations.","sort":759,"aliases":[]},{"id":761,"code":"21.05","level":2,"parent_id":730,"name":"Reflection and spherical mirrors","description":"Questions on laws of reflection, image formation by plane and spherical mirrors, sign conventions, and mirror formula applications.","sort":760,"aliases":["reflection","plane mirror","spherical mirrors","mirror formula","magnification"]},{"id":762,"code":"21.05.01","level":3,"parent_id":761,"name":"Multiple-reflection transmission through a parallel plate","description":"Questions on lateral shift of a ray passing through a glass slab or parallel plate, shift = t(1 - 1/μ), and multiple refractions.","sort":761,"aliases":[]},{"id":763,"code":"21.05.02","level":3,"parent_id":761,"name":"Marginal rays in a wide-aperture spherical mirror","description":"Questions on spherical aberration for wide-aperture mirrors, marginal vs paraxial ray focus difference, and caustic curve formation.","sort":762,"aliases":[]},{"id":764,"code":"21.05.03","level":3,"parent_id":761,"name":"Multiple reflections in inclined mirrors","description":"Questions on number of images formed by two inclined plane mirrors, formula n = 360/θ - 1, and image positions for symmetric/asymmetric object placement.","sort":763,"aliases":[]},{"id":765,"code":"21.05.04","level":3,"parent_id":761,"name":"Longitudinal magnification for an axial extended object","description":"Questions on longitudinal magnification m_L = -m^2 for axial extended objects in spherical mirrors, and relation between object length and image length.","sort":764,"aliases":[]},{"id":766,"code":"21.05.05","level":3,"parent_id":761,"name":"Parabolic mirror focusing without spherical aberration","description":"Questions on parabolic mirror property of focusing all parallel rays to a single focus, absence of spherical aberration, and applications in telescopes.","sort":765,"aliases":[]},{"id":767,"code":"21.05.06","level":3,"parent_id":761,"name":"Image speed for a moving object or mirror","description":"Questions on velocity of image when object or mirror moves, using differentiation of mirror formula, and relative speed calculations.","sort":766,"aliases":[]},{"id":768,"code":"21.05.07","level":3,"parent_id":761,"name":"Reflection at plane and spherical mirrors","description":"Questions on basic reflection at plane and spherical mirrors, image characteristics (real/virtual, erect/inverted, magnified/diminished), and ray tracing.","sort":767,"aliases":[]},{"id":769,"code":"21.05.08","level":3,"parent_id":761,"name":"Mirror formula and magnification","description":"Questions on mirror formula 1/f = 1/v + 1/u, magnification m = -v/u, and numerical problems involving object/image distances and focal length.","sort":768,"aliases":[]},{"id":770,"code":"21.06","level":2,"parent_id":730,"name":"Rectilinear Propagation of Light","description":"Questions on light traveling in straight lines in homogeneous media, formation of shadows, pinhole images, and eclipses.","sort":769,"aliases":["Shadows and pinhole imaging","Rectilinear propagation"]},{"id":771,"code":"21.06.01","level":3,"parent_id":770,"name":"Rectilinear light propagation and pinhole imaging","description":"Questions on pinhole camera image formation, image size calculation using similar triangles, and effect of pinhole size on image sharpness.","sort":770,"aliases":[]},{"id":772,"code":"21.06.02","level":3,"parent_id":770,"name":"Rectilinear propagation of light and shadow formation","description":"Questions on umbra and penumbra formation, shadow size calculations for extended sources, and conditions for total/partial shadow.","sort":771,"aliases":[]},{"id":773,"code":"21.06.03","level":3,"parent_id":770,"name":"Rectilinear propagation and pinhole imaging","description":"Questions on rectilinear propagation demonstrated through pinhole imaging, image inversion, and magnification in pinhole cameras.","sort":772,"aliases":[]},{"id":774,"code":"21.06.04","level":3,"parent_id":770,"name":"Shadow formation and eclipses","description":"Questions on solar and lunar eclipses, umbra/penumbra geometry, and angular size calculations for shadow formation.","sort":773,"aliases":[]},{"id":775,"code":"21.06.05","level":3,"parent_id":770,"name":"Opaque-body colour from absorption spectrum","description":"Questions on why an opaque body appears a certain color based on which wavelengths it reflects vs absorbs, and relation to absorption spectrum.","sort":774,"aliases":[]},{"id":776,"code":"21.06.06","level":3,"parent_id":770,"name":"Overlapping absorptive colour filters","description":"Questions on resultant color when light passes through multiple colored filters, subtractive color mixing, and transmitted wavelength determination.","sort":775,"aliases":[]},{"id":777,"code":"21.06.07","level":3,"parent_id":770,"name":"Photometry: luminance and Lambert law","description":"Questions on luminous intensity, illuminance, Lambert's cosine law E = I cosθ/r², and photometry calculations for point sources.","sort":776,"aliases":[]},{"id":778,"code":"21.06.08","level":3,"parent_id":770,"name":"Photometry: luminous vs radiant flux","description":"Questions comparing luminous flux (perceived brightness in lumens) and radiant flux (total emitted power in watts), often involving luminous efficiency or conversion between the two for a source.","sort":777,"aliases":[]},{"id":779,"code":"21.06.09","level":3,"parent_id":770,"name":"Radiant intensity and inverse-square photometry","description":"Questions applying radiant intensity (power per steradian) and the inverse-square law to calculate irradiance or illuminance at a distance from a point source.","sort":778,"aliases":[]},{"id":780,"code":"22","level":1,"parent_id":null,"name":"Wave Optics","description":"JEE chapter 22: Wave Optics. Concepts: Young's double slit experiment (YDSE); Wave Optics Basics and Huygens' Principle; Diffraction of Light; Interference of light: coherence and thin films; Polarisation of Light.","sort":779,"aliases":[]},{"id":781,"code":"22.01","level":2,"parent_id":780,"name":"Young's double slit experiment (YDSE)","description":"YDSE numericals: fringe width β = λD/d, bright/dark fringe positions from path difference d·y/D = nλ or (2n−1)λ/2, intensity I = 4I0cos²(φ/2) and I_max/I_min = ((√I1+√I2)/(√I1−√I2))², fringe shifts from a thin mica/slab (shift = (μ−1)tD/d), YDSE in a medium, white-light fringes, and effect of changing d, D, or λ.","sort":780,"aliases":["YDSE","young's double slit","fringe width","path difference","interference fringes"]},{"id":782,"code":"22.01.01","level":3,"parent_id":781,"name":"Interference and Young's double-slit experiment","description":"Questions on two-slit interference geometry: path difference Δ = d·sinθ ≈ yd/D, conditions y = nλD/d (bright) and y = (2n−1)λD/2 (dark), phase difference δ = 2πΔ/λ, and intensity I = 4I₀cos²(δ/2). Typical setups give slit separation d, screen distance D and ask for fringe positions, order n, or point brightness.","sort":781,"aliases":[]},{"id":783,"code":"22.01.02","level":3,"parent_id":781,"name":"YDSE fringe width and intensity distribution","description":"Numericals on fringe width β = λD/d and intensity distribution: how β or Imax/Imin changes when d, D, λ, or slit widths change, ratio of intensities at bright/dark fringes from amplitude ratio, and graph-reading questions on the I vs y cos² pattern.","sort":782,"aliases":[]},{"id":784,"code":"22.01.03","level":3,"parent_id":781,"name":"YDSE with mica sheets and path difference changes","description":"A thin sheet (mica, glass) of thickness t and refractive index μ placed over one slit adds path (μ−1)t, shifting the whole pattern by Δy = (μ−1)tD/d with fringe width unchanged. Questions ask number of fringes shifted n = (μ−1)t/λ, central fringe displacement, or t/μ from observed shift; variants include filling one slit region with water or a wedge.","sort":783,"aliases":[]},{"id":785,"code":"22.01.04","level":3,"parent_id":781,"name":"Lloyd's mirror and white light interference","description":"Lloyd's single-mirror setup: slit and its virtual image act as coherent sources with a π phase flip, so the fringe system is complementary — a dark fringe at the mirror edge and shifted bright/dark positions. Also white-light YDSE: central white fringe with a few colored fringes on either side, asking which color vanishes first or fringe overlap counts.","sort":784,"aliases":[]},{"id":786,"code":"22.01.05","level":3,"parent_id":781,"name":"YDSE in media and wavelength dependence","description":"YDSE performed inside a medium (e.g., apparatus immersed in water, λ → λ/μ) or with different wavelengths/colors: fringe width β = λD/d scales with λ, so patterns spread or compress, orders of different colors coincide at y = n₁λ₁D/d = n₂λ₂D/d. Questions compute new fringe width, coincidence orders, or wavelength from measured fringe spacing.","sort":785,"aliases":[]},{"id":787,"code":"22.02","level":2,"parent_id":780,"name":"Wave Optics Basics and Huygens' Principle","description":"Umbrella node for wave-optics fundamentals: wavefronts and Huygens' construction, superposition and coherence, laws of reflection/refraction from wave theory, validity of ray optics, and historic speed-of-light measurements. Tag here only for general/basic wave-optics questions not fitting a child node.","sort":786,"aliases":["wavefront","Huygens principle","reflection and refraction of wavefronts"]},{"id":788,"code":"22.02.01","level":3,"parent_id":787,"name":"Huygens' principle","description":"Conceptual questions on Huygens' principle itself: every point on a wavefront acts as a secondary source emitting spherical wavelets, the forward envelope is the new wavefront, and why backward wavelets are ignored. Includes statements/assumptions of the principle and its use to explain rectilinear propagation.","sort":787,"aliases":[]},{"id":789,"code":"22.02.02","level":3,"parent_id":787,"name":"Wavefronts and Huygens' principle","description":"Questions on wavefront shapes and propagation: spherical wavefront from a point source, cylindrical from a line source, plane far away; shape of wavefront after reflection from a mirror, refraction at a surface, or passage through a lens/prism; relation between wavefront and rays (always perpendicular).","sort":788,"aliases":[]},{"id":790,"code":"22.02.03","level":3,"parent_id":787,"name":"Reflection and refraction via Huygens' construction","description":"Derivation-style and numerical questions deriving laws of reflection and Snell's law sin i/sin r = v₁/v₂ = μ from Huygens' construction, including time taken across a slab, bending of light entering a denser/rarer medium, and frequency staying fixed while speed and wavelength change.","sort":789,"aliases":[]},{"id":791,"code":"22.02.04","level":3,"parent_id":787,"name":"Validity of ray optics and diffraction scale","description":"When ray optics fails: apertures/obstacles of size a comparable to λ show diffraction; Fresnel distance z_F = a²/λ marks the limit, and for z ≪ z_F ray optics is valid. Questions compute z_F for a given aperture/wavelength or the largest aperture for which straight-line propagation holds.","sort":790,"aliases":[]},{"id":792,"code":"22.02.05","level":3,"parent_id":787,"name":"Speed of light measurement (Fizeau's toothed wheel)","description":"Fizeau's toothed-wheel experiment: light through a gap travels distance d to a mirror and back; if the wheel (N teeth, n rotations/s) turns one tooth-to-gap spacing in time 2d/c, then c = 4dNn. Questions give wheel data (N, rps at first eclipse/reappearance, d) and ask for c, or reverse-calculate a parameter.","sort":791,"aliases":[]},{"id":793,"code":"22.03","level":2,"parent_id":780,"name":"Diffraction of Light","description":"Umbrella node for diffraction of light: single-slit patterns, diffraction gratings, Fresnel zones and straight-edge diffraction, resolving power, and X-ray/Bragg diffraction. Tag here for general diffraction questions not matching a child node.","sort":792,"aliases":["single slit diffraction","resolving power","diffraction grating"]},{"id":794,"code":"22.03.01","level":3,"parent_id":793,"name":"Acousto-optic (Debye-Sears) diffraction","description":"Debye–Sears (acousto-optic) effect: ultrasonic waves of wavelength λ_s in a liquid act as a grating of spacing λ_s, diffracting light at sinθ = nλ/λ_s. Questions use measured diffraction angle to find sound velocity v = fλ_s or the ultrasonic frequency.","sort":793,"aliases":[]},{"id":795,"code":"22.03.02","level":3,"parent_id":793,"name":"Debye-Scherrer powder diffraction","description":"Debye–Scherrer powder method: monochromatic X-rays on a polycrystalline powder produce concentric diffraction cones/rings on film at angles satisfying Bragg's law; questions extract interplanar spacing d from ring angles or identify lattice parameters from ring patterns.","sort":794,"aliases":[]},{"id":796,"code":"22.03.03","level":3,"parent_id":793,"name":"Diffraction at a single slit","description":"Single-slit Fraunhofer diffraction: minima at a·sinθ = nλ, secondary maxima at roughly a·sinθ = (2n+1)λ/2, central maximum angular width 2λ/a and linear width 2λD/a, intensity ∝ (sinα/α)² with central intensity I₀. Questions ask angular spread, effect of narrowing the slit or increasing λ, or widths of secondary maxima.","sort":795,"aliases":[]},{"id":797,"code":"22.03.04","level":3,"parent_id":793,"name":"Fresnel diffraction at a straight edge","description":"Fresnel diffraction at a straight edge: intensity falls gradually into the geometric shadow (never exactly zero) and a few alternating maxima/minima appear in the illuminated region, with the first maximum just outside the edge. Questions on the nature of bands, position of first bright band, and contrast with Fraunhofer fringes.","sort":796,"aliases":[]},{"id":798,"code":"22.03.05","level":3,"parent_id":793,"name":"Fresnel zones: radii and aperture diffraction","description":"Fresnel half-period zones: radius of the nth zone rₙ = √(nλb) (b = distance from zone plate/aperture to point), total zones in an aperture of radius a, and zone-plate focusing with focal length f = rₙ²/(nλ). Questions count zones exposed by a given aperture and the resulting intensity at the axial point.","sort":797,"aliases":[]},{"id":799,"code":"22.03.06","level":3,"parent_id":793,"name":"Polarization as a coherence gate in diffraction","description":"Questions where polarizers/analyzer placement before slits or an aperture controls the visibility or intensity of an interference/diffraction pattern — e.g., crossed polarizers over YDSE slits killing fringes, or polarizing filters gating which coherent components diffract. Treats polarization state as a switch on coherence.","sort":798,"aliases":[]},{"id":800,"code":"22.03.07","level":3,"parent_id":793,"name":"X-ray diffraction by an orthorhombic crystal","description":"X-ray diffraction from an orthorhombic crystal: interplanar spacing 1/d² = h²/a² + k²/b² + l²/c² combined with Bragg's law 2d·sinθ = nλ. Questions give lattice constants (a ≠ b ≠ c) and Miller indices, asking which planes diffract a given wavelength at a given angle.","sort":799,"aliases":[]},{"id":801,"code":"22.03.08","level":3,"parent_id":793,"name":"Bragg reflection","description":"Core Bragg reflection concept: constructive reflection from parallel crystal planes when 2d·sinθ = nλ, with θ the glancing angle and n the order. Questions check the condition itself, missing orders, or how the diffraction angle shifts with λ, d, or order.","sort":800,"aliases":[]},{"id":802,"code":"22.03.09","level":3,"parent_id":793,"name":"Bragg reflection from crystal planes","description":"Applied Bragg-law numericals on crystal planes: given successive-order angles θₙ (e.g., first and second order), find wavelength λ and spacing d via 2d·sinθₙ = nλ; identify planes, compute glancing angles for a target order, or determine crystal identity from measured reflections.","sort":801,"aliases":[]},{"id":803,"code":"22.03.10","level":3,"parent_id":793,"name":"Diffraction gratings and multi-slit interference","description":"Diffraction gratings and N-slit interference: principal maxima at d·sinθ = nλ (d = grating element = slit spacing + width), sharp maxima with N−1 minima between them, angular dispersion/width ∝ 1/N, and missing orders when grating maxima coincide with single-slit minima. Questions compute diffraction angles, number of orders visible, or resolving-related spacing.","sort":802,"aliases":[]},{"id":804,"code":"22.03.11","level":3,"parent_id":793,"name":"Fresnel zones: phasor (vibration-curve) method","description":"Vibration-curve (phasor) treatment of Fresnel zones: amplitudes of successive zones form a slowly spiraling phasor chain, giving resultant amplitude ≈ half the first zone's contribution (A ∝ a₁/2) for an unobstructed wavefront, and enhanced amplitude when a zone plate blocks alternate zones. Questions compare amplitudes/intensities with and without obstructions or zone plates.","sort":803,"aliases":[]},{"id":805,"code":"22.03.12","level":3,"parent_id":793,"name":"Resolving power","description":"Resolving power concept: Rayleigh criterion — two point sources are just resolved when the principal maximum of one falls on the first minimum of the other, giving limit of resolution θ_min = 1.22λ/D (circular aperture) or λ/a (slit). Questions on how λ and aperture size affect resolution and on computing minimum resolvable angle or separation.","sort":804,"aliases":[]},{"id":806,"code":"22.03.13","level":3,"parent_id":793,"name":"Resolving power of optical instruments","description":"Resolving power of specific instruments: telescope R = D/1.22λ (aperture-limited, e.g., resolving binary stars or car headlights at distance), microscope with numerical aperture (R = 2μ·sinθ/1.22λ), and the human eye. Numericals ask smallest resolvable separation, required aperture, or maximum distance for resolution.","sort":805,"aliases":[]},{"id":807,"code":"22.03.14","level":3,"parent_id":793,"name":"Fresnel diffraction: straight edge and Fresnel zones","description":"Questions on near-field diffraction by a straight edge (intensity variations inside/outside geometric shadow, fringe spacing near edge) and half-period Fresnel zones (zone radii ∝√n, effect of blocking zones, zone plate focusing).","sort":806,"aliases":[]},{"id":808,"code":"22.03.15","level":3,"parent_id":793,"name":"Bragg reflection and X-ray diffraction","description":"X-ray diffraction from crystal planes using Bragg's law 2d sinθ = nλ; questions give lattice spacing, glancing angle, or wavelength and ask for missing orders, maxima angles, or d of planes.","sort":807,"aliases":[]},{"id":809,"code":"22.04","level":2,"parent_id":780,"name":"Interference of light: coherence and thin films","description":"Parent node for interference requiring coherent sources (path difference < coherence length), superposition intensity I = I₁+I₂+2√(I₁I₂)cosδ, and thin-film interference; use when the question spans several subtopics.","sort":808,"aliases":["coherence","thin film interference","conditions for sustained interference"]},{"id":810,"code":"22.04.01","level":3,"parent_id":809,"name":"Thin-film interference in reflection","description":"Thin-film interference in reflected light: phase change of π on reflection off denser medium, condition 2μt cos r = (n+½)λ or nλ for bright/dark, questions on soap films, oil slicks, wedge-shaped films, and wavelength missing in reflection.","sort":809,"aliases":[]},{"id":811,"code":"22.04.02","level":3,"parent_id":809,"name":"Spatial coherence and Michelson stellar interferometry","description":"Spatial coherence and angular width of sources: visibility of fringes vs source size, Michelson stellar interferometer measuring angular diameter of stars (fringe disappearance when d ≈ λ/θ).","sort":810,"aliases":[]},{"id":812,"code":"22.04.03","level":3,"parent_id":809,"name":"Temporal coherence of a multi-line source","description":"Temporal coherence with sources emitting multiple wavelengths/lines: coherence length c/Δν, beat of fringe systems from different lines, fringe pattern washout at large path difference.","sort":811,"aliases":[]},{"id":813,"code":"22.04.04","level":3,"parent_id":809,"name":"Temporal and spatial coherence","description":"General coherence questions: coherence time and length (τ = λ²/cΔλ), distinction between temporal and spatial coherence, fringe visibility dependence on path difference and source size.","sort":812,"aliases":[]},{"id":814,"code":"22.04.05","level":3,"parent_id":809,"name":"Fabry-Perot etalon and multiple-beam interference","description":"Fabry-Perot etalon: multiple-beam interference with sharp fringes, finesse and resolving power, transmission maxima 2μt cosθ = nλ; questions on fringe sharpness vs two-beam interference.","sort":813,"aliases":[]},{"id":815,"code":"22.04.06","level":3,"parent_id":809,"name":"Holography","description":"Holography: recording interference of object and reference beams to store amplitude and phase, reconstruction producing 3-D image; conceptual questions on why coherent light is needed and what hologram records.","sort":814,"aliases":[]},{"id":816,"code":"22.05","level":2,"parent_id":780,"name":"Polarisation of Light","description":"Parent node for polarisation of transverse waves: polarisers, unpolarised vs polarised light, intensity after polarisers; use when question mixes Malus's law, Brewster angle, and crystal optics.","sort":815,"aliases":["polaroids","Brewster's law","Malus' law","polarized light","scattering of light","polarisation by scattering"]},{"id":817,"code":"22.05.01","level":3,"parent_id":816,"name":"Polarisation (Malus's law, Brewster's angle)","description":"Malus's law I = I₀cos²θ for light through polariser-analyser pairs (including unpolarised input giving I₀/2) and Brewster's angle tanθ_B = μ for polarised reflected light.","sort":816,"aliases":[]},{"id":818,"code":"22.05.02","level":3,"parent_id":816,"name":"Birefringent retarder plates","description":"Quarter-wave and half-wave plates: path/phase difference between ordinary and extraordinary rays (Δ = (μₒ−μₑ)t), converting plane-polarised to circular/elliptical polarisation, thickness calculations.","sort":817,"aliases":[]},{"id":819,"code":"22.05.03","level":3,"parent_id":816,"name":"Double refraction in a uniaxial crystal","description":"Double refraction in calcite/quartz: ordinary and extraordinary rays with different refractive indices and velocities, optic axis behaviour, questions on which ray obeys Snell's law and image doubling.","sort":818,"aliases":[]},{"id":820,"code":"22.05.04","level":3,"parent_id":816,"name":"Optical activity (natural optical rotation)","description":"Optical activity: rotation of plane of polarisation by solutions (specific rotation α = θ/(lc), θ = S·l·c for sugar solutions) and solids; questions on concentration, length, or specific rotation.","sort":819,"aliases":[]},{"id":821,"code":"22.05.05","level":3,"parent_id":816,"name":"Polarization from orthogonal field components","description":"Resultant polarisation state from two orthogonal field components: E = E₁cos(ωt) x̂ + E₂cos(ωt+δ) ŷ, determining linear/circular/elliptical polarisation from amplitude ratio and phase difference δ.","sort":820,"aliases":[]},{"id":822,"code":"22.05.06","level":3,"parent_id":816,"name":"Wollaston prism","description":"Wollaston prism: birefringent prism splitting unpolarised light into two orthogonally polarised beams angularly separated; questions on beam separation angle and construction from calcite/quartz.","sort":821,"aliases":[]},{"id":823,"code":"22.05.07","level":3,"parent_id":816,"name":"Rayleigh scattering by air molecules","description":"Rayleigh scattering: intensity ∝ 1/λ⁴ by particles much smaller than wavelength, explaining blue sky and red sunsets, polarisation of scattered light at 90°.","sort":822,"aliases":[]},{"id":824,"code":"22.05.08","level":3,"parent_id":816,"name":"Retarder plates and Babinet compensator","description":"Retarder plates (quarter/half-wave) and Babinet compensator: variable path difference across a wedge of birefringent material, measuring phase retardation and analysing elliptically polarised light.","sort":823,"aliases":[]},{"id":825,"code":"22.05.09","level":3,"parent_id":816,"name":"Kerr and Faraday magneto/electro-optic effects","description":"Kerr effect (field-induced birefringence, Δn ∝ E², Kerr cell as shutter) and Faraday effect (rotation θ = V·B·l in magneto-optic medium); questions on rotation angle or induced birefringence.","sort":824,"aliases":[]},{"id":826,"code":"22.05.10","level":3,"parent_id":816,"name":"Imperfect polarizers and principal transmittances","description":"Imperfect (partial) polarizers characterized by principal transmittances k₁ and k₂: transmitted intensity for polarised input I = I₀(k₁cos²θ + k₂sin²θ), maximum/minimum transmission, degree of polarisation.","sort":825,"aliases":[]},{"id":827,"code":"23","level":1,"parent_id":null,"name":"Dual Nature of Radiation & Matter","description":"JEE chapter 23: Dual Nature of Radiation & Matter. Concepts: Photoelectric Effect and Particle Nature of Light (Photons); Matter Waves and de Broglie Wavelength; Heisenberg Uncertainty Principle.","sort":826,"aliases":["Dual Nature of Radiation and Matter"]},{"id":828,"code":"23.01","level":2,"parent_id":827,"name":"Photoelectric Effect and Particle Nature of Light (Photons)","description":"Parent node for quantum nature of light: photoelectric effect with Einstein's equation, photon energy and momentum, work function/threshold, stopping potential, and Compton scattering; use for any photon–electron interaction question.","sort":827,"aliases":["photon energy and momentum","Compton effect","particle nature of light: photons and Compton effect","quantum nature of radiation"]},{"id":829,"code":"23.01.01","level":3,"parent_id":828,"name":"Photoelectric effect and Einstein's equation","description":"Einstein's photoelectric equation: K_max = hν − φ, stopping potential eV₀ = hν − φ, straight-line V₀ vs ν graphs whose slope is h/e and intercept −φ/e, saturation current, and effect of changing intensity or frequency on photocurrent.","sort":828,"aliases":[]},{"id":830,"code":"23.01.02","level":3,"parent_id":828,"name":"Contact potential in a vacuum photocell","description":"Contact potential difference in a vacuum photocell: the stopping potential measured includes the cathode–anode contact potential set by their work-function difference, shifting the apparent threshold; questions compare measured vs true stopping potentials or work functions of the two electrodes.","sort":829,"aliases":[]},{"id":831,"code":"23.01.03","level":3,"parent_id":828,"name":"Work function and threshold frequency","description":"Work function and threshold: φ = hν₀ = hc/λ₀, threshold frequency/wavelength of a metal, computing φ from stopping-potential or longest-wavelength data, and comparing emission of two metals for a given light frequency.","sort":830,"aliases":[]},{"id":832,"code":"23.01.04","level":3,"parent_id":828,"name":"Compton effect","description":"Compton scattering: wavelength shift Δλ = (h/m_ec)(1 − cosθ), Compton wavelength 2.43 pm, scattered photon wavelength and recoil electron kinetic energy/angle, maximum shift at θ = 180°, and why shift is independent of incident wavelength and target material.","sort":831,"aliases":[]},{"id":833,"code":"23.01.05","level":3,"parent_id":828,"name":"Particle nature of light (photons)","description":"Photon properties: E = hν = hc/λ, p = h/λ = E/c, number of photons per second from a source of given power/wavelength, radiation pressure and momentum transfer, and energy bookkeeping in photon absorption or emission events.","sort":832,"aliases":[]},{"id":834,"code":"23.01.06","level":3,"parent_id":828,"name":"Experimental observations and laws of the photoelectric effect","description":"Experimental laws of photoelectric emission: instantaneous emission, existence of a threshold frequency, K_max depends on frequency but not intensity, and saturation photocurrent proportional to intensity; questions match observations to wave-theory failures or identify which law a graph/statement illustrates.","sort":833,"aliases":[]},{"id":835,"code":"23.01.07","level":3,"parent_id":828,"name":"Photon flux, intensity and number of photons","description":"Questions computing photon count from intensity: N/sec = P/(hc/λ) = Pλ/(hc), photon flux at distance r from an isotropic source (P/(4πr²)·λ/hc), photons entering an eye or falling per m² per s, or energy of n photons vs a single photon of wavelength λ.","sort":834,"aliases":[]},{"id":836,"code":"23.01.08","level":3,"parent_id":828,"name":"Planck's black-body spectrum and its Rayleigh-Jeans and Wien limits","description":"Questions on Planck's law u(ν,T) = 8πhν³/c³·1/(e^{hν/kT}−1), recovering the Rayleigh–Jeans limit 8πν²kT/c³ at low ν (ultraviolet catastrophe) and Wien's exponential limit at high ν, plus Wien displacement λ_m T = b and shifts of peak wavelength with temperature.","sort":835,"aliases":[]},{"id":837,"code":"23.01.09","level":3,"parent_id":828,"name":"Black-body photon gas: thermodynamics and number-density spectra","description":"Cavity/black-body radiation thermodynamics: Stefan–Boltzmann law P = σAT⁴, energy density u = aT⁴, radiation pressure p = u/3, photon number density ∝ T³, and adiabatic expansion of photon gas (T ∝ 1/R) in cosmology-style problems.","sort":836,"aliases":[]},{"id":838,"code":"23.02","level":2,"parent_id":827,"name":"Matter Waves and de Broglie Wavelength","description":"Parent node for matter waves: any question computing or comparing de Broglie wavelengths λ = h/p = h/√(2mE) for electrons, protons, neutrons, alpha particles, or macroscopic objects, including particles accelerated through potential V (λ = h/√(2meV) ≈ 12.27/√V Å) and thermal particles (λ = h/√(3mkT)).","sort":837,"aliases":["wave nature of matter","de Broglie hypothesis","matter waves","Davisson-Germer experiment"]},{"id":839,"code":"23.02.01","level":3,"parent_id":838,"name":"1D infinite well via de Broglie standing waves","description":"Deriving particle-in-a-box quantization by fitting standing de Broglie waves: n(λ/2) = L gives p = nh/2L and E_n = n²h²/(8mL²); typical questions find ground-state energy of an electron in an atom-sized box or energy spacing between levels.","sort":838,"aliases":[]},{"id":840,"code":"23.02.02","level":3,"parent_id":838,"name":"de Broglie hypothesis","description":"Core de Broglie hypothesis questions: λ = h/mv, wavelength of an electron accelerated through V volts, ratio of wavelengths of particles with same KE or same momentum, electron vs photon wavelength comparison at equal energy, and why macroscopic objects show no wave effects.","sort":839,"aliases":[]},{"id":841,"code":"23.02.03","level":3,"parent_id":838,"name":"Davisson-Germer experiment","description":"Davisson–Germer electron diffraction from a nickel crystal: 54 V electrons giving a 50° scattering peak, combining Bragg's law 2d sinθ = nλ with λ = h/√(2meV) to verify wave nature of electrons; questions ask for the peak angle, accelerating voltage, or crystal spacing.","sort":840,"aliases":[]},{"id":842,"code":"23.02.04","level":3,"parent_id":838,"name":"Matter-wave refraction and electron-optical refractive index","description":"Refraction of de Broglie waves at a potential step or region of changed potential energy: electron-optical refractive index μ = v₂/v₁ = √((E−U₂)/(E−U₁)) or μ = √(1 + eV/E), applying Snell's law to electron beams crossing potential boundaries.","sort":841,"aliases":[]},{"id":843,"code":"23.02.05","level":3,"parent_id":838,"name":"Quantum confinement and quantum-dot fluorescence","description":"Quantum confinement in quantum dots: energy levels scale as 1/L² so smaller dots emit shorter wavelengths (blue shift); questions relate dot size to fluorescence color (e.g., CdSe dots), emission wavelength changes, and confinement energy estimates.","sort":842,"aliases":[]},{"id":844,"code":"23.03","level":2,"parent_id":827,"name":"Heisenberg Uncertainty Principle","description":"Parent node for uncertainty: Δx·Δp ≥ ħ/2 and ΔE·Δt ≥ ħ/2 estimates, e.g., minimum momentum/velocity uncertainty of an electron confined to an atom or a proton in a nucleus, and the classic argument for why electrons cannot exist inside the nucleus.","sort":843,"aliases":["zero-point energy","Uncertainty principle and zero-point energy"]},{"id":845,"code":"23.03.01","level":3,"parent_id":844,"name":"Uncertainty principle and zero-point energy","description":"Using Δp ~ ħ/Δx to estimate minimum (zero-point) kinetic energy of a confined particle: electron in a nucleus or atom (~MeV vs ~eV argument), particle in a box ground-state energy ~ħ²/(2mL²), harmonic oscillator E₀ = ħω/2, and why atoms don't collapse.","sort":844,"aliases":[]},{"id":846,"code":"23.03.02","level":3,"parent_id":844,"name":"Energy-time uncertainty principle","description":"Energy–time uncertainty ΔE·Δt ≥ ħ/2 applied to natural linewidth of excited atomic states (ΔE ≈ ħ/τ), lifetime–energy-spread estimates, short-lived particle mass widths, and order-of-magnitude virtual-process arguments.","sort":845,"aliases":[]},{"id":847,"code":"24","level":1,"parent_id":null,"name":"Atoms & Nuclei","description":"JEE chapter 24: Atoms & Nuclei. Concepts: Hydrogen Spectrum and Spectral Series; Bohr Model of the Hydrogen Atom; Atomic Models: Thomson and Rutherford; X-rays: Production, Spectra, and Moseley's Law; Nuclear Physics: Composition and Basic Properties of Nuclei; Nuclear reactions and Q-value; Radioactivity and Decay Laws; Mass-energy equivalence and nuclear binding energy; Special relativity (non-NCERT): relativistic energy, momentum and kinematics; Nuclear Fission and Fusion.","sort":846,"aliases":["Atoms","Nuclei"]},{"id":848,"code":"24.01","level":2,"parent_id":847,"name":"Hydrogen Spectrum and Spectral Series","description":"Questions on the emission/absorption spectrum of atomic hydrogen, including identifying series, calculating wavelengths, and relating transitions to energy levels.","sort":847,"aliases":["Lyman, Balmer, Paschen, Brackett, Pfund series","Rydberg formula"]},{"id":849,"code":"24.01.01","level":3,"parent_id":848,"name":"Energy levels and spectral series","description":"Questions requiring use of the hydrogen energy-level formula E_n = -13.6/n^2 eV to find transition energies, photon energies, or initial/final quantum numbers.","sort":848,"aliases":[]},{"id":850,"code":"24.01.02","level":3,"parent_id":848,"name":"Rydberg formula and spectral series","description":"Questions applying 1/λ = R(1/n1^2 - 1/n2^2) to compute wavelengths, wavenumbers, or Rydberg constant values for hydrogen and hydrogen-like ions.","sort":849,"aliases":[]},{"id":851,"code":"24.01.03","level":3,"parent_id":848,"name":"Lyman, Balmer, Paschen, Brackett, and Pfund series","description":"Questions asking to identify which spectral series (Lyman, Balmer, Paschen, Brackett, Pfund) a transition belongs to based on n1, or to match series to spectral regions.","sort":850,"aliases":[]},{"id":852,"code":"24.01.04","level":3,"parent_id":848,"name":"Hydrogen spectrum line wavelengths and series limits","description":"Questions involving calculation of specific line wavelengths, series limits (n2→∞), or minimum/maximum wavelengths within a given spectral series.","sort":851,"aliases":[]},{"id":853,"code":"24.01.05","level":3,"parent_id":848,"name":"Ionization energy and ground-state energy of hydrogen","description":"Questions on the ionization energy of hydrogen (13.6 eV), ground-state energy, or energy required to remove an electron from a given excited state.","sort":852,"aliases":[]},{"id":854,"code":"24.02","level":2,"parent_id":847,"name":"Bohr Model of the Hydrogen Atom","description":"Questions on Bohr's postulates, derivation of orbit radius/velocity/energy, and application of Bohr model formulas to hydrogen atom problems.","sort":853,"aliases":["Bohr's postulates","energy levels and radii","de Broglie explanation of Bohr's second postulate"]},{"id":855,"code":"24.02.01","level":3,"parent_id":854,"name":"Bohr model for hydrogen-like ions","description":"Questions applying Bohr model formulas with atomic number Z to He+, Li2+, or other hydrogen-like ions, including scaling of radius, velocity, and energy.","sort":854,"aliases":[]},{"id":856,"code":"24.02.02","level":3,"parent_id":854,"name":"Limitations of Bohr's model","description":"Questions testing understanding of Bohr model failures: inability to explain fine structure, multi-electron atoms, relative intensities, or wave nature of electrons.","sort":855,"aliases":[]},{"id":857,"code":"24.02.03","level":3,"parent_id":854,"name":"Quantization of orbital angular momentum","description":"Questions using the quantization condition mvr = nh/2π to find allowed angular momenta, quantum numbers, or related orbit parameters.","sort":856,"aliases":[]},{"id":858,"code":"24.02.04","level":3,"parent_id":854,"name":"Reduced-mass correction to Bohr and Rydberg formulas","description":"Questions requiring use of reduced mass μ = mM/(m+M) in Bohr or Rydberg formulas for hydrogen, deuterium, positronium, or muonic atoms.","sort":857,"aliases":[]},{"id":859,"code":"24.02.05","level":3,"parent_id":854,"name":"Bohr radius, velocity, and energy of electron orbits","description":"Questions calculating Bohr radius (0.529 Å), orbital velocity (2.18×10^6/n m/s), or total energy (-13.6/n^2 eV) for specific orbits.","sort":858,"aliases":[]},{"id":860,"code":"24.02.06","level":3,"parent_id":854,"name":"Excitation and de-excitation transitions in Bohr atom","description":"Questions on energy absorbed/emitted when an electron jumps between Bohr orbits, including excitation energy, de-excitation photon emission, and transition counting.","sort":859,"aliases":[]},{"id":861,"code":"24.02.07","level":3,"parent_id":854,"name":"Atomic magnetic moment and Lande g-factor","description":"Questions on orbital magnetic moment μ = eL/2m, Bohr magneton, and Lande g-factor for atomic states.","sort":860,"aliases":[]},{"id":862,"code":"24.02.08","level":3,"parent_id":854,"name":"Atomic terms of two non-equivalent electrons","description":"Questions on determining atomic term symbols (e.g., ^1S, ^3P) for two non-equivalent electrons using L-S coupling rules.","sort":861,"aliases":[]},{"id":863,"code":"24.02.09","level":3,"parent_id":854,"name":"Classical rigid-sphere model of electron spin","description":"Questions treating electron spin classically as a rotating charged sphere, calculating spin angular momentum or magnetic moment from rigid-body rotation.","sort":862,"aliases":[]},{"id":864,"code":"24.02.10","level":3,"parent_id":854,"name":"Collision ionization and energy-transfer efficiency","description":"Questions on ionization of atoms by electron/particle collision, including threshold kinetic energy, energy transfer efficiency, and post-collision electron energies.","sort":863,"aliases":[]},{"id":865,"code":"24.02.11","level":3,"parent_id":854,"name":"Coulomb energy of a one-dimensional ionic chain","description":"Questions calculating electrostatic potential energy of a linear chain of alternating positive and negative ions using Coulomb's law and Madelung-type sums.","sort":864,"aliases":[]},{"id":866,"code":"24.02.12","level":3,"parent_id":854,"name":"Elastic hard-sphere scattering geometry","description":"Questions on scattering angle, impact parameter, and cross-section geometry for elastic collisions between hard spheres.","sort":865,"aliases":[]},{"id":867,"code":"24.02.13","level":3,"parent_id":854,"name":"Impulse approximation for fast charged-particle collisions","description":"Questions using the impulse approximation (momentum transfer = ∫F dt) to analyze fast charged-particle collisions with atomic electrons, including energy loss.","sort":866,"aliases":[]},{"id":868,"code":"24.02.14","level":3,"parent_id":854,"name":"L-S (Russell-Saunders) coupling","description":"Questions on Russell-Saunders (L-S) coupling: combining orbital and spin angular momenta to find J values, term symbols, and selection rules.","sort":867,"aliases":[]},{"id":869,"code":"24.02.15","level":3,"parent_id":854,"name":"Larmor precession of atomic angular momentum","description":"Questions on Larmor precession frequency ω = eB/2m of atomic magnetic moments in external magnetic fields, including precession period calculations.","sort":868,"aliases":[]},{"id":870,"code":"24.02.16","level":3,"parent_id":854,"name":"Molecular bonding and polarity","description":"Questions on dipole moments, electronegativity differences, and polar vs nonpolar character of diatomic or simple polyatomic molecules.","sort":869,"aliases":[]},{"id":871,"code":"24.02.17","level":3,"parent_id":854,"name":"Normal vibrational modes of a polyatomic molecule","description":"Questions determining the number and types of normal vibrational modes (stretching, bending) of polyatomic molecules using 3N-5 or 3N-6 rules.","sort":870,"aliases":[]},{"id":872,"code":"24.02.18","level":3,"parent_id":854,"name":"Optically pumped excited-state population","description":"Questions on optical pumping: using light to excite atoms to higher states, population inversion, and steady-state excited-state populations under pumping.","sort":871,"aliases":[]},{"id":873,"code":"24.02.19","level":3,"parent_id":854,"name":"Pauli exclusion in atomic electron configurations","description":"Questions on applying the Pauli exclusion principle to determine allowed electron configurations, maximum occupancy of shells/subshells, and identifying invalid configurations in atoms.","sort":872,"aliases":[]},{"id":874,"code":"24.02.20","level":3,"parent_id":854,"name":"Photoionization and field ionization of Rydberg atoms","description":"Questions on threshold wavelengths/frequencies or field strengths needed to ionize atoms, especially Rydberg atoms, using energy-level or field-ionization formulas.","sort":873,"aliases":[]},{"id":875,"code":"24.02.21","level":3,"parent_id":854,"name":"Quantum harmonic oscillator expectation values","description":"Questions computing expectation values such as ⟨x⟩, ⟨x²⟩, ⟨p⟩, or ⟨p²⟩ for quantum harmonic oscillator states using ladder operators or wavefunctions.","sort":874,"aliases":[]},{"id":876,"code":"24.02.22","level":3,"parent_id":854,"name":"Quantum oscillator mean energy (Boltzmann distribution)","description":"Questions using the Boltzmann distribution to find the mean thermal energy of a quantum harmonic oscillator, often comparing with classical kT at high or low temperature.","sort":875,"aliases":[]},{"id":877,"code":"24.02.23","level":3,"parent_id":854,"name":"Raman Stokes and anti-Stokes lines in diatomic molecules","description":"Questions on Raman scattering in diatomic molecules, identifying Stokes and anti-Stokes line frequencies from vibrational energy spacing and incident photon energy.","sort":876,"aliases":[]},{"id":878,"code":"24.02.24","level":3,"parent_id":854,"name":"Rock-salt (NaCl) ionic crystal structure","description":"Questions on the NaCl rock-salt structure, including lattice constant, nearest-neighbor distances, coordination number, and packing or density calculations.","sort":877,"aliases":[]},{"id":879,"code":"24.02.25","level":3,"parent_id":854,"name":"Sommerfeld electronic heat capacity of a metal","description":"Questions on the Sommerfeld model of electronic heat capacity in metals, using C = γT with γ expressed through the density of states at the Fermi energy.","sort":878,"aliases":[]},{"id":880,"code":"24.02.26","level":3,"parent_id":854,"name":"Spin magnetism and absent orbital moment in an s-state","description":"Questions on why s-state electrons have zero orbital angular momentum, leaving only spin magnetic moment, and related Landé g-factor or magnetic moment calculations.","sort":879,"aliases":[]},{"id":881,"code":"24.02.27","level":3,"parent_id":854,"name":"Standing-wave mode density in a continuous medium","description":"Questions counting standing-wave modes in a continuous medium, often using density of states in frequency or k-space for a 1D/2D/3D cavity or solid.","sort":880,"aliases":[]},{"id":882,"code":"24.02.28","level":3,"parent_id":854,"name":"Stimulated-emission correction to resonant absorption","description":"Questions on absorption corrected for stimulated emission, using Einstein B coefficients and population differences to find net absorption rate or cross-section.","sort":881,"aliases":[]},{"id":883,"code":"24.02.29","level":3,"parent_id":854,"name":"Thermionic emission (Richardson-Dushman law)","description":"Questions on thermionic emission from metal surfaces using the Richardson-Dushman law J = AT² exp(−φ/kT), including work function and temperature dependence.","sort":882,"aliases":[]},{"id":884,"code":"24.02.30","level":3,"parent_id":854,"name":"Zero-point vibrational energy of a Debye crystal","description":"Questions on zero-point vibrational energy of a Debye crystal, often computing total zero-point energy from the Debye frequency or Debye temperature.","sort":883,"aliases":[]},{"id":885,"code":"24.02.31","level":3,"parent_id":854,"name":"Diatomic potential-energy curves","description":"Questions on diatomic potential-energy curves, including equilibrium bond length, dissociation energy, and interpreting Morse or Lennard-Jones potential parameters.","sort":884,"aliases":[]},{"id":886,"code":"24.02.32","level":3,"parent_id":854,"name":"Diatomic rotation-vibration band (P/R branches)","description":"Questions on rotation-vibration spectra of diatomic molecules, identifying P-branch and R-branch transition frequencies and using the rotational-vibrational energy formula.","sort":885,"aliases":[]},{"id":887,"code":"24.02.33","level":3,"parent_id":854,"name":"Diatomic rotational energy levels (rigid rotor)","description":"Questions on rigid-rotor rotational energy levels of diatomic molecules, using E_J = BJ(J+1) to find transition frequencies, moment of inertia, or rotational constant.","sort":886,"aliases":[]},{"id":888,"code":"24.02.34","level":3,"parent_id":854,"name":"Diatomic vibrational energy levels","description":"Questions on vibrational energy levels of diatomic molecules treated as harmonic oscillators, using E_v = (v+1/2)ħω to find transition energies or force constants.","sort":887,"aliases":[]},{"id":889,"code":"24.02.35","level":3,"parent_id":854,"name":"3D Debye molar heat capacity","description":"Questions on the 3D Debye model molar heat capacity, including low-temperature T³ law, high-temperature Dulong-Petit limit, and numerical integration of the Debye function.","sort":888,"aliases":[]},{"id":890,"code":"24.02.36","level":3,"parent_id":854,"name":"Debye temperature from the cutoff frequency","description":"Questions relating Debye temperature to the cutoff frequency via k_B Θ_D = ħ ω_D, often finding Θ_D from given sound speed and atomic density.","sort":889,"aliases":[]},{"id":891,"code":"24.02.37","level":3,"parent_id":854,"name":"Discrete-chain mode density (nonlinear dispersion)","description":"Questions on mode density for a discrete linear chain with nonlinear dispersion, counting normal modes and identifying van Hove singularities or cutoff behavior.","sort":890,"aliases":[]},{"id":892,"code":"24.02.38","level":3,"parent_id":854,"name":"Einstein A/B coefficients","description":"Questions on Einstein A and B coefficients, including relations between spontaneous emission, stimulated emission, and absorption coefficients and their frequency dependence.","sort":891,"aliases":[]},{"id":893,"code":"24.02.39","level":3,"parent_id":854,"name":"Laser physics","description":"Questions on laser principles such as population inversion, threshold gain, cavity modes, and basic laser rate equations or output characteristics.","sort":892,"aliases":[]},{"id":894,"code":"24.02.40","level":3,"parent_id":854,"name":"Free-electron Fermi gas at T=0 in a metal","description":"Questions on the free-electron Fermi gas at T=0 in metals, computing Fermi energy, Fermi wavevector, Fermi velocity, or total ground-state energy from electron density.","sort":893,"aliases":[]},{"id":895,"code":"24.02.41","level":3,"parent_id":854,"name":"Free-electron Fermi-gas density of states at T=0","description":"Questions on the density of states of a free-electron Fermi gas at T=0, using g(E) ∝ E^{1/2} to find states per energy interval or related quantities.","sort":894,"aliases":[]},{"id":896,"code":"24.02.42","level":3,"parent_id":854,"name":"Effective nuclear charge and electron screening","description":"Questions on effective nuclear charge and electron screening, using Slater's rules or screening concepts to estimate ionization energies, radii, or energy levels.","sort":895,"aliases":[]},{"id":897,"code":"24.02.43","level":3,"parent_id":854,"name":"Electron shell and subshell filling capacities","description":"Questions on electron shell and subshell filling capacities, using 2n² for shells and 2(2l+1) for subshells to determine maximum electron occupancy.","sort":896,"aliases":[]},{"id":898,"code":"24.02.44","level":3,"parent_id":854,"name":"Hund's rules","description":"Questions on Hund's rules for determining ground-state term symbols of multi-electron atoms, including maximizing spin, maximizing orbital angular momentum, and J value for less or more than half-filled subshells.","sort":897,"aliases":[]},{"id":899,"code":"24.02.45","level":3,"parent_id":854,"name":"Stern-Gerlach force in an inhomogeneous field","description":"Questions on the force experienced by magnetic dipole moments of atoms in an inhomogeneous magnetic field, as in the Stern-Gerlach experiment, and spatial quantization of angular momentum.","sort":898,"aliases":[]},{"id":900,"code":"24.02.46","level":3,"parent_id":854,"name":"Zeeman effect in a weak external magnetic field","description":"Questions on the splitting of atomic spectral lines in a weak external magnetic field, including normal and anomalous Zeeman effect, magnetic quantum number, and energy shifts.","sort":899,"aliases":[]},{"id":901,"code":"24.02.47","level":3,"parent_id":854,"name":"Vector model of atomic angular-momentum coupling","description":"Questions on the vector model of coupling of orbital and spin angular momenta in atoms, including LS coupling, jj coupling, resultant J, and vector diagrams for angular momentum addition.","sort":900,"aliases":[]},{"id":902,"code":"24.02.48","level":3,"parent_id":854,"name":"Alkali spectral quantum defect","description":"Questions on the quantum defect in alkali metal spectra, where energy levels are modified from hydrogen-like values due to core penetration, and calculation of effective principal quantum number.","sort":901,"aliases":[]},{"id":903,"code":"24.02.49","level":3,"parent_id":854,"name":"Boltzmann population ratio of atomic energy levels","description":"Questions using the Boltzmann distribution to find the ratio of populations of two atomic energy levels at a given temperature, including degeneracy factors and exponential energy dependence.","sort":902,"aliases":[]},{"id":904,"code":"24.02.50","level":3,"parent_id":854,"name":"Hyperfine splitting and the hydrogen 21-cm line","description":"Questions on the hyperfine interaction between electron and nuclear spins, the 21-cm transition in hydrogen, and its frequency/wavelength.","sort":903,"aliases":[]},{"id":905,"code":"24.02.51","level":3,"parent_id":854,"name":"Relativistic fine structure of hydrogen","description":"Questions on relativistic corrections to hydrogen energy levels, including the fine structure constant and splitting of Bohr orbits.","sort":904,"aliases":[]},{"id":906,"code":"24.02.52","level":3,"parent_id":854,"name":"Spectroscopic term symbols for hydrogen-like atoms","description":"Questions on writing term symbols (n, L, J, multiplicity) for hydrogen-like atoms and identifying allowed transitions.","sort":905,"aliases":[]},{"id":907,"code":"24.02.53","level":3,"parent_id":854,"name":"Spin-orbit coupling and its effective magnetic field","description":"Questions on spin-orbit interaction energy, the effective magnetic field seen by the electron, and resulting level splitting.","sort":906,"aliases":[]},{"id":908,"code":"24.02.54","level":3,"parent_id":854,"name":"Classical two-electron Bohr model of helium","description":"Questions on applying Bohr-like quantization to a classical two-electron helium model, including effective nuclear charge and ionization energy.","sort":907,"aliases":[]},{"id":909,"code":"24.02.55","level":3,"parent_id":854,"name":"Bohr quantization in a non-Coulomb central potential","description":"Questions on applying Bohr quantization condition to non-Coulomb central potentials, such as harmonic oscillator or logarithmic potentials.","sort":908,"aliases":[]},{"id":910,"code":"24.02.56","level":3,"parent_id":854,"name":"Hydrogen most-probable radius scaling with n","description":"Questions on the most probable radius of the hydrogen electron, its dependence on principal quantum number n, and radial probability density.","sort":909,"aliases":[]},{"id":911,"code":"24.03","level":2,"parent_id":847,"name":"Atomic Models: Thomson and Rutherford","description":"Questions comparing Thomson and Rutherford atomic models, their experimental basis, and limitations.","sort":910,"aliases":["alpha-particle scattering experiment","nuclear model of atom"]},{"id":912,"code":"24.03.01","level":3,"parent_id":911,"name":"Thomson atomic model and electron SHM","description":"Questions on the Thomson plum-pudding model, electron simple harmonic motion inside a uniform positive sphere, and oscillation frequency.","sort":911,"aliases":[]},{"id":913,"code":"24.03.02","level":3,"parent_id":911,"name":"Rutherford's nuclear model","description":"Questions on Rutherford's nuclear model, its postulates, and implications for atomic structure and stability.","sort":912,"aliases":[]},{"id":914,"code":"24.03.03","level":3,"parent_id":911,"name":"Alpha-particle scattering and impact parameter","description":"Questions on alpha-particle scattering, impact parameter, scattering angle, and Rutherford scattering formula.","sort":913,"aliases":[]},{"id":915,"code":"24.03.04","level":3,"parent_id":911,"name":"Distance of closest approach in Rutherford scattering","description":"Questions on the distance of closest approach of an alpha particle to a nucleus, using conservation of energy and Coulomb repulsion.","sort":914,"aliases":[]},{"id":916,"code":"24.04","level":2,"parent_id":847,"name":"X-rays: Production, Spectra, and Moseley's Law","description":"Questions on X-ray production, continuous and characteristic spectra, and Moseley's law.","sort":915,"aliases":["characteristic X-ray spectrum","Moseley's law as application of Bohr theory","cutoff wavelength","continuous X-ray spectrum"]},{"id":917,"code":"24.04.01","level":3,"parent_id":916,"name":"X-ray physics","description":"Questions on basic X-ray physics, including production mechanisms, properties, and applications.","sort":916,"aliases":[]},{"id":918,"code":"24.04.02","level":3,"parent_id":916,"name":"X-rays and Moseley's law","description":"Questions on Moseley's law, its formula relating X-ray frequency to atomic number, and its use in determining atomic numbers.","sort":917,"aliases":[]},{"id":919,"code":"24.04.03","level":3,"parent_id":916,"name":"X-rays: production and spectra","description":"Questions on X-ray production methods and the features of X-ray spectra, including cutoff wavelength and characteristic lines.","sort":918,"aliases":[]},{"id":920,"code":"24.04.04","level":3,"parent_id":916,"name":"X-rays: production and absorption","description":"Questions on X-ray production and absorption, including attenuation coefficients and half-value thickness.","sort":919,"aliases":[]},{"id":921,"code":"24.04.05","level":3,"parent_id":916,"name":"X-ray production and spectra","description":"Questions on X-ray production and spectra, including tube voltage, cutoff wavelength, and characteristic peaks.","sort":920,"aliases":[]},{"id":922,"code":"24.04.06","level":3,"parent_id":916,"name":"Continuous and characteristic X-rays","description":"Questions on continuous (bremsstrahlung) and characteristic X-rays, their origins, and spectral features.","sort":921,"aliases":[]},{"id":923,"code":"24.04.07","level":3,"parent_id":916,"name":"Moseley's law","description":"Questions on Moseley's law, its derivation, and applications to X-ray spectra and atomic number determination.","sort":922,"aliases":[]},{"id":924,"code":"24.04.08","level":3,"parent_id":916,"name":"X-ray absorption and attenuation","description":"Questions on X-ray absorption, attenuation, Beer-Lambert law, and half-value thickness in materials.","sort":923,"aliases":[]},{"id":925,"code":"24.04.09","level":3,"parent_id":916,"name":"Bragg diffraction of X-rays","description":"Questions on Bragg diffraction of X-rays, Bragg's law, and crystal structure determination.","sort":924,"aliases":[]},{"id":926,"code":"24.05","level":2,"parent_id":847,"name":"Nuclear Physics: Composition and Basic Properties of Nuclei","description":"Umbrella node for basic nuclear properties: composition (protons, neutrons), nuclear size and density, nuclear magnetic moment, stability systematics, quark structure of nucleons, and elementary particle classification. Tag here only if a question spans several of these subtopics without fitting a specific one.","sort":925,"aliases":["nuclear structure","nuclear size and density","nuclear force (short range, saturation, charge independence)"]},{"id":927,"code":"24.05.01","level":3,"parent_id":926,"name":"Composition and size of the nucleus","description":"Questions computing nuclear radius R = R₀A^(1/3) (R₀ ≈ 1.2 fm), showing nuclear density is independent of A (~10^17 kg/m³), and identifying mass number A, atomic number Z, neutron number N, and isotope/isobar/isotone/isomer relations.","sort":926,"aliases":[]},{"id":928,"code":"24.05.02","level":3,"parent_id":926,"name":"Neutron-beam attenuation by nuclear interactions","description":"Questions on attenuation of a neutron beam passing through matter: I = I₀e^(−nσx) with number density n and nuclear cross-section σ, computing transmitted intensity, fraction absorbed, or mean free path λ = 1/(nσ).","sort":927,"aliases":[]},{"id":929,"code":"24.05.03","level":3,"parent_id":926,"name":"Nuclear magnetic moment and NMR spin-flip transition","description":"Questions on nuclear magnetic moments in units of nuclear magneton μ_N = eh/4πm_p, and NMR-type spin-flip resonance where photon energy hν = 2μB flips a proton/nucleus spin in field B, asking for the resonant frequency.","sort":928,"aliases":[]},{"id":930,"code":"24.05.04","level":3,"parent_id":926,"name":"Nuclear structure and stability","description":"Questions on why certain N/Z ratios give stable nuclei: the stability belt (N ≈ Z for light nuclei, N > Z for heavy), comparison of binding energy per nucleon, and predicting whether a nuclide is stable or which decay it would undergo.","sort":929,"aliases":[]},{"id":931,"code":"24.05.05","level":3,"parent_id":926,"name":"Quark composition of hadrons","description":"Questions on quark content of hadrons: proton = uud, neutron = udd, mesons as quark–antiquark pairs, and deducing charge or baryon number of a hadron from its quark combination (e.g. Δ++, π+, or a given combination's properties).","sort":930,"aliases":[]},{"id":932,"code":"24.05.06","level":3,"parent_id":926,"name":"Elementary particle classification: hadrons, leptons and quantum numbers","description":"Questions classifying particles into hadrons (baryons, mesons) vs leptons, and using conservation of charge, baryon number, and lepton number to test whether a particle reaction is allowed or to identify an unknown particle in it.","sort":931,"aliases":[]},{"id":933,"code":"24.06","level":2,"parent_id":847,"name":"Nuclear reactions and Q-value","description":"Umbrella node for nuclear reactions: balancing A and Z, computing Q-values, threshold energies, product kinetic energies, compound-nucleus excitation, and feasibility checks. Tag here only for general reaction questions not fitting a specific subtopic.","sort":932,"aliases":["nuclear reactions","Q-value","threshold energy of a nuclear reaction"]},{"id":934,"code":"24.06.01","level":3,"parent_id":933,"name":"Balancing nuclear reactions by mass and charge","description":"Questions where a nuclear equation has a missing product or projectile and must be completed by conserving mass number and atomic number, e.g. finding X in ⁷Li(p,X)⁷Be or identifying the particle emitted in a given transmutation.","sort":933,"aliases":[]},{"id":935,"code":"24.06.02","level":3,"parent_id":933,"name":"Nuclear reaction Q-value and product kinematics","description":"Questions computing Q = (Σm_initial − Σm_final)c² in energy units (931.5 MeV/u) and then the kinetic energies of the products via momentum conservation, e.g. splitting the released energy between an alpha particle and daughter nucleus.","sort":934,"aliases":[]},{"id":936,"code":"24.06.03","level":3,"parent_id":933,"name":"Nuclear reaction threshold energy","description":"Questions on the minimum projectile kinetic energy needed to drive an endothermic (Q < 0) reaction in the lab frame, E_th = −Q(1 + m_projectile/m_target), including why projectile energy must exceed |Q| due to momentum conservation.","sort":935,"aliases":[]},{"id":937,"code":"24.06.04","level":3,"parent_id":933,"name":"Compound-nucleus excitation energy from particle capture","description":"Questions on capture reactions forming a compound nucleus (e.g. ¹³C + α → ¹⁴N* + n or X + n → compound), computing the excitation energy of the compound nucleus as the captured particle's binding energy plus available kinetic energy (minus recoil).","sort":936,"aliases":[]},{"id":938,"code":"24.06.05","level":3,"parent_id":933,"name":"Reaction/decay feasibility by conservation laws","description":"Questions deciding whether a proposed reaction or decay can occur by checking conservation of charge, mass number, baryon number, lepton number, and energy–momentum, e.g. ruling out p → n + e⁺ in free space or verifying a fission/fusion channel is energetically allowed.","sort":937,"aliases":[]},{"id":939,"code":"24.07","level":2,"parent_id":847,"name":"Radioactivity and Decay Laws","description":"Umbrella node for radioactivity: nature of α, β, γ emissions, the exponential decay law, half-life and mean life, activity, decay chains, dating, and recoil kinematics. Tag here only for general radioactivity questions not fitting a specific subtopic.","sort":938,"aliases":["alpha, beta, gamma decay","half-life and mean life"]},{"id":940,"code":"24.07.01","level":3,"parent_id":939,"name":"Radioactivity (alpha, beta, gamma; decay law and half-life)","description":"Questions using N = N₀e^(−λt), t₁/₂ = ln2/λ, mean life τ = 1/λ, and activity A = λN (in becquerel/curie), plus identifying α (He²⁺), β⁻/β⁺, and γ emissions and their penetrating powers or the changes they cause in A and Z.","sort":939,"aliases":[]},{"id":941,"code":"24.07.02","level":3,"parent_id":939,"name":"Alpha-decay recoil kinematics","description":"Questions on two-body alpha decay kinematics: momentum conservation gives the daughter recoil, with T_α = Q·M_daughter/(M_daughter + M_α) and recoil energy T_d = Q·M_α/(M_daughter + M_α), often asking for recoil speed or energy split given the Q-value.","sort":940,"aliases":[]},{"id":942,"code":"24.07.03","level":3,"parent_id":939,"name":"Biological effects and dose of ionizing radiation","description":"Questions on radiation dose quantities: absorbed dose in gray (J/kg), dose equivalent in sievert with quality/w weighting factors, exposure in roentgen, and simple computations of energy deposited or dose received from a given activity or absorbed energy.","sort":941,"aliases":[]},{"id":943,"code":"24.07.04","level":3,"parent_id":939,"name":"Electron energy in neutron beta decay","description":"Questions on neutron beta decay n → p + e⁻ + ν̄: computing the maximum (endpoint) electron kinetic energy from the Q-value ≈ 0.78 MeV and explaining the continuous electron spectrum due to the antineutrino sharing energy.","sort":942,"aliases":[]},{"id":944,"code":"24.07.05","level":3,"parent_id":939,"name":"Multi-step alpha/beta decay bookkeeping","description":"Questions on decay series bookkeeping: given parent and final nucleus (e.g. ²³⁸U to ²⁰⁶Pb), find the number of α and β⁻ decays from ΔA = −4n_α and ΔZ = −2n_α + n_β, or track A and Z through a stated sequence of emissions.","sort":943,"aliases":[]},{"id":945,"code":"24.07.06","level":3,"parent_id":939,"name":"Radioactive parent-daughter chain accumulation","description":"Questions on parent–daughter accumulation: N_d(t) = λ_pN_p0/(λ_d − λ_p)(e^(−λ_p t) − e^(−λ_d t)), secular equilibrium when the parent is much longer-lived (activity ratio → 1), and time to reach maximum daughter activity.","sort":944,"aliases":[]},{"id":946,"code":"24.07.07","level":3,"parent_id":939,"name":"Radiometric dating","description":"Questions on radiometric dating: carbon-14 age from t = (1/λ)ln(N₀/N) using present C-14 activity or ¹⁴C/¹²C ratio versus the living-sample value, and analogous uranium–lead or rubidium–strontium age calculations.","sort":945,"aliases":[]},{"id":947,"code":"24.07.08","level":3,"parent_id":939,"name":"Radionuclide buildup under constant production and decay","description":"Questions on radionuclides produced at constant rate R (e.g. in a reactor): dN/dt = R − λN giving N(t) = (R/λ)(1 − e^(−λt)), saturation activity R at large t, and number of nuclei or activity after a given irradiation time.","sort":946,"aliases":[]},{"id":948,"code":"24.07.09","level":3,"parent_id":939,"name":"Recoil from light or massless particle emission","description":"Questions on recoil of a nucleus when it emits a nearly massless particle (γ photon or neutrino): recoil momentum p = E/c and tiny recoil energy E²/(2Mc²), e.g. finding the photon energy shift or recoil speed in a nuclear gamma emission.","sort":947,"aliases":[]},{"id":949,"code":"24.07.10","level":3,"parent_id":939,"name":"Two-component radioactive decay curve stripping","description":"Activity or count-rate data that is the sum of two decaying isotopes, N(t)=N1·e^(−λ1 t)+N2·e^(−λ2 t), with very different half-lives. Questions ask to strip the curve: fit the long-time linear tail (log plot) of the long-lived component, subtract it, and extract the short-lived component's half-life or initial activity.","sort":948,"aliases":[]},{"id":950,"code":"24.08","level":2,"parent_id":847,"name":"Mass-energy equivalence and nuclear binding energy","description":"Parent node for E=mc² applied to nuclei: mass defect Δm = Z·m_p + N·m_n − M_nucleus, binding energy BE = Δm·c² with 1 u = 931.5 MeV, BE per nucleon and the BE/A curve. Tag here for general mass–energy bookkeeping in nuclear reactions when no more specific child fits.","sort":949,"aliases":["mass defect","binding energy per nucleon"]},{"id":951,"code":"24.08.01","level":3,"parent_id":950,"name":"Mass-energy and nuclear binding energy","description":"Numerical computation of mass defect and binding energy: given nuclear/atomic masses find Δm, total BE and BE per nucleon, compare stability via the BE/A curve (peak ~8.8 MeV near Fe-56), or find Q-value of a reaction from mass tables using 931.5 MeV/u.","sort":950,"aliases":[]},{"id":952,"code":"24.08.02","level":3,"parent_id":950,"name":"Pair production","description":"Photon energy converting into rest mass: γ → e⁺ + e⁻ with threshold 2m_e c² = 1.022 MeV (plus recoil correction near a nucleus), and the reverse e⁺e⁻ annihilation into two 0.511 MeV photons. Questions give photon wavelength/energy and ask if pair production is possible or how many pairs form.","sort":951,"aliases":[]},{"id":953,"code":"24.08.03","level":3,"parent_id":950,"name":"Liquid-drop model of nuclear binding energy","description":"Semi-empirical mass formula (Weizsäcker): volume a_V·A, surface −a_S·A^(2/3), Coulomb −a_C·Z(Z−1)/A^(1/3), asymmetry −a_A·(A−2Z)²/A, pairing ±δ terms. Questions ask which term explains a trend (e.g., why BE/A peaks, why heavy nuclei need neutrons, Coulomb term scaling with Z²/A^(1/3)).","sort":952,"aliases":[]},{"id":954,"code":"24.09","level":2,"parent_id":847,"name":"Special relativity (non-NCERT): relativistic energy, momentum and kinematics","description":"Parent node for non-NCERT special relativity problems: γ-factor kinematics, E²=p²c²+m²c⁴, relativistic collisions and decays, Doppler, Lorentz transforms. Tag here only when the question is generic relativity and no specific child node matches.","sort":953,"aliases":["special relativity","relativistic kinematics","relativistic energy and momentum","Lorentz factor","four-momentum"]},{"id":955,"code":"24.09.01","level":3,"parent_id":954,"name":"Endpoint energy in many-body relativistic decay","description":"Maximum (endpoint) kinetic energy of one product in a decay with three or more bodies, e.g. M → m + (cluster of rest mass μ): the endpoint occurs when the cluster moves as one unit, giving K_max = [(M² − (m+μ)²)/(2M)]c². Typical archetypes: β-decay electron endpoint, three-body breakup endpoint spectra.","sort":954,"aliases":[]},{"id":956,"code":"24.09.02","level":3,"parent_id":954,"name":"Equal-energy massless-product decay opening angle","description":"Decay of a moving particle into two photons (or two massless products) of equal energy E, relating their lab opening angle θ to the parent: invariant m²c⁴ = 2E²(1−cosθ), i.e. cos(θ/2) = mc²/(2E) type relations. Questions give parent energy/momentum and ask for the angle, or vice versa (e.g., π⁰ decay in flight).","sort":955,"aliases":[]},{"id":957,"code":"24.09.03","level":3,"parent_id":954,"name":"Relativistic energy-momentum relation","description":"Use of E² = p²c² + m²c⁴ (and E = pc for photons) in single- and multi-particle settings: find p from E, invariant mass of a photon pair or system, momentum of a particle given total energy, or show a massless particle must carry momentum E/c.","sort":956,"aliases":[]},{"id":958,"code":"24.09.04","level":3,"parent_id":954,"name":"Relativistic kinetic energy and classical limit","description":"Relativistic kinetic energy K = (γ−1)mc² versus classical ½mv²: compute K at given v, find v from K, or estimate the speed where the classical formula errs by a given percent (expansion K ≈ ½mv² + 3mv⁴/8c²). Common with electrons accelerated through potential differences.","sort":957,"aliases":[]},{"id":959,"code":"24.09.05","level":3,"parent_id":954,"name":"Relativistic perfectly inelastic collisions","description":"Two relativistic particles stick together (perfectly inelastic): conserve total E and p to find the composite's velocity and rest mass, e.g. equal masses m with one at rest give M = m√(2(1+γ)) and v = c²p/E. Questions give γ or speed and ask for the merged mass or energy converted to heat.","sort":958,"aliases":[]},{"id":960,"code":"24.09.06","level":3,"parent_id":954,"name":"Relativistic photon rocket with variable rest mass","description":"Photon rocket: a body of initial rest mass M₀ ejects photons (often until rest mass M remains) and momentum conservation gives v/c = (M₀² − M²)/(M₀² + M²). Questions ask final speed, remaining mass fraction to reach a given v, or energy radiated.","sort":959,"aliases":[]},{"id":961,"code":"24.09.07","level":3,"parent_id":954,"name":"Relativistic threshold energy for particle production","description":"Threshold energy for endothermic reactions in the lab frame, E_th = [(Σm_final)² − (Σm_initial)²]c²/(2m_target), e.g. γ + p → p + π⁰, p + p → p + p + p + p̄, or photodisintegration of a nucleus. Questions give rest masses and ask the minimum projectile energy, noting the target-recoil CM correction.","sort":960,"aliases":[]},{"id":962,"code":"24.09.08","level":3,"parent_id":954,"name":"Relativistic two-body decay in flight at a lab angle","description":"Two-body decay of a moving (in-flight) parent where one daughter is emitted at a lab angle θ: use momentum components plus E²−p²c² invariants to find the daughter's lab energy E(θ), e.g. E = (E*E_parent ± p*c·... )/... style angular dependence. Questions give parent energy and angle, ask daughter energy or the angle range.","sort":961,"aliases":[]},{"id":963,"code":"24.09.09","level":3,"parent_id":954,"name":"Relativistic two-body particle decay kinematics","description":"Two-body decay of a parent at rest, M → m1 + m2: daughter energies E1 = (M² + m1² − m2²)c²/(2M), E2 = (M² + m2² − m1²)c²/(2M), equal-and-opposite momenta p = √(λ-type expression)·c/(2M), and the resulting speeds. Archetypes: Λ → p + π⁻, π → μ + ν, nucleus emitting a gamma plus recoil.","sort":962,"aliases":[]},{"id":964,"code":"24.09.10","level":3,"parent_id":954,"name":"Single-particle relativistic energy-momentum relation","description":"Single-particle relativity algebra: given any two of (v, γ, p, E, K) find the rest using E = γmc², p = γmv, K = (γ−1)mc², β = pc/E. Typical items: electron with p = x MeV/c, find total energy and speed; particle whose E is n times rest energy, find v.","sort":963,"aliases":[]},{"id":965,"code":"24.09.11","level":3,"parent_id":954,"name":"Special-relativity postulates and Galilean relativity failure","description":"Conceptual/derivation questions on the two postulates (relativity principle, invariant c) and why Galilean transformations/velocity addition fail for light: Michelson–Morley-type reasoning, u = c ± v contradictions, need for a new transformation. Mostly qualitative or first-line derivations.","sort":964,"aliases":[]},{"id":966,"code":"24.09.12","level":3,"parent_id":954,"name":"Time dilation, length contraction and proper time","description":"Numerical time dilation Δt = γΔt₀ and length contraction L = L₀/γ: muon survival to Earth's surface, moving clock vs lab clock readings, proper time/proper length identification, a rod or spaceship appearing shortened, pion lifetimes in accelerators.","sort":965,"aliases":[]},{"id":967,"code":"24.09.13","level":3,"parent_id":954,"name":"Relativistic velocity addition and four-vectors","description":"Velocity addition u′ = (u − v)/(1 − uv/c²) and its transverse components u′_⊥ = u_⊥/(γ(1 − uv_x/c²)), plus four-vector (E/c, p) invariance E²−p²c² = m²c⁴. Questions: head-on light beams, relative speed of two particles, transforming momentum-energy between frames.","sort":966,"aliases":[]},{"id":968,"code":"24.09.14","level":3,"parent_id":954,"name":"Lorentz transformations, invariant interval and Minkowski diagrams","description":"Lorentz transformation x′ = γ(x − vt), t′ = γ(t − vx/c²) (and inverse), invariant interval s² = c²t² − x², spacelike/timelike/lightlike separation, Minkowski diagrams with worldlines and light cones, simultaneity loss (Δt′ = −γvΔx/c²). Questions ask event coordinates in another frame or classify intervals.","sort":967,"aliases":[]},{"id":969,"code":"24.09.15","level":3,"parent_id":954,"name":"Relativistic Doppler effect and the twin paradox","description":"Relativistic Doppler shift ν′ = ν√((1±β)/(1∓β)) for receding/approaching sources (redshift of galaxies, moving mirror problems) and twin-paradox aging: travelling twin at speed v for time T returns younger by the γ factor, with turnaround-frame bookkeeping.","sort":968,"aliases":[]},{"id":970,"code":"24.09.16","level":3,"parent_id":954,"name":"Relativistic force and longitudinal/transverse mass","description":"Relativistic dynamics with force: F = dp/dt, F∥ = γ³ma∥, F⊥ = γma⊥, longitudinal mass γ³m and transverse mass γm. Questions ask acceleration of a charged particle under given force parallel/perpendicular to v, or work–energy via F·d leading to (γ−1)mc².","sort":969,"aliases":[]},{"id":971,"code":"24.09.17","level":3,"parent_id":954,"name":"Relativistic motion under constant proper acceleration","description":"Hyperbolic motion under constant proper acceleration a: worldline x² − c²t² = (c²/a)², v(t) = at/√(1+(at/c)²) approaching c, coordinate vs proper time, rapidity additivity. Questions give proper acceleration and ask speed/distance after a lab or proper time.","sort":970,"aliases":[]},{"id":972,"code":"24.09.18","level":3,"parent_id":954,"name":"Equivalence principle and gravitational time dilation","description":"Equivalence-principle arguments and gravitational time dilation: clock rate shifts Δf/f ≈ gh/c² near Earth or Δf/f = ΔΦ/c² generally, elevator thought experiments, light bending intuition, comparing clocks at tower top vs bottom or GPS satellite corrections.","sort":971,"aliases":[]},{"id":973,"code":"24.10","level":2,"parent_id":847,"name":"Nuclear Fission and Fusion","description":"Fission and fusion energetics: U-235 neutron-induced fission (~200 MeV, ~2–3 neutrons, chain reaction, mass defect from fragment masses), D–T and D–D fusion Q-values, energy per nucleon comparison via the BE/A curve, and why fusion releases more per nucleon. Questions compute energy released per reaction or per kg of fuel.","sort":972,"aliases":["nuclear energy","chain reaction"]},{"id":974,"code":"24.10.01","level":3,"parent_id":973,"name":"Fission chain-reaction population growth","description":"Questions on neutron multiplication in fission: N = N₀kⁿ after n generations, reproduction factor k, critical mass/size, and counting how many U-235 nuclei fission or how many neutrons exist after a given number of generations.","sort":973,"aliases":[]},{"id":975,"code":"24.10.02","level":3,"parent_id":973,"name":"Reactor power, fission rate and neutron generation","description":"Reactor arithmetic: fission rate = P/(200 MeV per fission), fuel burn-up (grams of U-235 per day/MWh), energy released per fission, and role of moderator/control rods in keeping k = 1.","sort":974,"aliases":[]},{"id":976,"code":"25","level":1,"parent_id":null,"name":"Semiconductor Electronics","description":"JEE chapter 25: Semiconductor Electronics. Concepts: Transistors; Logic gates; Diode circuits and special-purpose diodes; Semiconductors; p-n junction and diode characteristics.","sort":975,"aliases":[]},{"id":977,"code":"25.01","level":2,"parent_id":976,"name":"Transistors","description":"Parent node for bipolar junction transistors: n-p-n and p-n-p structure, biasing of emitter-base and collector-base junctions, CE/CB/CC configurations, input and output characteristic curves, and current-gain relations Ie = Ib + Ic, α = Ic/Ie, β = Ic/Ib, β = α/(1−α).","sort":976,"aliases":["transistor as amplifier","transistor oscillator","feedback amplifiers"]},{"id":978,"code":"25.01.01","level":3,"parent_id":977,"name":"Transistor as a current amplifier","description":"Transistor amplifier calculations: given some of Ib, Ic, Ie, α, β find the rest; CE amplifier voltage gain Av = β·(Rout/Rin), collector/emitter resistor drops, and transfer-ratio or current-amplification numericals.","sort":977,"aliases":[]},{"id":979,"code":"25.01.02","level":3,"parent_id":977,"name":"Transistor as a switch (cutoff and saturation)","description":"Transistor used as a switch: cutoff state (input 0, both junctions reverse biased, Ic ≈ ICEO, output high) vs saturation state (input 1, both junctions forward biased, VCE ≈ 0, output low), and choosing base resistor so the transistor saturates for a given load like an LED or bulb.","sort":978,"aliases":[]},{"id":980,"code":"25.01.03","level":3,"parent_id":977,"name":"Electronic oscillators and LC feedback","description":"Transistor oscillator questions: LC tank circuit with feedback from collector to emitter/base (inductive or capacitive coupling), oscillation frequency f = 1/(2π√LC), and the role of positive feedback and energy compensation in sustaining undamped oscillations.","sort":979,"aliases":[]},{"id":981,"code":"25.01.04","level":3,"parent_id":977,"name":"Triode valve characteristics and amplification factor","description":"Vacuum-tube triode problems: plate characteristics curves, plate resistance rp, mutual conductance (transconductance) gm, and amplification factor μ = gm × rp, including finding these from slopes of characteristic graphs.","sort":980,"aliases":[]},{"id":982,"code":"25.02","level":2,"parent_id":976,"name":"Logic gates","description":"Parent node for digital logic: OR, AND, NOT, NAND, NOR, XOR/XNOR gates, their symbols and truth tables, realization of gates from diodes/transistors, and building compound gates from NAND/NOR as universal gates.","sort":981,"aliases":[]},{"id":983,"code":"25.02.01","level":3,"parent_id":982,"name":"Logic gates, truth tables, and Boolean algebra","description":"Problems combining gates: write the truth table or Boolean expression of a gate network, simplify expressions using Boolean algebra and De Morgan's laws, evaluate output for given input bits, and identify which single gate a circuit is equivalent to.","sort":982,"aliases":[]},{"id":984,"code":"25.03","level":2,"parent_id":976,"name":"Diode circuits and special-purpose diodes","description":"Parent node for p-n diode applications: rectifier circuits, Zener voltage regulation, LED, photodiode, and solar cell, plus basic diode circuit analysis with forward/reverse biased diodes in networks.","sort":983,"aliases":["rectifiers","special-purpose diodes","diodes","p-n junction diode applications","rectifiers (half-wave and full-wave)","ripple and rectification efficiency","clippers and clampers","Zener diode voltage regulator","LED","photodiode","solar cell"]},{"id":985,"code":"25.03.01","level":3,"parent_id":984,"name":"Special-purpose diodes (Zener, LED, photodiode, solar cell)","description":"Questions on Zener diode (breakdown mechanisms, voltage regulator with series resistor, finding R or load current), LED (emitted wavelength λ ≈ 1240/Eg(eV) nm and color from band gap), photodiode (reverse saturation current proportional to light intensity, used in detectors), and solar cell (photovoltaic effect, working).","sort":984,"aliases":[]},{"id":986,"code":"25.03.02","level":3,"parent_id":984,"name":"Rectifiers (half-wave and full-wave)","description":"Rectifier problems: half-wave and full-wave (center-tap and bridge) circuits, output waveforms, ripple frequency f vs 2f, peak inverse voltage, rectification efficiency, and the smoothing role of a filter capacitor across the load.","sort":985,"aliases":[]},{"id":987,"code":"25.04","level":2,"parent_id":976,"name":"Semiconductors","description":"Parent node for semiconductor basics: energy bands, intrinsic and doped (n-type/p-type) materials, charge carriers, conductivity and its temperature dependence, and Hall effect in semiconductors.","sort":986,"aliases":["energy bands in solids"]},{"id":988,"code":"25.04.01","level":3,"parent_id":987,"name":"Intrinsic and extrinsic semiconductors","description":"Doping questions: pentavalent donors (P, As, Sb) giving n-type with electrons as majority carriers, trivalent acceptors (B, Al, In) giving p-type with holes as majority, donor/acceptor energy levels near the bands, and the mass-action law ne·nh = ni² (e.g., finding minority carrier density).","sort":987,"aliases":[]},{"id":989,"code":"25.04.02","level":3,"parent_id":987,"name":"Intrinsic semiconductor conductivity versus temperature","description":"Intrinsic conductivity vs temperature: σ = σ0·exp(−Eg/2kT) so conductivity rises (resistance falls) with heating — opposite to metals; numericals on resistance change with temperature and extracting band gap from the slope of ln σ vs 1/T.","sort":988,"aliases":[]},{"id":990,"code":"25.04.03","level":3,"parent_id":987,"name":"Intrinsic/extrinsic semiconductor conductivity plot","description":"Interpretation of conductivity/resistivity vs temperature graphs for semiconductors: identifying the impurity-ionization region, the extrinsic saturation plateau (carrier density fixed by dopants), and the high-T intrinsic rise, plus comparing intrinsic and doped samples on the same plot.","sort":989,"aliases":[]},{"id":991,"code":"25.04.04","level":3,"parent_id":987,"name":"Energy bands in solids","description":"Band theory classification of solids: valence band, conduction band, and forbidden gap; conductors (overlapping/partially filled bands), semiconductors (Eg ≈ 1 eV: Si 1.1 eV, Ge 0.7 eV), insulators (Eg ≈ 6 eV, diamond ~5.4 eV); questions compare gaps, explain conductivity, or identify material type from Eg.","sort":990,"aliases":[]},{"id":992,"code":"25.04.05","level":3,"parent_id":987,"name":"Finite-temperature Fermi-Dirac occupation","description":"Fermi–Dirac distribution problems: f(E) = 1/[1 + e^((E−EF)/kT)], probability that a state ΔE above the Fermi level is occupied at temperature T, the f = 1/2 occupation at EF for T > 0, and how the occupation tail spreads with temperature.","sort":991,"aliases":[]},{"id":993,"code":"25.04.06","level":3,"parent_id":987,"name":"Two-carrier Hall coefficient and zero-Hall condition","description":"Hall effect with both electrons and holes: Hall coefficient RH = (pμp² − nμn²)/[e(pμp + nμn)²], the zero-Hall condition nμn² = pμp², sign of RH deciding carrier type, and finding carrier concentration from Hall voltage V_H = IB/(net).","sort":992,"aliases":[]},{"id":994,"code":"25.04.07","level":3,"parent_id":987,"name":"Photoconductivity and photoconductive relaxation","description":"Photoconductivity problems: illumination generates electron–hole pairs raising conductivity by Δσ = e(Δn·μn + Δp·μp), steady-state balance of generation rate and recombination (lifetime τ), and the exponential relaxation of conductivity toward equilibrium when light is switched off/on.","sort":993,"aliases":[]},{"id":995,"code":"25.05","level":2,"parent_id":976,"name":"p-n junction and diode characteristics","description":"Parent node for the p-n junction diode: depletion region and barrier potential formation, drift vs diffusion currents, forward and reverse bias behavior, the V-I characteristic with knee/turn-on voltage (~0.2–0.7 V), reverse saturation current and breakdown, and the diode equation I = I0(e^{eV/kT} − 1).","sort":994,"aliases":["p-n junction formation and biasing","depletion layer","barrier potential","forward and reverse bias","diode I-V characteristics","dynamic resistance","knee voltage"]},{"id":996,"code":"25.05.01","level":3,"parent_id":995,"name":"p-n junction and diode","description":"Junction physics and diode I-V numericals: barrier potential and depletion width dependence on doping and bias, dynamic resistance from the characteristic curve, current from the exponential diode equation at a given voltage, and identifying forward/reverse bias in circuit diagrams.","sort":995,"aliases":[]},{"id":997,"code":"27","level":1,"parent_id":null,"name":"Communication Systems","description":"JEE chapter 27 (outside the current JEE syllabus): Communication Systems. Concepts: Elements of a communication system, signals and bandwidth; Modulation and demodulation (amplitude modulation); Propagation of electromagnetic waves (ground, sky and space wave).","sort":996,"aliases":[]},{"id":998,"code":"27.01","level":2,"parent_id":997,"name":"Elements of a communication system, signals and bandwidth","description":"Parent node for communication system questions: block diagram of transmitter-channel-receiver, message signal bandwidths, and bandwidth of transmission media.","sort":997,"aliases":["communication systems","block diagram of a communication system","transmitter, receiver and channel","transducer","noise","attenuation","repeater","analog and digital signals","bandwidth of signals","channel bandwidth","coaxial cable and optical fibre bandwidth"]},{"id":999,"code":"27.01.01","level":3,"parent_id":998,"name":"Elements of a communication system (transmitter, receiver, channel)","description":"Questions on the block diagram of a communication system: identifying the role of transducer, amplifier, modulator, antenna, channel, and demodulator, or tracing signal flow from source to receiver.","sort":998,"aliases":[]},{"id":1000,"code":"27.01.02","level":3,"parent_id":998,"name":"Bandwidth of signals and transmission media","description":"Numerical/conceptual questions on bandwidth of speech, music, and video signals, bandwidth of coaxial cable, optical fiber, and free space, and how many channels fit in a given frequency band.","sort":999,"aliases":[]},{"id":1001,"code":"27.02","level":2,"parent_id":997,"name":"Modulation and demodulation (amplitude modulation)","description":"Parent node for amplitude modulation: need for modulation, AM waveform, modulation index, sidebands, and production/detection of AM waves.","sort":1000,"aliases":["modulation","amplitude modulation","modulation index","sidebands","demodulation","need for modulation"]},{"id":1002,"code":"27.02.01","level":3,"parent_id":1001,"name":"Amplitude modulation: modulation index and sidebands","description":"Questions computing modulation index μ = Am/Ac, sideband frequencies ωc ± ωm, total power Pt = Pc(1 + μ²/2), and constraints like μ ≤ 1 or antenna height reduction from modulation.","sort":1001,"aliases":[]},{"id":1003,"code":"27.02.02","level":3,"parent_id":1001,"name":"Production and detection of amplitude modulated wave","description":"Questions on circuits for generating and recovering AM signals: square-law modulator combining carrier and message, and detection using a diode envelope detector with RC, or LC tuned receiver circuits.","sort":1002,"aliases":[]},{"id":1004,"code":"27.03","level":2,"parent_id":997,"name":"Propagation of electromagnetic waves (ground, sky and space wave)","description":"Parent node for EM wave propagation modes: ground, sky, and space wave; which frequency ranges use which mode, critical frequency, and line-of-sight range calculations.","sort":1003,"aliases":["ground wave","sky wave","space wave","ionospheric propagation","critical frequency","line-of-sight range","antenna height"]},{"id":1005,"code":"27.03.01","level":3,"parent_id":1004,"name":"Ground wave propagation","description":"Questions on ground wave propagation: low-frequency (up to ~1–2 MHz) waves gliding over Earth's surface, attenuation with distance, and use in medium-wave AM broadcasting.","sort":1004,"aliases":[]},{"id":1006,"code":"27.03.02","level":3,"parent_id":1004,"name":"Sky wave propagation and ionosphere","description":"Questions on sky wave propagation: reflection from ionospheric layers, critical frequency fc = 9√Nmax, skip distance, maximum usable frequency, and why shortwave (2–30 MHz) bands are used for long-distance radio.","sort":1005,"aliases":[]},{"id":1007,"code":"27.03.03","level":3,"parent_id":1004,"name":"Space wave propagation and line-of-sight distance","description":"Questions on space wave/line-of-sight propagation: horizon distance d = √(2Rh) from antenna height, TV broadcast coverage area, microwave links, and satellite communication ranges.","sort":1006,"aliases":[]}]}