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PhysicsJEE Main & Advanced

Physics Formulas

Complete formula sheets with vector mechanics, circuits, wave phenomena, field laws, and modern quantum constants.

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Chapter Directory (24 Chapters)

Click any chapter to view dedicated formula sheet & concept diagrams
Mechanics462 Qs

Units, Dimensions & Errors

Dimensional analysis, dimensional formulas of physical quantities, propagation of errors, vernier callipers and screw gauge.

8 Formulas
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Mechanics560 Qs

Kinematics & Motion in 1D & 2D

Uniformly accelerated motion, projectile motion, trajectory parabola, circular kinematics, and relative velocity.

8 Formulas Diagram
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Mechanics348 Qs

Laws of Motion & Friction

Newton's laws of motion, free body diagrams, static and kinetic friction, banking of roads, and pseudo-force.

7 Formulas
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Mechanics343 Qs

Work, Power & Energy

Work-energy theorem, conservative forces, potential energy, elastic collisions in 1D/2D, and power.

7 Formulas
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Mechanics271 Qs

Center of Mass & Momentum Conservation

Center of mass coordinates, continuous mass distribution, momentum conservation, and explosion of bodies.

7 Formulas
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Mechanics665 Qs

Rotational Motion & Rigid Bodies

Moment of inertia, torque, angular momentum conservation, rolling motion without slipping, and rotational kinetic energy.

8 Formulas Diagram
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Mechanics519 Qs

Gravitation & Satellite Motion

Newton's law, variation of g with height/depth/latitude, gravitational potential energy, orbital and escape speed.

8 Formulas
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Mechanics413 Qs

Properties of Matter & Elasticity

Hooke's law, Young's modulus, bulk modulus, shear modulus, elastic potential energy, and thermal stress.

8 Formulas
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Mechanics368 Qs

Fluid Mechanics & Surface Tension

Pascal's law, Archimedes principle, continuity equation, Bernoulli's theorem, viscosity (Stokes law, Poiseuille), and surface tension.

7 Formulas
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Mechanics511 Qs

Simple Harmonic Motion (SHM)

Kinematics and dynamics of SHM, spring-mass systems, simple pendulum, energy conservation, and superpositions.

8 Formulas Diagram
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Mechanics642 Qs

Wave Motion & Sound Waves

Wave equation, speed of sound in gases, standing waves in strings and organ pipes, beats, and Doppler effect.

7 Formulas
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Thermal Physics1,411 Qs

Heat & Thermodynamics

First law of thermodynamics, work done in isobaric/isothermal/adiabatic processes, Carnot engine, and heat transfer laws.

10 Formulas Diagram
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Electromagnetism817 Qs

Electrostatics

Coulomb's law, Gauss's theorem, electric field, potential, dipoles, and electrostatic potential energy.

8 Formulas Diagram
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Electromagnetism431 Qs

Capacitance & Dielectrics

Parallel plate capacitors, dielectric insertion, stored energy, common potential, and transient RC circuits.

8 Formulas Diagram
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Electromagnetism902 Qs

Current Electricity

Ohm's law, drift velocity, resistivity, Kirchhoff's laws, Wheatstone bridge, potentiometer, and electrical power.

9 Formulas Diagram
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Electromagnetism480 Qs

Magnetic Effects of Current

Biot-Savart law, Ampere's circuital law, magnetic field of straight wire, circular loop, solenoid, and Lorentz magnetic force.

8 Formulas
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Electromagnetism490 Qs

Electromagnetic Induction (EMI)

Faraday's law, Lenz's law, motional EMF, self & mutual inductance, and LR growth/decay circuits.

9 Formulas
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Electromagnetism606 Qs

Alternating Current (AC)

RMS and peak values, reactance, impedance of LCR circuits, resonance, Q-factor, and power factor.

8 Formulas Diagram
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Electromagnetism299 Qs

Electromagnetic Waves (EM Waves)

Maxwell equations, displacement current, wave speed in vacuum, Poynting vector, and electromagnetic spectrum.

8 Formulas
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Optics698 Qs

Ray Optics & Optical Instruments

Snell's law, total internal reflection, lens maker's formula, thin lenses, prism formula, and microscopes/telescopes.

11 Formulas Diagram
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Optics578 Qs

Wave Optics & Interference

Huygens principle, Young's Double Slit Experiment (YDSE), single slit diffraction, and polarization Brewster law.

8 Formulas Diagram
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Modern Physics535 Qs

Dual Nature of Radiation & Matter

Photoelectric effect, Einstein's equation, work function, cut-off frequency, and de Broglie matter waves.

7 Formulas
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Modern Physics822 Qs

Atoms & Nuclei

Bohr atomic model, energy levels, Rydberg formula, radioactive decay law, half-life, and nuclear binding energy.

8 Formulas
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Modern Physics462 Qs

Semiconductors & Logic Gates

P-N junction diode, Zener breakdown diode, rectifiers, and truth tables of Boolean logic gates (AND, OR, NOT, NAND, NOR).

8 Formulas
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Formulas Reference (All)

24 Chapters match
MechanicsHigh Yield (1-2 Qs in JEE)462 PYQs

Units, Dimensions & Errors

Dimensional analysis, dimensional formulas of physical quantities, propagation of errors, vernier callipers and screw gauge.

Chapter

Propagation of Errors (Relative & Percentage)

Z=ApBqCr  ⟹  ΔZZ=pΔAA+qΔBB+rΔCCZ = \frac{A^p B^q}{C^r} \implies \frac{\Delta Z}{Z} = p \frac{\Delta A}{A} + q \frac{\Delta B}{B} + r \frac{\Delta C}{C}

Fractional and percentage error estimation in derived quantities.

Errors always add up; never subtract error terms even if quantity is in the denominator.

Vernier Calliper & Screw Gauge Least Count

LCvernier=1 MSD−1 VSD=1 MSDN,LCscrew gauge=PitchTotal Circular Scale Divisions\text{LC}_{\text{vernier}} = 1 \text{ MSD} - 1 \text{ VSD} = \frac{1 \text{ MSD}}{N}, \quad \text{LC}_{\text{screw gauge}} = \frac{\text{Pitch}}{\text{Total Circular Scale Divisions}}

Instrument precision formulas for laboratory physics where N divisions of vernier scale coincide with (N - 1) main scale divisions.

Pitch is distance advanced in one complete rotation of the thimble (usually 0.5 mm or 1.0 mm).

Zero Error & True Reading Correction

True Reading=Observed Reading−(Zero Error),Observed Reading=MSR+(VSR×LC)\text{True Reading} = \text{Observed Reading} - (\text{Zero Error}), \quad \text{Observed Reading} = \text{MSR} + (\text{VSR} \times \text{LC})

Correction for instrumental zero offset. If zero is to the right/above, error is positive; if to the left/below, error is negative.

Positive zero error is SUBTRACTED; negative zero error is ADDED (since -(-e) = +e).

Important Dimensional Formulas

[G]=[M−1L3T−2],[h]=[ML2T−1],[ε0]=[M−1L−3T4A2],[μ0]=[MLT−2A−2][G] = [M^{-1}L^3T^{-2}], \quad [h] = [ML^2T^{-1}], \quad [\varepsilon_0] = [M^{-1}L^{-3}T^4A^2], \quad [\mu_0] = [MLT^{-2}A^{-2}]

Fundamental universal constants tested in JEE Main dimensional matching.

Speed of light relation: c = 1 / √(μ₀ ε₀) has dimension [LT⁻¹].

Basic Mathematics Used in Physics (Approximations)

For θ≪1 rad:sin⁡θ≈θ,tan⁡θ≈θ,cos⁡θ≈1−θ22;(1+x)n≈1+nx(∣x∣≪1)\text{For } \theta \ll 1 \text{ rad}: \sin\theta \approx \theta, \quad \tan\theta \approx \theta, \quad \cos\theta \approx 1 - \frac{\theta^2}{2}; \quad (1 + x)^n \approx 1 + nx \quad (|x| \ll 1)

Essential mathematical approximations for small oscillations, optics angles, thermal expansions, and gravitation height variations.

Angle θ must be converted to radians before applying small angle trigonometric approximations.

Significant Figures & Calculation Rules

Add/Sub: Least decimal places;Mult/Div: Least significant figures;σmean=σN\text{Add/Sub: Least decimal places}; \quad \text{Mult/Div: Least significant figures}; \quad \sigma_{\text{mean}} = \frac{\sigma}{\sqrt{N}}

Precision rules for experimental physics observations and standard deviation of sample mean.

All non-zero digits are significant; zeroes between non-zero digits are significant; trailing zeroes after decimal are significant.

Propagation of Experimental Errors

Z=ApBqCr  ⟹  ΔZZ=p(ΔAA)+q(ΔBB)+r(ΔCC)Z = \frac{A^p B^q}{C^r} \implies \frac{\Delta Z}{Z} = p\left(\frac{\Delta A}{A}\right) + q\left(\frac{\Delta B}{B}\right) + r\left(\frac{\Delta C}{C}\right)

Logarithmic differentiation error combination. Fractional and percentage errors add up in quadrature or worst-case linear bound.

In addition or subtraction (Z = A ± B), absolute errors always add: ΔZ = ΔA + ΔB (never subtract errors).

Spherometer Radius of Curvature Formula

R=l26h+h2,Pitch=Distance movedRotations,LC=PitchNR = \frac{l^2}{6h} + \frac{h}{2}, \quad \text{Pitch} = \frac{\text{Distance moved}}{\text{Rotations}}, \quad \text{LC} = \frac{\text{Pitch}}{N}

Laboratory measurement of radius of curvature of convex/concave optical lenses where l is mean leg separation and h is central sagitta elevation.

For very small sagitta (h << l), R ≈ l² / (6h).
MechanicsHigh Yield (2-3 Qs in JEE)560 PYQs

Kinematics & Motion in 1D & 2D

Uniformly accelerated motion, projectile motion, trajectory parabola, circular kinematics, and relative velocity.

Chapter

Equations of Motion (Constant Acceleration)

v=u+at,s=ut+12at2,v2=u2+2as,sn=u+a2(2n−1)v = u + at, \quad s = ut + \frac{1}{2}at^2, \quad v^2 = u^2 + 2as, \quad s_n = u + \frac{a}{2}(2n - 1)

Standard 1D and vector kinematic equations.

s_n is the displacement specifically in the n-th second.

Projectile Motion Core Parameters

T=2usin⁡θg,H=u2sin⁡2θ2g,R=u2sin⁡2θgT = \frac{2u\sin\theta}{g}, \quad H = \frac{u^2\sin^2\theta}{2g}, \quad R = \frac{u^2\sin 2\theta}{g}

Time of flight, maximum height, and horizontal range on flat ground.

Range is identical for complementary launch angles θ and (90° - θ).

Equation of Projectile Trajectory

y=xtan⁡θ−gx22u2cos⁡2θ=xtan⁡θ(1−xR)y = x\tan\theta - \frac{gx^2}{2u^2\cos^2\theta} = x\tan\theta \left(1 - \frac{x}{R}\right)

Parabolic trajectory equation linking horizontal coordinate x to vertical coordinate y.

The form y = x tan θ (1 - x/R) saves massive calculation time when Range R is known.

Centripetal & Tangential Acceleration

ac=v2r=ω2r,at=dvdt=αr,atotal=ac2+at2a_c = \frac{v^2}{r} = \omega^2 r, \quad a_t = \frac{dv}{dt} = \alpha r, \quad a_{total} = \sqrt{a_c^2 + a_t^2}

Kinematics of circular motion with radial and tangential acceleration components.

Uniform circular motion has a_t = 0 (constant speed, changing velocity direction).

Relative Velocity & River-Boat Crossing

v⃗AB=v⃗A−v⃗B,tmin=dvbr,Zero Drift Condition: sin⁡θ=vrvbr  (vbr>vr)\vec{v}_{AB} = \vec{v}_A - \vec{v}_B, \quad t_{min} = \frac{d}{v_{br}}, \quad \text{Zero Drift Condition: } \sin\theta = \frac{v_r}{v_{br}} \; (v_{br} > v_r)

Relative velocity in 2D and crossing a river of width d.

For shortest time, boat must steer perpendicular to the river bank.

Projectile on an Inclined Plane

T=2usin⁡(α−β)gcos⁡β,R=u2gcos⁡2β[sin⁡(2α−β)−sin⁡β],Rmax=u2g(1+sin⁡β)T = \frac{2u\sin(\alpha - \beta)}{g\cos\beta}, \quad R = \frac{u^2}{g\cos^2\beta}[\sin(2\alpha - \beta) - \sin\beta], \quad R_{\text{max}} = \frac{u^2}{g(1 + \sin\beta)}

Launch angle α from horizontal onto plane inclined at angle β. Replace g with components g cos β and g sin β.

Maximum range up the plane occurs when α = π/4 + β/2; down the plane replace β with -β.

Radius of Curvature of Trajectory

ρ=v2a⊥=[1+(dy/dx)2]3/2∣d2y/dx2∣,ρtop=u2cos⁡2θg\rho = \frac{v^2}{a_\perp} = \frac{[1 + (dy/dx)^2]^{3/2}}{|d^2y/dx^2|}, \quad \rho_{\text{top}} = \frac{u^2\cos^2\theta}{g}

Radius of the osculating circle at any point along a curved path where a_⊥ is normal acceleration.

At the highest point of standard projectile motion, speed is u cos θ and normal acceleration is g.

Rain-Man & Wind-Air Relative Motion

v⃗r/m=v⃗r−v⃗m,tan⁡θ=∣v⃗m∣∣v⃗r∣\vec{v}_{r/m} = \vec{v}_r - \vec{v}_m, \quad \tan\theta = \frac{|\vec{v}_m|}{|\vec{v}_r|}

Apparent direction of rainfall relative to a moving observer.

To protect from rain, hold umbrella in the direction of apparent velocity v_{r/m}.
MechanicsHigh Yield (2 Qs in JEE)348 PYQs

Laws of Motion & Friction

Newton's laws of motion, free body diagrams, static and kinetic friction, banking of roads, and pseudo-force.

Chapter

Newton's Second Law & Momentum

F⃗net=dp⃗dt=ma⃗,J⃗=∫F⃗ dt=Δp⃗\vec{F}_{net} = \frac{d\vec{p}}{dt} = m\vec{a}, \quad \vec{J} = \int \vec{F} \, dt = \Delta \vec{p}

Net force equals rate of change of linear momentum; Impulse J equals change in momentum.

Area under Force vs Time (F-t) graph gives impulse.

Static & Kinetic Friction

fs≤μsN,fk=μkN(μs>μk)f_s \le \mu_s N, \quad f_k = \mu_k N \quad (\mu_s > \mu_k)

Limiting static friction and constant sliding kinetic friction.

Angle of repose θ equals angle of friction: tan θ = μ_s.

Banking of Roads — v_max, v_min & Optimum Speed

vopt=rgtan⁡θ,vmax=rg⋅tan⁡θ+μs1−μstan⁡θ,vmin=rg⋅tan⁡θ−μs1+μstan⁡θv_{\text{opt}} = \sqrt{rg\tan\theta}, \quad v_{\text{max}} = \sqrt{rg\cdot\frac{\tan\theta + \mu_s}{1 - \mu_s\tan\theta}}, \quad v_{\text{min}} = \sqrt{rg\cdot\frac{\tan\theta - \mu_s}{1 + \mu_s\tan\theta}}

On a banked road of radius r and bank angle θ: at optimum speed v_opt lateral friction is zero. Between v_min and v_max the car negotiates the bend safely. Below v_min, car slides inward (friction acts up the bank); above v_max, car skids outward (friction acts down the bank).

v_min exists only when tan θ > μ_s. For tan θ ≤ μ_s, even at rest the car won't slide inward. On unbanked road (θ = 0): v_max = √(μ_s rg) and v_min = 0.

Connected Bodies & Pulley Systems

a=Net driving forceTotal mass,T=2m1m2m1+m2g  (Atwood machine)a = \frac{\text{Net driving force}}{\text{Total mass}}, \quad T = \frac{2 m_1 m_2}{m_1 + m_2} g \; (\text{Atwood machine})

Standard acceleration and string tension formulas.

Tension in light inextensible string is uniform throughout unless passing over massive pulley.

Two-Block System & Slipping Threshold

acommon=Fm1+m2,Fmax=μm1g(1+m2m1)=μ(m1+m2)ga_{\text{common}} = \frac{F}{m_1 + m_2}, \quad F_{\text{max}} = \mu m_1 g \left(1 + \frac{m_2}{m_1}\right) = \mu(m_1 + m_2)g

Maximum external pulling force F applied on bottom block before top block m₁ starts to slip relative to m₂.

If applied force F ≤ F_max, both blocks accelerate together as a single combined unit.

Pseudo-Force & Apparent Weight in Elevator

F⃗pseudo=−ma⃗frame,N=m(g±a)(+ for upward a,  − for downward a)\vec{F}_{\text{pseudo}} = -m \vec{a}_{\text{frame}}, \quad N = m(g \pm a) \quad (+ \text{ for upward } a, \; - \text{ for downward } a)

Inertial force applied to masses when observed from a non-inertial frame accelerating at a_frame.

In free fall (a = g downwards), apparent weight N = 0 (weightlessness).

Virtual Work & String Constraint Relations

∑T⃗⋅x⃗=0  ⟹  ∑T⃗⋅v⃗=0  ⟹  ∑T⃗⋅a⃗=0\sum \vec{T} \cdot \vec{x} = 0 \implies \sum \vec{T} \cdot \vec{v} = 0 \implies \sum \vec{T} \cdot \vec{a} = 0

Total virtual work done by tension in an inextensible, massless string system is identically zero.

Wedge constraint: Relative velocity components perpendicular to the common inclined contact face must be equal.
MechanicsHigh Yield (2 Qs in JEE)343 PYQs

Work, Power & Energy

Work-energy theorem, conservative forces, potential energy, elastic collisions in 1D/2D, and power.

Chapter

Work-Energy Theorem

Wnet=Wconservative+Wnon-conservative+Wexternal=ΔK=Kf−KiW_{\text{net}} = W_{\text{conservative}} + W_{\text{non-conservative}} + W_{\text{external}} = \Delta K = K_f - K_i

Net work done by all forces on a particle equals the change in its kinetic energy.

Most reliable problem-solving tool in mechanics.

Conservative Force & Potential Energy

F⃗=−∇⃗U=−(∂U∂xi^+∂U∂yj^+∂U∂zk^),ΔU=−Wcons\vec{F} = -\vec{\nabla} U = -\left(\frac{\partial U}{\partial x}\hat{i} + \frac{\partial U}{\partial y}\hat{j} + \frac{\partial U}{\partial z}\hat{k}\right), \quad \Delta U = -W_{\text{cons}}

Force derived from scalar potential energy gradient.

Stable equilibrium occurs where dU/dx = 0 and d²U/dx² > 0 (minimum potential energy).

1D Elastic Collision Velocities

v1=(m1−m2m1+m2)u1+(2m2m1+m2)u2,e=v2−v1u1−u2=1v_1 = \left(\frac{m_1 - m_2}{m_1 + m_2}\right)u_1 + \left(\frac{2m_2}{m_1 + m_2}\right)u_2, \quad e = \frac{v_2 - v_1}{u_1 - u_2} = 1

Final velocities after head-on perfectly elastic collision.

If m₁ = m₂ in elastic collision, velocities are completely exchanged: v₁ = u₂ and v₂ = u₁.

Instantaneous & Average Power

P=dWdt=F⃗⋅v⃗,Pavg=WtotalΔtP = \frac{dW}{dt} = \vec{F} \cdot \vec{v}, \quad P_{\text{avg}} = \frac{W_{\text{total}}}{\Delta t}

Rate of doing work (1 Watt = 1 Joule/second = 1 kg m² s⁻³).

1 horsepower (hp) ≈ 746 Watts.

Vertical Circular Motion (String & Light Rod)

vbottom≥5gR,vtop≥gR,Tbottom−Ttop=6mg,T(θ)=mv2R+mgcos⁡θv_{\text{bottom}} \ge \sqrt{5gR}, \quad v_{\text{top}} \ge \sqrt{gR}, \quad T_{\text{bottom}} - T_{\text{top}} = 6mg, \quad T(\theta) = \frac{mv^2}{R} + mg\cos\theta

Critical thresholds for looping the vertical circle without slackening of string.

For a massless rigid rod, speed at top can be 0, so critical bottom speed is only √(4gR) = 2√(gR).

Inelastic Collision & Kinetic Energy Loss

ΔK=12(m1m2m1+m2)(1−e2)(u1−u2)2,v1=m1u1+m2u2−em2(u1−u2)m1+m2\Delta K = \frac{1}{2} \left(\frac{m_1 m_2}{m_1 + m_2}\right) (1 - e^2) (u_1 - u_2)^2, \quad v_1 = \frac{m_1 u_1 + m_2 u_2 - e m_2(u_1 - u_2)}{m_1 + m_2}

Loss of mechanical kinetic energy dissipated as heat and sound during collision with coefficient of restitution e.

For perfectly inelastic collision (e = 0), both bodies stick together with maximum possible KE loss.

Spring Potential Energy & Work

Us=12kx2,Wspring=−ΔUs=12k(xi2−xf2)U_s = \frac{1}{2} k x^2, \quad W_{\text{spring}} = -\Delta U_s = \frac{1}{2} k (x_i^2 - x_f^2)

Elastic potential energy stored in an ideal spring deformed by displacement x.

Effective spring constant for parallel springs: k_p = k₁ + k₂; for series springs: 1/k_s = 1/k₁ + 1/k₂.
MechanicsHigh Yield (1-2 Qs in JEE)271 PYQs

Center of Mass & Momentum Conservation

Center of mass coordinates, continuous mass distribution, momentum conservation, and explosion of bodies.

Chapter

Center of Mass Coordinates

R⃗cm=∑mir⃗i∑mi=∫r⃗ dm∫dm\vec{R}_{cm} = \frac{\sum m_i \vec{r}_i}{\sum m_i} = \frac{\int \vec{r} \, dm}{\int dm}

Mass-weighted average position vector of a multi-particle or continuous system.

If net external force is zero, center of mass moves with constant velocity.

CM of Standard Uniform Symmetrical Bodies

ycm=2Rπ  (Semicircular ring),ycm=4R3π  (Semicircular disc),ycm=3R8  (Solid hemisphere)y_{cm} = \frac{2R}{\pi} \; (\text{Semicircular ring}), \quad y_{cm} = \frac{4R}{3\pi} \; (\text{Semicircular disc}), \quad y_{cm} = \frac{3R}{8} \; (\text{Solid hemisphere})

Standard heights of center of mass measured from the diametric base.

Hollow hemisphere has y_{cm} = R / 2.

Conservation of Linear Momentum

F⃗ext=0  ⟹  P⃗system=Mv⃗cm=constant\vec{F}_{ext} = 0 \implies \vec{P}_{system} = M \vec{v}_{cm} = \text{constant}

Total linear momentum remains constant in the absence of external forces.

Explosions and rifle recoil conserve total momentum.

CM of Cones & Hemispherical Shells

ycm=h4  (Solid Cone),ycm=h3  (Hollow Cone),ycm=R2  (Hemispherical Shell)y_{cm} = \frac{h}{4} \; (\text{Solid Cone}), \quad y_{cm} = \frac{h}{3} \; (\text{Hollow Cone}), \quad y_{cm} = \frac{R}{2} \; (\text{Hemispherical Shell})

Height of center of mass measured from flat circular base along axis of symmetry.

Memory mnemonic from base: Shell (R/2) > Solid hemisphere (3R/8); Hollow cone (h/3) > Solid cone (h/4).

Cavity Problems (Negative Mass Concept)

R⃗cm=Morigr⃗1−Mremovedr⃗2Morig−Mremoved\vec{R}_{cm} = \frac{M_{\text{orig}}\vec{r}_1 - M_{\text{removed}}\vec{r}_2}{M_{\text{orig}} - M_{\text{removed}}}

Calculates the shift in center of mass when a symmetrical portion or hole is carved out from a uniform body.

For circular disc with circular hole of radius r at distance d: x_{cm} = - (r² d) / (R² - r²).

Reduced Mass (Two-Body Connected System)

μ=m1m2m1+m2,T=2πμk\mu = \frac{m_1 m_2}{m_1 + m_2}, \quad T = 2\pi \sqrt{\frac{\mu}{k}}

Reduces two interacting bodies connected by a spring into an equivalent single particle problem.

When a spring connects m₁ and m₂ on a frictionless surface, natural frequency is ω = √(k / μ).

Variable Mass & Rocket Propulsion

Fthrust=vrel(−dmdt),v=v0+vrelln⁡(m0m)−gtF_{\text{thrust}} = v_{\text{rel}} \left(-\frac{dm}{dt}\right), \quad v = v_0 + v_{\text{rel}} \ln\left(\frac{m_0}{m}\right) - gt

Tsiolkovsky rocket equation for mass ejection rate dm/dt with exhaust velocity v_rel.

For upward acceleration just after liftoff: a = (v_rel / m₀)(|dm/dt|) - g.
MechanicsTop Priority (3-4 Qs in JEE)665 PYQs

Rotational Motion & Rigid Bodies

Moment of inertia, torque, angular momentum conservation, rolling motion without slipping, and rotational kinetic energy.

Chapter

Moment of Inertia of Standard Bodies

Iring=MR2,  Idisc=12MR2,  Isolid sphere=25MR2,  Irod=112ML2I_{\text{ring}} = MR^2, \; I_{\text{disc}} = \frac{1}{2}MR^2, \; I_{\text{solid sphere}} = \frac{2}{5}MR^2, \; I_{\text{rod}} = \frac{1}{12}ML^2

Moments of inertia about central symmetry axes.

Hollow sphere has I = (2/3)MR².

Parallel & Perpendicular Axis Theorems

I=Icm+Md2  (Parallel),Iz=Ix+Iy  (Perpendicular, 2D laminas)I = I_{cm} + Md^2 \; (\text{Parallel}), \quad I_z = I_x + I_y \; (\text{Perpendicular, 2D laminas})

Translating and rotating axes of inertia.

Parallel axis theorem requires I_{cm} through center of mass, NOT any random point.

Torque & Angular Momentum

τ⃗=r⃗×F⃗=Iα⃗,L⃗=r⃗×p⃗=Iω⃗,τ⃗ext=dL⃗dt\vec{\tau} = \vec{r} \times \vec{F} = I\vec{\alpha}, \quad \vec{L} = \vec{r} \times \vec{p} = I\vec{\omega}, \quad \vec{\tau}_{ext} = \frac{d\vec{L}}{dt}

Rotational analogs of Newton's second law.

If net external torque is zero, angular momentum is conserved (I₁ω₁ = I₂ω₂).

Pure Rolling Motion Kinetic Energy

Ktotal=Ktrans+Krot=12Mvcm2(1+k2R2)K_{total} = K_{trans} + K_{rot} = \frac{1}{2}Mv_{cm}^2 \left(1 + \frac{k^2}{R^2}\right)

Total kinetic energy of body rolling without slipping with radius of gyration k.

For pure rolling: v_{cm} = Rω and a_{cm} = Rα.

Acceleration on an Inclined Plane

a=gsin⁡θ1+k2R2,μmin=tan⁡θ1+R2k2a = \frac{g\sin\theta}{1 + \frac{k^2}{R^2}}, \quad \mu_{min} = \frac{\tan\theta}{1 + \frac{R^2}{k^2}}

Linear acceleration of a rolling body down an incline of angle θ.

Body with smallest k²/R² reaches the bottom first (Solid Sphere > Disc > Hollow Sphere > Ring).

Angular Momentum in Combined Motion (Translation + Rotation)

L⃗O=r⃗cm×Mv⃗cm+Icmω⃗,K=12IIAOR ω2\vec{L}_O = \vec{r}_{cm} \times M\vec{v}_{cm} + I_{cm}\vec{\omega}, \quad K = \frac{1}{2}I_{\text{IAOR}}\,\omega^2

Total angular momentum about origin O decomposes into orbital CM term plus spin intrinsic term.

Pure rolling can be treated as pure rotation about the Instantaneous Axis of Rotation (IAOR) passing through the contact point.

Rolling Kinematics & Point Velocity on Wheel

vP=2vcmsin⁡(θ2)  ⟹  vtop=2vcm,vbottom=0,acontact=vcm2R  (towards center)v_P = 2 v_{cm} \sin\left(\frac{\theta}{2}\right) \implies v_{\text{top}} = 2 v_{cm}, \quad v_{\text{bottom}} = 0, \quad a_{\text{contact}} = \frac{v_{cm}^2}{R} \; (\text{towards center})

Speed of any point on perimeter of rolling cylinder at angle θ from bottom contact point.

The bottom contact point is instantaneously at rest (v = 0), but its instantaneous acceleration is non-zero (directed toward the center).

Toppling vs Sliding of a Block

Toppling before sliding condition: μ>bh,Ftopple=mgb2h\text{Toppling before sliding condition: } \mu > \frac{b}{h}, \quad F_{\text{topple}} = \frac{mg b}{2h}

Critical threshold for block of base width b and height h pushed by horizontal force at top edge.

Normal reaction shifts forward by distance x = (F·h)/(mg) as horizontal force F increases, reaching the front corner (x = b/2) at the verge of toppling.
MechanicsHigh Yield (2 Qs in JEE)519 PYQs

Gravitation & Satellite Motion

Newton's law, variation of g with height/depth/latitude, gravitational potential energy, orbital and escape speed.

Chapter

Newton's Gravitation Law & Potential Energy

F=Gm1m2r2,U=−GMmr,V=−GMrF = G \frac{m_1 m_2}{r^2}, \quad U = -G \frac{M m}{r}, \quad V = -\frac{GM}{r}

Universal inverse-square gravitational force and potential.

Potential V is negative everywhere, approaching 0 at infinity.

Variation of g with Altitude & Depth

gh=g(1+hR)2≈g(1−2hR),gd=g(1−dR)g_h = \frac{g}{\left(1 + \frac{h}{R}\right)^2} \approx g\left(1 - \frac{2h}{R}\right), \quad g_d = g\left(1 - \frac{d}{R}\right)

Decrease of gravitational acceleration above and below Earth's surface.

At Earth's center (d = R), g = 0.

Escape Velocity & Orbital Velocity

ve=2GMR=2gR≈11.2 km/s,vo=GMR=gR≈7.92 km/sv_e = \sqrt{\frac{2GM}{R}} = \sqrt{2gR} \approx 11.2 \text{ km/s}, \quad v_o = \sqrt{\frac{GM}{R}} = \sqrt{gR} \approx 7.92 \text{ km/s}

Speeds required to escape Earth and maintain circular low-orbit.

Escape speed is exactly √2 times orbital speed: v_e = √2 v_o.

Kepler's Third Law (Harmonic Law)

T2=(4π2GM)r3  ⟺  T2∝r3T^2 = \left(\frac{4\pi^2}{GM}\right) r^3 \iff T^2 \propto r^3

Square of satellite orbital period is proportional to cube of semi-major axis.

Geostationary satellite: T = 24 hours, altitude h ≈ 36,000 km.

Satellite Total Energy & Binding Energy

Etotal=−GMm2r=−K=12U,Binding Energy=GMm2rE_{\text{total}} = -\frac{GMm}{2r} = -K = \frac{1}{2}U, \quad \text{Binding Energy} = \frac{GMm}{2r}

Mechanical energy partition for stable circular orbit.

Negative total energy signifies a bound gravitational state.

Gravitational Field & Potential Inside Solid Sphere & Shell

Shell (r<R):I=0,  V=−GMR;Solid Sphere (r≤R):I=−GMrR3,  V(r)=−GM2R3(3R2−r2)\text{Shell } (r < R): I = 0, \; V = -\frac{GM}{R}; \quad \text{Solid Sphere } (r \le R): I = -\frac{GMr}{R^3}, \; V(r) = -\frac{GM}{2R^3}(3R^2 - r^2)

Internal field and potential for uniform spherical distributions. Field inside hollow shell is zero everywhere.

Potential at the center of solid planet is 1.5 times the surface potential: V_center = -1.5 GM/R.

Variation of g with Earth's Rotation & Latitude

g′=g−ω2Rcos⁡2λ  ⟹  gequator=g−ω2R,gpole=gg' = g - \omega^2 R \cos^2\lambda \implies g_{\text{equator}} = g - \omega^2 R, \quad g_{\text{pole}} = g

Apparent gravitational acceleration considering Earth's diurnal rotation at latitude angle λ.

If Earth stopped rotating, effective weight would increase at the equator by m·ω²R, but would remain unchanged at the poles.

Orbital Shift Energy & Hyperbolic Escape Speed

ΔE=GMm2(1r1−1r2),v∞=v2−ve2(v>ve)\Delta E = \frac{GMm}{2}\left(\frac{1}{r_1} - \frac{1}{r_2}\right), \quad v_\infty = \sqrt{v^2 - v_e^2} \quad (v > v_e)

Energy required to boost a satellite between circular orbits of radii r₁ and r₂, and residual interstellar speed at infinity.

For elliptic orbit of semi-major axis a: total energy is E = -GMm / (2a).
MechanicsHigh Yield (1-2 Qs in JEE)413 PYQs

Properties of Matter & Elasticity

Hooke's law, Young's modulus, bulk modulus, shear modulus, elastic potential energy, and thermal stress.

Chapter

Hooke's Law & Moduli of Elasticity

Y=StressStrain=F/AΔL/L,B=−ΔPΔV/V,η=F/AθY = \frac{\text{Stress}}{\text{Strain}} = \frac{F / A}{\Delta L / L}, \quad B = -\frac{\Delta P}{\Delta V / V}, \quad \eta = \frac{F / A}{\theta}

Young's, Bulk, and Shear moduli.

Compressibility K = 1 / B.

Elastic Potential Energy Stored in Wire

U=12×Force×ΔL,u=UVolume=12×Stress×Strain=12Y(Strain)2U = \frac{1}{2} \times \text{Force} \times \Delta L, \quad u = \frac{U}{\text{Volume}} = \frac{1}{2} \times \text{Stress} \times \text{Strain} = \frac{1}{2} Y (\text{Strain})^2

Work done in stretching a wire stored as elastic energy.

Area under Stress-Strain curve up to elastic limit gives energy density.

Thermal Stress in Clamped Wire

Stress=YαΔT,F=YAαΔT\text{Stress} = Y \alpha \Delta T, \quad F = Y A \alpha \Delta T

Force developed when temperature of clamped wire changes by ΔT.

α is the linear coefficient of thermal expansion.

Poisson's Ratio & Relations Between Elastic Moduli

σ=−Δr/rΔL/L,Y=3B(1−2σ)=2η(1+σ),9Y=1B+3η\sigma = -\frac{\Delta r / r}{\Delta L / L}, \quad Y = 3B(1 - 2\sigma) = 2\eta(1 + \sigma), \quad \frac{9}{Y} = \frac{1}{B} + \frac{3}{\eta}

Lateral contraction strain to longitudinal extension strain ratio, and master algebraic interrelations among Y, B, η, and σ.

Theoretical limits of σ are -1 to 0.5; practical engineering materials lie between 0 and 0.5.

Elongation of Wire Under Self-Weight

ΔLself=MgL2AY=ρgL22Y\Delta L_{\text{self}} = \frac{M g L}{2AY} = \frac{\rho g L^2}{2Y}

Elongation produced purely by the wire's own gravitational weight (mass M, density ρ, length L).

Elongation under self-weight is exactly HALF the elongation produced by hanging a mass M at the bottom end.

Depression of a Loaded Beam

δ=WL348YIg=WL34bd3Y(Rectangular bar with breadth b, depth d)\delta = \frac{W L^3}{48 Y I_g} = \frac{W L^3}{4 b d^3 Y} \quad (\text{Rectangular bar with breadth } b, \text{ depth } d)

Sagging depression δ at the center of a beam of length L supported at both ends loaded with weight W.

To minimize sagging, depth d is made large compared to breadth b (I-shaped girders in construction).

Bulk Modulus, Compressibility & Fluid Density with Depth

B=−VdPdV,K=1B,ρ(h)=ρ0(1+ρ0ghB)B = -V\frac{dP}{dV}, \quad K = \frac{1}{B}, \quad \rho(h) = \rho_0\left(1 + \frac{\rho_0 g h}{B}\right)

Volumetric elasticity and fractional increase in liquid density at depth h under hydrostatic pressure.

Minus sign indicates volume decreases as pressure increases.

Torsion of a Cylinder (Twisting Couple)

C=πηr42L,W=12Cθ2=πηr4θ24LC = \frac{\pi \eta r^4}{2L}, \quad W = \frac{1}{2}C\theta^2 = \frac{\pi \eta r^4 \theta^2}{4L}

Restoring couple C per unit angle of twist in a circular shaft of shear modulus η, radius r, and length L.

A hollow cylinder is much more resistant to twisting than a solid cylinder of identical mass and length.
MechanicsHigh Yield (2 Qs in JEE)368 PYQs

Fluid Mechanics & Surface Tension

Pascal's law, Archimedes principle, continuity equation, Bernoulli's theorem, viscosity (Stokes law, Poiseuille), and surface tension.

Chapter

Continuity Equation & Bernoulli's Principle

A1v1=A2v2,P+12ρv2+ρgh=constantA_1 v_1 = A_2 v_2, \quad P + \frac{1}{2}\rho v^2 + \rho g h = \text{constant}

Mass and energy conservation in ideal streamline fluid flow.

Torricelli's speed of efflux: v = √(2gh).

Stokes' Law & Terminal Velocity

Fv=6πηrv,vt=29r2(ρ−σ)gηF_v = 6\pi\eta r v, \quad v_t = \frac{2}{9}\frac{r^2(\rho - \sigma)g}{\eta}

Viscous drag force on sphere and steady terminal fall speed.

Terminal velocity is proportional to the square of sphere radius (v_t ∝ r²).

Excess Pressure in Drops & Bubbles

ΔPdrop=2TR,ΔPsoap bubble=4TR\Delta P_{\text{drop}} = \frac{2T}{R}, \quad \Delta P_{\text{soap bubble}} = \frac{4T}{R}

Pressure difference across curved liquid interface due to surface tension T.

Soap bubble has 2 free surfaces, so excess pressure is doubled.

Capillary Rise (Jurin's Law)

h=2Tcos⁡θρgrh = \frac{2T\cos\theta}{\rho g r}

Height of liquid column climbing a capillary tube of radius r with contact angle θ.

For pure water in clean glass, contact angle θ ≈ 0° (cos θ = 1).

Torricelli's Law & Efflux Range

v=2gh,R=2h(H−h),Rmax=H  (at h=H/2),tempty=Aa2Hgv = \sqrt{2gh}, \quad R = 2\sqrt{h(H - h)}, \quad R_{\text{max}} = H \; (\text{at } h = H/2), \quad t_{\text{empty}} = \frac{A}{a}\sqrt{\frac{2H}{g}}

Speed of efflux from orifice at depth h below surface, horizontal range on ground, and total emptying time of tank.

Two holes at depths h and (H - h) yield identical horizontal ranges.

Poiseuille's Viscous Flow & Reynolds Number

Q=πΔPr48ηL,Re=ρvDηQ = \frac{\pi \Delta P r^4}{8\eta L}, \quad R_e = \frac{\rho v D}{\eta}

Volume flow rate Q through a circular capillary pipe and dimensionless Reynolds number determining flow stability.

Flow is laminar for R_e < 2000, unstable for 2000 < R_e < 3000, and turbulent for R_e > 3000.

Surface Energy & Droplet Coalescence

W=TΔA,R=n1/3r,ΔUreleased=4πTr2(n−n2/3),ΔT=3Tsρ(1r−1R)W = T \Delta A, \quad R = n^{1/3}r, \quad \Delta U_{\text{released}} = 4\pi T r^2 (n - n^{2/3}), \quad \Delta T = \frac{3T}{s \rho}\left(\frac{1}{r} - \frac{1}{R}\right)

Work done in changing surface area, and heat released when n identical droplets coalesce into a single large drop.

When droplets coalesce, surface area decreases, releasing energy and warming the liquid.
MechanicsHigh Yield (2 Qs in JEE)511 PYQs

Simple Harmonic Motion (SHM)

Kinematics and dynamics of SHM, spring-mass systems, simple pendulum, energy conservation, and superpositions.

Chapter

Kinematics of SHM

x=Asin⁡(ωt+ϕ),v=ωA2−x2,a=−ω2xx = A\sin(\omega t + \phi), \quad v = \omega\sqrt{A^2 - x^2}, \quad a = -\omega^2 x

Displacement, velocity, and restoring acceleration in simple harmonic motion.

Maximum velocity v_max = ωA; maximum acceleration a_max = ω²A.

Energy in Simple Harmonic Motion

K=12mω2(A2−x2),U=12mω2x2,Etotal=12mω2A2=constantK = \frac{1}{2}m\omega^2(A^2 - x^2), \quad U = \frac{1}{2}m\omega^2 x^2, \quad E_{total} = \frac{1}{2}m\omega^2 A^2 = \text{constant}

Continuous interchange between kinetic and potential energy.

At x = A/√2, kinetic energy equals potential energy (K = U = E/2).

Time Period of Spring-Mass System

T=2πmk,kseries=k1k2k1+k2,kparallel=k1+k2T = 2\pi\sqrt{\frac{m}{k}}, \quad k_{\text{series}} = \frac{k_1 k_2}{k_1 + k_2}, \quad k_{\text{parallel}} = k_1 + k_2

Oscillation period of spring with mass m.

Cutting a spring into n equal parts increases stiffness of each part to n*k.

Simple Pendulum Time Period

T=2πLgT = 2\pi\sqrt{\frac{L}{g}}

Small-angle oscillation period of ideal pendulum.

Seconds pendulum has T = 2 s and length L ≈ 1 meter.

Superposition of Two Collinear SHMs

Ares=A12+A22+2A1A2cos⁡δ,tan⁡θ=A2sin⁡δA1+A2cos⁡δA_{\text{res}} = \sqrt{A_1^2 + A_2^2 + 2 A_1 A_2 \cos\delta}, \quad \tan\theta = \frac{A_2\sin\delta}{A_1 + A_2\cos\delta}

Vector phasor addition of two SHMs of identical frequency with phase difference δ.

Constructive interference at δ = 0 gives A = A₁ + A₂.

Physical (Compound) Pendulum & Minimum Time Period

T=2πImgd=2πk2+d2gd  ⟹  Tmin=2π2kg(when d=k)T = 2\pi\sqrt{\frac{I}{mgd}} = 2\pi\sqrt{\frac{k^2 + d^2}{gd}} \implies T_{\text{min}} = 2\pi\sqrt{\frac{2k}{g}} \quad (\text{when } d = k)

Rigid body swinging about horizontal suspension axis at distance d from CM (k is radius of gyration about CM).

Points of suspension and oscillation are interchangeable: interchanging d and k²/d yields the exact same time period.

Damped Harmonic Oscillations & Quality Factor

A(t)=A0e−bt2m,ω′=km−b24m2,E(t)=E0e−btm,Q=ω0mbA(t) = A_0 e^{-\frac{b t}{2m}}, \quad \omega' = \sqrt{\frac{k}{m} - \frac{b^2}{4m^2}}, \quad E(t) = E_0 e^{-\frac{b t}{m}}, \quad Q = \frac{\omega_0 m}{b}

Oscillations with viscous damping force F_d = -b v. Amplitude decays exponentially with damping constant b/(2m).

Critical damping occurs when b² = 4mk (no oscillations occur; body returns to equilibrium most rapidly).

Liquid Column in U-Tube & Floating Cylinder

TU-tube=2πhg=2πL2g,Tcylinder=2πmAρg=2πhsubmergedgT_{\text{U-tube}} = 2\pi\sqrt{\frac{h}{g}} = 2\pi\sqrt{\frac{L}{2g}}, \quad T_{\text{cylinder}} = 2\pi\sqrt{\frac{m}{A\rho g}} = 2\pi\sqrt{\frac{h_{\text{submerged}}}{g}}

Restoring buoyant and gravitational head pressures creating natural simple harmonic oscillations.

Both systems oscillate with effective length equal to the equilibrium submerged/arm depth h.
MechanicsHigh Yield (2 Qs in JEE)642 PYQs

Wave Motion & Sound Waves

Wave equation, speed of sound in gases, standing waves in strings and organ pipes, beats, and Doppler effect.

Chapter

General Wave Equation & Wave Speed

y(x,t)=Asin⁡(kx±ωt+ϕ),v=ωk=fλ=Tμ  (String)y(x, t) = A\sin(kx \pm \omega t + \phi), \quad v = \frac{\omega}{k} = f\lambda = \sqrt{\frac{T}{\mu}} \; (\text{String})

Wave traveling with wavenumber k = 2π/λ and angular frequency ω = 2πf.

Minus sign (-ωt) signifies propagation in positive x-direction.

Speed of Sound (Laplace Correction)

v=γPρ=γRTMv = \sqrt{\frac{\gamma P}{\rho}} = \sqrt{\frac{\gamma R T}{M}}

Speed of acoustic waves in ideal gas under adiabatic compressions.

Speed of sound increases with temperature: v ∝ √T (in Kelvin).

Standing Waves in Organ Pipes

fn=nv2L  (n=1,2,3…,Open Pipe),fn=(2n−1)v4L  (n=1,2,3…,Closed Pipe)f_n = n \frac{v}{2L} \; (n=1,2,3\dots, \text{Open Pipe}), \quad f_n = (2n - 1) \frac{v}{4L} \; (n=1,2,3\dots, \text{Closed Pipe})

Harmonics in resonant air columns.

Fundamental frequency of open pipe is twice that of closed pipe of identical length.

Beats Frequency & Doppler Effect

fbeat=∣f1−f2∣,f′=f0(v±vov∓vs)f_{\text{beat}} = |f_1 - f_2|, \quad f' = f_0 \left( \frac{v \pm v_o}{v \mp v_s} \right)

Periodic amplitude modulation from interfering sound waves, and frequency shift due to relative motion.

Use upper signs when observer and source approach each other.

Sound Intensity & Loudness Level (Decibels)

I=12ρvω2A2=2π2ρvf2A2,β=10log⁡10(II0) dB  (I0=10−12 W/m2)I = \frac{1}{2}\rho v \omega^2 A^2 = 2\pi^2 \rho v f^2 A^2, \quad \beta = 10\log_{10}\left(\frac{I}{I_0}\right) \text{ dB} \; (I_0 = 10^{-12} \text{ W/m}^2)

Energy transmission per unit area per second and logarithmic decibel scale.

A 10-fold increase in intensity corresponds to a +10 dB loudness increase.

Resonance Tube & End Correction

e=0.6r,λ=2(L2−L1),v=2f(L2−L1)e = 0.6 r, \quad \lambda = 2(L_2 - L_1), \quad v = 2f(L_2 - L_1)

Acoustic end correction for organ pipe of inner radius r. Eliminates end correction error by subtracting first and second resonance lengths.

End correction applies at every open end: open pipe has 2e, closed pipe has 1e.

Standing Waves on Stretched Fixed String

fn=n2LTμ=nf1(n=1,2,3… )f_n = \frac{n}{2L}\sqrt{\frac{T}{\mu}} = n f_1 \quad (n = 1, 2, 3\dots)

Natural vibration frequencies of string clamped at both ends under tension T with linear mass density μ.

Nodes occur at fixed ends; anti-nodes occur at centers of vibration loops.
Thermal PhysicsTop Priority (4-5 Qs in JEE)1,411 PYQs

Heat & Thermodynamics

First law of thermodynamics, work done in isobaric/isothermal/adiabatic processes, Carnot engine, and heat transfer laws.

Chapter

First Law of Thermodynamics (Physics)

ΔQ=ΔU+W,ΔU=nCvΔT\Delta Q = \Delta U + W, \quad \Delta U = n C_v \Delta T

Energy conservation: heat supplied ΔQ equals increase in internal energy plus work done by the system.

Internal energy ΔU depends ONLY on initial and final temperatures, independent of process path.

Work Done in Thermodynamic Processes

Wisochoric=0,Wisobaric=PΔV=nRΔT,Wisothermal=nRTln⁡V2V1,Wadiabatic=P1V1−P2V2γ−1W_{\text{isochoric}} = 0, \quad W_{\text{isobaric}} = P\Delta V = nR\Delta T, \quad W_{\text{isothermal}} = nRT\ln\frac{V_2}{V_1}, \quad W_{\text{adiabatic}} = \frac{P_1 V_1 - P_2 V_2}{\gamma - 1}

Formulas for area under P-V indicator diagram.

In adiabatic process, PV^γ = constant and TV^(γ-1) = constant.

Molar Heat Capacities & Mayer's Relation

Cp−Cv=R,γ=CpCv=1+2f,Cv=f2RC_p - C_v = R, \quad \gamma = \frac{C_p}{C_v} = 1 + \frac{2}{f}, \quad C_v = \frac{f}{2}R

Degrees of freedom f: Monoatomic = 3 (γ = 5/3), Diatomic = 5 (γ = 7/5), Polyatomic non-linear = 6 (γ = 4/3).

At high temperatures, vibrational degrees of freedom add 2 extra degrees.

Carnot Engine Efficiency & Refrigerator COP

η=1−TCTH=WQH,βrefrigerator=QCW=TCTH−TC\eta = 1 - \frac{T_C}{T_H} = \frac{W}{Q_H}, \quad \beta_{\text{refrigerator}} = \frac{Q_C}{W} = \frac{T_C}{T_H - T_C}

Maximum theoretical efficiency of reversible heat engine operating between reservoirs T_H and T_C.

COP of refrigerator relates to engine efficiency: β = (1 - η) / η.

Heat Transfer Laws (Conduction & Radiation)

dQdt=kA(T1−T2)L,Pradiation=σeA(T4−T04),σ=5.67×10−8 W m−2K−4\frac{dQ}{dt} = \frac{kA(T_1 - T_2)}{L}, \quad P_{\text{radiation}} = \sigma e A (T^4 - T_0^4), \quad \sigma = 5.67 \times 10^{-8} \text{ W m}^{-2}\text{K}^{-4}

Fourier conduction law, Stefan-Boltzmann radiation law for net radiation from body at T in surrounding T₀. ε = emissivity (0 ≤ ε ≤ 1; black body ε = 1).

Thermal resistance R_th = L / (kA). Series thermal resistance R_s = R₁ + R₂. Parallel: 1/R_p = 1/R₁ + 1/R₂.

Wien's Displacement Law & Black Body Radiation

λmT=b=2.898×10−3 m⋅K,Eλ∝1λ5⋅1ehc/λkBT−1  (Planck’s Law)\lambda_m T = b = 2.898 \times 10^{-3} \text{ m·K}, \quad E_\lambda \propto \frac{1}{\lambda^5} \cdot \frac{1}{e^{hc/\lambda k_B T} - 1} \; (\text{Planck's Law})

Wien's law: wavelength of peak emission intensity is inversely proportional to absolute temperature. As temperature increases, peak shifts to shorter wavelengths (blue shift).

Sun (T ≈ 5800 K): λ_m ≈ 500 nm (visible green). Human body (T ≈ 310 K): λ_m ≈ 9.4 μm (infrared). Used in NEET & JEE to identify stellar temperatures from emission spectra.

Van der Waals Gas & Critical Constants

(P+aVm2)(Vm−b)=RT,Tc=8a27Rb,Pc=a27b2,Vc=3b,Zc=PcVcRTc=38\left(P + \frac{a}{V_m^2}\right)(V_m - b) = RT, \quad T_c = \frac{8a}{27Rb}, \quad P_c = \frac{a}{27b^2}, \quad V_c = 3b, \quad Z_c = \frac{P_c V_c}{RT_c} = \frac{3}{8}

Van der Waals constants: a accounts for intermolecular attraction (pressure correction), b accounts for finite molecular volume (volume correction). Critical compressibility factor Z_c = 3/8 for any VdW gas.

Boyle temperature T_B = a/(Rb) where the gas behaves most ideally over a wide pressure range. T_B = (27/8) T_c. At T < T_c, the gas can be liquefied by pressure alone.

Kinetic Theory of Gases (KTG) Speeds

vrms=3RTM,vavg=8RTπM,vmp=2RTM,vrms:vavg:vmp=3:8/π:2v_{\text{rms}} = \sqrt{\frac{3RT}{M}}, \quad v_{\text{avg}} = \sqrt{\frac{8RT}{\pi M}}, \quad v_{\text{mp}} = \sqrt{\frac{2RT}{M}}, \quad v_{\text{rms}} : v_{\text{avg}} : v_{\text{mp}} = \sqrt{3} : \sqrt{8/\pi} : \sqrt{2}

Root-mean-square, average, and most probable speeds of ideal gas molecules of molar mass M.

Total translational kinetic energy of n moles of any gas is E = (3/2) nRT = (3/2) PV.

Mean Free Path & Pressure of Ideal Gas

λ=12πnd2=kBT2πd2P,P=13ρvrms2=23Ev\lambda = \frac{1}{\sqrt{2}\pi n d^2} = \frac{k_B T}{\sqrt{2}\pi d^2 P}, \quad P = \frac{1}{3}\rho v_{\text{rms}}^2 = \frac{2}{3}E_v

Average distance traveled by a molecule of diameter d between successive collisions in molecular density n.

Mean free path λ is directly proportional to temperature T and inversely proportional to pressure P.

Newton's Law of Cooling

dTdt=−K(T−T0)  ⟹  T1−T2t=K(T1+T22−T0)\frac{dT}{dt} = -K(T - T_0) \implies \frac{T_1 - T_2}{t} = K\left(\frac{T_1 + T_2}{2} - T_0\right)

Rate of loss of heat by a body is directly proportional to temperature difference between body and surroundings.

Valid strictly when (T - T₀) is small (typically ≤ 30 °C). Exact solution is T(t) = T₀ + (T_initial - T₀) e^(-Kt).
ElectromagnetismTop Priority (3-4 Qs in JEE)817 PYQs

Electrostatics

Coulomb's law, Gauss's theorem, electric field, potential, dipoles, and electrostatic potential energy.

Chapter

Coulomb's Law & Permittivity

F=14πε0εrq1q2r2,14πε0≈9×109 N m2/C2F = \frac{1}{4\pi\varepsilon_0 \varepsilon_r} \frac{q_1 q_2}{r^2}, \quad \frac{1}{4\pi\varepsilon_0} \approx 9 \times 10^9 \text{ N m}^2/\text{C}^2

Inverse-square force between point charges in medium of relative permittivity ε_r.

Force in medium of dielectric constant K is F_air / K.

Gauss's Law & Electric Flux

ΦE=∮E⃗⋅dA⃗=Qenclosedε0\Phi_E = \oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enclosed}}}{\varepsilon_0}

Total electric flux through closed Gaussian surface equals enclosed charge divided by ε₀.

Charges outside the Gaussian surface contribute zero net flux.

Electric Field of Standard Geometries

Ewire=2kλr,Esheet=σ2ε0,Ering, axis=kqx(R2+x2)3/2E_{\text{wire}} = \frac{2k\lambda}{r}, \quad E_{\text{sheet}} = \frac{\sigma}{2\varepsilon_0}, \quad E_{\text{ring, axis}} = \frac{kqx}{(R^2 + x^2)^{3/2}}

Field formulas derived via Gauss's law for high-symmetry distributions.

Maximum field on ring axis occurs at x = R / √2.

Electric Dipole Fields & Torque

Eaxial=2kpr3,Eequatorial=kpr3,τ⃗=p⃗×E⃗,U=−p⃗⋅E⃗E_{\text{axial}} = \frac{2kp}{r^3}, \quad E_{\text{equatorial}} = \frac{kp}{r^3}, \quad \vec{\tau} = \vec{p} \times \vec{E}, \quad U = -\vec{p} \cdot \vec{E}

Field and potential energy of electric dipole p = q(2a) at distant points (r >> a).

Axial field is exactly twice the equatorial field: E_axial = 2 E_eq.

Relation Between Field and Potential

E⃗=−∇⃗V=−(∂V∂xi^+∂V∂yj^+∂V∂zk^),ΔV=−∫E⃗⋅dr⃗\vec{E} = -\vec{\nabla} V = -\left(\frac{\partial V}{\partial x}\hat{i} + \frac{\partial V}{\partial y}\hat{j} + \frac{\partial V}{\partial z}\hat{k}\right), \quad \Delta V = -\int \vec{E} \cdot d\vec{r}

Electric field points in direction of steepest decrease of potential.

Equipotential surfaces are always perpendicular to electric field lines.

Uniform Solid Dielectric Sphere (Field & Potential)

r≤R:  E(r)=kQrR3=ρr3ε0,V(r)=kQ2R3(3R2−r2)  ⟹  Vcenter=1.5 Vsurfacer \le R: \; E(r) = \frac{k Q r}{R^3} = \frac{\rho r}{3\varepsilon_0}, \quad V(r) = \frac{kQ}{2R^3}(3R^2 - r^2) \implies V_{\text{center}} = 1.5 \, V_{\text{surface}}

Linear internal electric field and parabolic internal potential distribution for uniform volume charge density ρ.

Potential at the exact center of a uniformly charged solid non-conducting sphere is 1.5 times the surface potential.

Electrostatic Self-Energy of Spheres

Uconducting shell=kQ22R,Usolid insulating sphere=35kQ2RU_{\text{conducting shell}} = \frac{k Q^2}{2R}, \quad U_{\text{solid insulating sphere}} = \frac{3}{5}\frac{k Q^2}{R}

Work done in assembling a continuous spherical charge distribution against mutual electrostatic repulsion.

For concentric shells: total energy is sum of individual self-energies plus mutual interaction energy.

Earthing of Concentric Conducting Shells

Vearthed shell=0  ⟹  ∑ikqiReffective=0V_{\text{earthed shell}} = 0 \implies \sum_{i} \frac{k q_i}{R_{\text{effective}}} = 0

When a conducting shell is grounded to Earth, charge flows to/from Earth until its net electrostatic potential becomes exactly zero.

Earthing a shell does NOT mean its charge is zero; it means its net potential V = 0.
ElectromagnetismHigh Yield (2-3 Qs in JEE)431 PYQs

Capacitance & Dielectrics

Parallel plate capacitors, dielectric insertion, stored energy, common potential, and transient RC circuits.

Chapter

Parallel Plate Capacitance

C=ε0Ad,Cdielectric=ε0Ad−t+tKC = \frac{\varepsilon_0 A}{d}, \quad C_{\text{dielectric}} = \frac{\varepsilon_0 A}{d - t + \frac{t}{K}}

Capacitance with vacuum and with dielectric slab of thickness t and constant K.

Filling entire gap with dielectric increases capacitance to K*C₀.

Stored Energy & Energy Density

U=12CV2=Q22C=12QV,uE=12ε0E2U = \frac{1}{2}CV^2 = \frac{Q^2}{2C} = \frac{1}{2}QV, \quad u_E = \frac{1}{2}\varepsilon_0 E^2

Electrostatic potential energy stored in the electric field between capacitor plates.

Energy density u_E has units of Joules per cubic meter (J/m³).

Combinations of Capacitors

Cparallel=∑Ci,1Cseries=∑1CiC_{\text{parallel}} = \sum C_i, \quad \frac{1}{C_{\text{series}}} = \sum \frac{1}{C_i}

Series combination has same charge on each capacitor; parallel has same potential difference.

Notice rules are opposite to resistor combinations.

Charge Sharing & Heat Loss

Vcommon=C1V1+C2V2C1+C2,ΔH=12C1C2C1+C2(V1−V2)2V_{\text{common}} = \frac{C_1 V_1 + C_2 V_2}{C_1 + C_2}, \quad \Delta H = \frac{1}{2}\frac{C_1 C_2}{C_1 + C_2}(V_1 - V_2)^2

Common potential and heat dissipated when two charged capacitors are connected.

Heat loss ΔH is always positive (energy is lost, never gained).

Transient RC Circuit (Charging & Discharging)

q(t)=Q0(1−e−t/RC)  (Charging),q(t)=Q0e−t/RC  (Discharging),τ=RCq(t) = Q_0(1 - e^{-t/RC}) \; (\text{Charging}), \quad q(t) = Q_0 e^{-t/RC} \; (\text{Discharging}), \quad \tau = RC

Exponential charging and discharging with capacitive time constant τ = RC.

At t = τ, capacitor charges to ~63.2% of Q₀; at t = 5τ, it is ~99.3% charged.

Attractive Force Between Plates & Electrostatic Pressure

F=Q22ε0A=12QE,Pelec=FA=σ22ε0=12ε0E2F = \frac{Q^2}{2\varepsilon_0 A} = \frac{1}{2}QE, \quad P_{\text{elec}} = \frac{F}{A} = \frac{\sigma^2}{2\varepsilon_0} = \frac{1}{2}\varepsilon_0 E^2

Mutual electrostatic attraction between oppositely charged plates, and electrostatic outward pulling pressure.

The factor of 1/2 arises because the charge on one plate experiences the electric field E/2 produced solely by the other plate.

Spherical & Cylindrical Capacitor Geometries

Cspherical=4πε0(abb−a),Cisolated sphere=4πε0R,CcylindricalL=2πε0ln⁡(b/a)C_{\text{spherical}} = 4\pi\varepsilon_0 \left(\frac{ab}{b - a}\right), \quad C_{\text{isolated sphere}} = 4\pi\varepsilon_0 R, \quad \frac{C_{\text{cylindrical}}}{L} = \frac{2\pi\varepsilon_0}{\ln(b/a)}

Capacitance formulas for concentric spherical shells and coaxial cylindrical conductors of radii a and b (b > a).

For Earth (R ≈ 6400 km), isolated spherical capacitance is C ≈ 711 μF.

Dielectric Slab Partially Inserted (Series & Parallel)

Slab (thickness t, full area):  C=ε0Ad−t+t/K(series stacking)Slab (partial area A1, rest A2):  C=ε0KA1d+ε0A2d=ε0d(KA1+A2)\text{Slab (thickness } t\text{, full area):}\; C = \frac{\varepsilon_0 A}{d - t + t/K} \quad \text{(series stacking)}\\[4pt] \text{Slab (partial area } A_1\text{, rest } A_2\text{):}\; C = \frac{\varepsilon_0 K A_1}{d} + \frac{\varepsilon_0 A_2}{d} = \frac{\varepsilon_0}{d}(K A_1 + A_2)

When dielectric occupies only part of the gap thickness (series): effective gap is reduced by t(1 - 1/K). When dielectric covers part of the plate area (parallel): two sub-capacitors add directly.

Battery connected while inserting: V = const → Q increases, C increases, E between plates unchanged. Battery disconnected: Q = const → V decreases, C increases, E decreases. Energy is LOST when battery connected, REDUCED when disconnected.
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Current Electricity

Ohm's law, drift velocity, resistivity, Kirchhoff's laws, Wheatstone bridge, potentiometer, and electrical power.

Chapter

Drift Velocity & Current Density

I=neAvd,vd=eEτm,j=σE=Eρ,ρ=mne2τI = n e A v_d, \quad v_d = \frac{e E \tau}{m}, \quad j = \sigma E = \frac{E}{\rho}, \quad \rho = \frac{m}{n e^2 \tau}

Microscopic electron transport relations.

Relaxation time τ decreases with temperature in metals, increasing resistivity ρ.

Resistance & Temperature Coefficient

R=ρLA,RT=R0(1+αΔT)R = \rho \frac{L}{A}, \quad R_T = R_0(1 + \alpha \Delta T)

Geometric resistance formula and thermal resistance variation.

Stretching a wire to n times its length increases resistance by a factor of n² (volume constant).

Kirchhoff's Laws (KCL & KVL)

∑Inode=0  (Charge Conservation),∑ΔVloop=0  (Energy Conservation)\sum I_{\text{node}} = 0 \; (\text{Charge Conservation}), \quad \sum \Delta V_{\text{loop}} = 0 \; (\text{Energy Conservation})

Fundamental nodal current and loop voltage laws for circuit analysis.

Always maintain consistent sign conventions across resistors and battery terminals.

Grouping of Cells (EMF & Internal Resistance)

Eeq=∑Ei/ri∑1/ri  (Parallel),1req=∑1ri,Eeq=∑Ei  (Series)E_{eq} = \frac{\sum E_i / r_i}{\sum 1 / r_i} \; (\text{Parallel}), \quad \frac{1}{r_{eq}} = \sum \frac{1}{r_i}, \quad E_{eq} = \sum E_i \; (\text{Series})

Equivalent battery combination formulas.

Maximum power transfer occurs when external load resistance equals internal resistance (R = r).

Wheatstone Bridge & Meter Bridge

PQ=RS  ⟹  IG=0,Meter Bridge: RS=l100−l\frac{P}{Q} = \frac{R}{S} \implies I_G = 0, \quad \text{Meter Bridge: } \frac{R}{S} = \frac{l}{100 - l}

Null deflection condition for measuring unknown resistances.

Sensitivity is maximum when all four branch resistances are of comparable magnitude.

Potentiometer Principle

E1E2=l1l2,r=R(l1l2−1)\frac{E_1}{E_2} = \frac{l_1}{l_2}, \quad r = R\left(\frac{l_1}{l_2} - 1\right)

Compares EMFs and determines internal resistance without drawing current from the test cell.

Potentiometer is superior to voltmeter because it draws zero current at balance point.

Maximum Power Transfer Theorem & Heating

Pload=E2R(R+r)2  ⟹  Pmax=E24r(when R=r),H=I2RtP_{\text{load}} = \frac{E^2 R}{(R + r)^2} \implies P_{\text{max}} = \frac{E^2}{4r} \quad (\text{when } R = r), \quad H = I^2 R t

Maximum electrical power is transferred to load when load resistance equals the internal resistance of the power source.

Efficiency of power transfer at maximum power output is exactly 50% (half energy dissipated in internal resistance r).

Seebeck Effect & Thermoelectric Inversion

E=αT+12βT2,Tn=−αβ,Ti−Tn=Tn−T0  ⟹  Ti=2Tn−T0E = \alpha T + \frac{1}{2}\beta T^2, \quad T_n = -\frac{\alpha}{\beta}, \quad T_i - T_n = T_n - T_0 \implies T_i = 2T_n - T_0

Thermoelectric EMF generated across thermocouple junctions at temperatures T and T₀ (T_n is neutral temperature, T_i is inversion temperature).

At neutral temperature T_n, thermoelectric power (dE/dT) is zero and EMF reaches its peak maximum.

Joule's Heating Law & Electrical Power

P=VI=I2R=V2R,H=Pt=I2Rt=V2tR,kWh=1 unit=3.6×106 JP = VI = I^2 R = \frac{V^2}{R}, \quad H = Pt = I^2 R t = \frac{V^2 t}{R}, \quad \text{kWh} = 1 \text{ unit} = 3.6 \times 10^6 \text{ J}

Heat produced by current I through resistance R in time t. All three forms of power (P = VI, I²R, V²/R) are equivalent by Ohm's law.

For resistors in series, same I → more heat in larger R. For parallel, same V → more heat in smaller R. Domestic electrical energy billing unit 'one unit' = 1 kWh.
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Magnetic Effects of Current

Biot-Savart law, Ampere's circuital law, magnetic field of straight wire, circular loop, solenoid, and Lorentz magnetic force.

Chapter

Lorentz Magnetic Force

F⃗m=q(v⃗×B⃗),r=mv⊥qB,T=2πmqB\vec{F}_m = q(\vec{v} \times \vec{B}), \quad r = \frac{m v_\perp}{qB}, \quad T = \frac{2\pi m}{qB}

Force on moving charge in magnetic field and helical circular radius.

Time period and cyclotron frequency are completely independent of particle speed v.

Biot-Savart Law

dB⃗=μ04πI(dl⃗×r^)r2,μ04π=10−7 T m/Ad\vec{B} = \frac{\mu_0}{4\pi} \frac{I (d\vec{l} \times \hat{r})}{r^2}, \quad \frac{\mu_0}{4\pi} = 10^{-7} \text{ T m/A}

Fundamental differential magnetic field generated by current element.

Direction is given by right-hand palm rule.

Magnetic Field of Standard Geometries

Bwire=μ0I2πd,Bring, center=μ0I2R,Bring, axis=μ0IR22(R2+x2)3/2B_{\text{wire}} = \frac{\mu_0 I}{2\pi d}, \quad B_{\text{ring, center}} = \frac{\mu_0 I}{2R}, \quad B_{\text{ring, axis}} = \frac{\mu_0 I R^2}{2(R^2 + x^2)^{3/2}}

Field formulas derived from Biot-Savart integration.

For ideal long solenoid, B = μ₀ n I inside, and 0 outside.

Force Between Two Parallel Current Wires

FL=μ0I1I22πd\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}

Attractive if currents flow in same direction; repulsive if in opposite directions.

Used to define the international Ampere standard.

Magnetic Dipole Moment & Torque on Loop

M⃗=NIA⃗,τ⃗=M⃗×B⃗,U=−M⃗⋅B⃗\vec{M} = N I \vec{A}, \quad \vec{\tau} = \vec{M} \times \vec{B}, \quad U = -\vec{M} \cdot \vec{B}

Magnetic dipole behavior of planar current loop.

Gyromagnetic ratio for revolving electron: M / L = e / (2m).

Ampère's Circuital Law, Solenoid & Thick Wire

∮B⃗⋅dl⃗=μ0Ienc,Bsolenoid=μ0nI,Bthick wire(r≤R)=μ0Ir2πR2\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}}, \quad B_{\text{solenoid}} = \mu_0 n I, \quad B_{\text{thick wire}}(r \le R) = \frac{\mu_0 I r}{2\pi R^2}

Circuital law line integral and internal fields for infinite solenoid, toroid, and solid cylindrical conductors.

Field at the exact ends of a semi-infinite solenoid is exactly half the interior field: B_end = (1/2)μ₀ n I.

Moving Coil Galvanometer, Ammeter & Voltmeter

θ=(NABC)I,Shunt: S=IgRgI−Ig,Multiplier: R=VIg−Rg\theta = \left(\frac{NAB}{C}\right)I, \quad \text{Shunt: } S = \frac{I_g R_g}{I - I_g}, \quad \text{Multiplier: } R = \frac{V}{I_g} - R_g

Torsional suspension equilibrium and conversion into ammeter (low parallel shunt S) and voltmeter (high series resistance R).

Ideal ammeter has zero resistance; ideal voltmeter has infinite resistance.

Bohr Magneton & Revolving Electron

μl=(e2m)L=nμB,μB=eℏ2m=eh4πm≈9.27×10−24 A⋅m2\mu_l = \left(\frac{e}{2m}\right)L = n\mu_B, \quad \mu_B = \frac{e\hbar}{2m} = \frac{eh}{4\pi m} \approx 9.27 \times 10^{-24}\,\text{A}\cdot\text{m}^2

Magnetic moment of an electron orbiting a nucleus in the n-th Bohr orbit.

Gyromagnetic ratio e/(2m) is a universal constant independent of orbit radius and velocity.
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Electromagnetic Induction (EMI)

Faraday's law, Lenz's law, motional EMF, self & mutual inductance, and LR growth/decay circuits.

Chapter

Faraday's & Lenz's Law

ε=−dΦBdt,ΦB=B⃗⋅A⃗=BAcos⁡θ\varepsilon = -\frac{d\Phi_B}{dt}, \quad \Phi_B = \vec{B} \cdot \vec{A} = B A \cos\theta

Induced electromotive force opposing changing magnetic flux.

Total induced charge q = ΔΦ / R is independent of the rate of change of flux.

Motional EMF

ε=(v⃗×B⃗)⋅l⃗=Bvl  (Translational),ε=12BωL2  (Rotating Rod)\varepsilon = (\vec{v} \times \vec{B}) \cdot \vec{l} = B v l \; (\text{Translational}), \quad \varepsilon = \frac{1}{2}B \omega L^2 \; (\text{Rotating Rod})

Voltage induced across conductors moving through magnetic field.

Induced current direction given by Fleming's Right-Hand Rule.

Self & Mutual Inductance

Φ=LI  ⟹  ε=−LdIdt,Lsolenoid=μ0n2Al=μ0N2Al,M=kL1L2\Phi = L I \implies \varepsilon = -L \frac{dI}{dt}, \quad L_{\text{solenoid}} = \mu_0 n^2 A l = \frac{\mu_0 N^2 A}{l}, \quad M = k\sqrt{L_1 L_2}

Inertia of electric current in coils and coupling coefficient k.

Stored magnetic energy density: u_B = B² / (2μ₀).

Transient LR Circuit (Current Growth & Decay)

I(t)=VR(1−e−t/τ)  (Growth),I(t)=I0e−t/τ  (Decay),τ=LRI(t) = \frac{V}{R}(1 - e^{-t / \tau}) \; (\text{Growth}), \quad I(t) = I_0 e^{-t / \tau} \; (\text{Decay}), \quad \tau = \frac{L}{R}

Inductive time constant governing current evolution.

At t = 0, inductor acts as open circuit; at t -> ∞, acts as ideal short wire.

Non-Conservative Induced Electric Field

∮E⃗⋅dl⃗=−dΦBdt  ⟹  E(r)={r2dBdt(r≤R)R22rdBdt(r>R)\oint \vec{E} \cdot d\vec{l} = -\frac{d\Phi_B}{dt} \implies E(r) = \begin{cases} \frac{r}{2}\frac{dB}{dt} & (r \le R) \\ \frac{R^2}{2r}\frac{dB}{dt} & (r > R) \end{cases}

Electric field produced by time-varying magnetic flux inside and outside a cylindrical field region of radius R.

Unlike electrostatic fields, induced electric field lines form closed continuous loops and work done along a closed path is non-zero.

Magnetic Force & Power Dissipated in Moving Loop

Fmagnetic=B2L2vR,Pmechanical=PJoule=B2L2v2R,v(t)=v0e−t/τF_{\text{magnetic}} = \frac{B^2 L^2 v}{R}, \quad P_{\text{mechanical}} = P_{\text{Joule}} = \frac{B^2 L^2 v^2}{R}, \quad v(t) = v_0 e^{-t / \tau}

Retarding Lorentz braking force on a conducting rod sliding across uniform field rails with load resistance R.

Mechanical power input equals rate of Joule heat dissipation in the circuit, verifying energy conservation.

Inductance Combinations & Stored Magnetic Energy

UB=12LI2,uB=B22μ0,Lseries=L1+L2±2M,Lparallel=L1L2−M2L1+L2∓2MU_B = \frac{1}{2} L I^2, \quad u_B = \frac{B^2}{2\mu_0}, \quad L_{\text{series}} = L_1 + L_2 \pm 2M, \quad L_{\text{parallel}} = \frac{L_1 L_2 - M^2}{L_1 + L_2 \mp 2M}

Total magnetic energy stored in current-carrying inductor and equivalent inductance of coupled coils.

Use +2M when magnetic fluxes aid each other; use -2M when fluxes oppose each other.

AC Generator (Rotating Coil) EMF

ε=NBAωsin⁡(ωt)=ε0sin⁡(ωt),ε0=NBAω,εrms=ε02=NBAω2\varepsilon = NBA\omega\sin(\omega t) = \varepsilon_0\sin(\omega t), \quad \varepsilon_0 = NBA\omega, \quad \varepsilon_{\text{rms}} = \frac{\varepsilon_0}{\sqrt{2}} = \frac{NBA\omega}{\sqrt{2}}

EMF produced by N-turn coil of area A rotating with angular velocity ω in uniform magnetic field B. Peak EMF ε₀ = NBAω. Coil plane is parallel to B at maximum EMF; perpendicular to B at zero EMF.

Maximum EMF occurs when coil plane is parallel to B (flux = 0, rate of change of flux is maximum). Zero EMF when coil is perpendicular to B (flux maximum, rate of change = 0). Phase difference between flux and EMF is 90°.

Eddy Currents — Power Loss & Applications

Peddy∝B2f2t2/ρresistivity,Skin depth: δ=2ρωμP_{\text{eddy}} \propto B^2 f^2 t^2 / \rho_{\text{resistivity}}, \quad \text{Skin depth: } \delta = \sqrt{\frac{2\rho}{\omega \mu}}

Eddy currents are induced in bulk conductors by changing B. Loss is reduced by laminating the core (thin silicon steel sheets increase resistivity path). Power loss ∝ (frequency)² × (thickness)² × (B)².

Applications: electromagnetic braking (train brakes), induction furnace (heating by eddy currents), galvanometer damping, speedometers. Silicon steel (high resistivity) is used in transformer cores to minimise eddy current losses.
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Alternating Current (AC)

RMS and peak values, reactance, impedance of LCR circuits, resonance, Q-factor, and power factor.

Chapter

RMS & Average Values of AC

Irms=I02≈0.707I0,Iavg=2I0π≈0.637I0  (half cycle)I_{\text{rms}} = \frac{I_0}{\sqrt{2}} \approx 0.707 I_0, \quad I_{\text{avg}} = \frac{2I_0}{\pi} \approx 0.637 I_0 \; (\text{half cycle})

Root-mean-square effective heating current and half-wave rectification average.

Domestic AC supply of 220 V means V_rms = 220 V (peak V₀ ≈ 311 V).

Inductive & Capacitive Reactance

XL=ωL=2πfL,XC=1ωC=12πfCX_L = \omega L = 2\pi f L, \quad X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C}

Frequency-dependent opposition to current.

Inductor blocks high frequencies (X_L ∝ f); capacitor blocks DC (X_C -> ∞ at f = 0).

Series LCR Impedance & Phase Angle

Z=R2+(XL−XC)2,tan⁡ϕ=XL−XCRZ = \sqrt{R^2 + (X_L - X_C)^2}, \quad \tan\phi = \frac{X_L - X_C}{R}

Total impedance vector and phase angle between voltage and current.

Current leads voltage if X_C > X_L (capacitive); lags if X_L > X_C (inductive).

Resonance & Quality Factor in LCR

ω0=1LC  ⟹  f0=12πLC,Zmin=R,Q=ω0LR=1RLC\omega_0 = \frac{1}{\sqrt{LC}} \implies f_0 = \frac{1}{2\pi\sqrt{LC}}, \quad Z_{\text{min}} = R, \quad Q = \frac{\omega_0 L}{R} = \frac{1}{R}\sqrt{\frac{L}{C}}

Condition when inductive and capacitive reactances cancel out.

At resonance, current is maximum, impedance is purely resistive, and power factor cos φ = 1.

Average Power & Power Factor

Pavg=VrmsIrmscos⁡ϕ,Power Factor cos⁡ϕ=RZP_{\text{avg}} = V_{\text{rms}} I_{\text{rms}} \cos\phi, \quad \text{Power Factor } \cos\phi = \frac{R}{Z}

Real electrical power dissipated in AC circuit.

Wattless current component is I_rms * sin φ.

Transformer Equation & Efficiency

VsVp=NsNp=IpIs,η=PoutPin=VsIsVpIp\frac{V_s}{V_p} = \frac{N_s}{N_p} = \frac{I_p}{I_s}, \quad \eta = \frac{P_{\text{out}}}{P_{\text{in}}} = \frac{V_s I_s}{V_p I_p}

Voltage transformation ratio in mutual induction transformer.

Step-up transformer increases voltage but decreases current.

LC Oscillations & Total Stored Energy

q(t)=Q0cos⁡(ω0t),I(t)=−ω0Q0sin⁡(ω0t),Etotal=q22C+12LI2=Q022Cq(t) = Q_0 \cos(\omega_0 t), \quad I(t) = -\omega_0 Q_0 \sin(\omega_0 t), \quad E_{\text{total}} = \frac{q^2}{2C} + \frac{1}{2}LI^2 = \frac{Q_0^2}{2C}

Harmonic energy exchange between electrical field in capacitor and magnetic field in inductor.

Analogy with mechanical SHM: mass m ↔ inductance L, spring constant k ↔ 1/C, velocity v ↔ current I.

Bandwidth & Half-Power Frequencies in LCR

Δω=ω2−ω1=RL,Q=ω0Δω=ω0LR=1RLC\Delta\omega = \omega_2 - \omega_1 = \frac{R}{L}, \quad Q = \frac{\omega_0}{\Delta\omega} = \frac{\omega_0 L}{R} = \frac{1}{R}\sqrt{\frac{L}{C}}

Frequency width between half-power points where current drops to I_max / √2. Smaller bandwidth implies sharper resonance.

High Q circuits are required in radio receivers for tuning sharply to a single carrier frequency.
ElectromagnetismHigh Yield (1 Q in JEE)299 PYQs

Electromagnetic Waves (EM Waves)

Maxwell equations, displacement current, wave speed in vacuum, Poynting vector, and electromagnetic spectrum.

Chapter

Wave Speed & Field Amplitude Ratio

c=1μ0ε0=E0B0=ErmsBrms≈3×108 m/sc = \frac{1}{\sqrt{\mu_0 \varepsilon_0}} = \frac{E_0}{B_0} = \frac{E_{\text{rms}}}{B_{\text{rms}}} \approx 3 \times 10^8 \text{ m/s}

Fundamental connection between permittivity, permeability, and optical field amplitudes.

In a medium of refractive index n: v = c / n = 1 / √(μ ε).

Energy Density & Intensity of EM Waves

uavg=12ε0E02=B022μ0,I=uavgc=12ε0cE02u_{\text{avg}} = \frac{1}{2}\varepsilon_0 E_0^2 = \frac{B_0^2}{2\mu_0}, \quad I = u_{\text{avg}} c = \frac{1}{2}\varepsilon_0 c E_0^2

Energy is divided equally between electric and magnetic fields.

Poynting vector S = (1/μ₀)(E × B) represents directional power flux.

Displacement Current

Id=ε0dΦEdt=CdVdtI_d = \varepsilon_0 \frac{d\Phi_E}{dt} = C \frac{dV}{dt}

Maxwell's addition to Ampere's circuital law inside capacitor dielectric gap.

Displacement current between capacitor plates equals conduction current in lead wires.

Poynting Vector (Directional Energy Flux)

S⃗=1μ0(E⃗×B⃗),I=⟨S⟩=E0B02μ0=E022μ0c=12ε0cE02\vec{S} = \frac{1}{\mu_0}(\vec{E} \times \vec{B}), \quad I = \langle S \rangle = \frac{E_0 B_0}{2\mu_0} = \frac{E_0^2}{2\mu_0 c} = \frac{1}{2}\varepsilon_0 c E_0^2

Instantaneous directional energy flow per unit area carried by propagating electromagnetic waves.

Wave propagates strictly in the direction of vector cross product (E × B).

Radiation Pressure & Momentum

p=Uc,Prad={Ic(Complete Absorption)2Ic(Complete Reflection)p = \frac{U}{c}, \quad P_{\text{rad}} = \begin{cases} \frac{I}{c} & (\text{Complete Absorption}) \\ \frac{2I}{c} & (\text{Complete Reflection}) \end{cases}

Mechanical force per unit area exerted by electromagnetic radiation on physical surfaces.

For oblique incidence at angle θ: P_rad = (I/c) cos²θ (absorption); P_rad = (2I/c) cos²θ (reflection).

Free Space Impedance & EM Spectrum Hierarchy

Z0=μ0ε0=E0H0≈376.7 Ω≈120π ΩZ_0 = \sqrt{\frac{\mu_0}{\varepsilon_0}} = \frac{E_0}{H_0} \approx 376.7 \,\Omega \approx 120\pi \,\Omega

Intrinsic wave impedance of vacuum. Frequency order: Radio < Micro < IR < Visible < UV < X-ray < Gamma.

Memory mnemonic for increasing frequency / decreasing wavelength: R-M-I-V-U-X-G (Radio, Microwave, IR, Visible, UV, X-ray, Gamma).

Maxwell's Four Fundamental Equations

∮E⃗⋅dA⃗=Qε0,  ∮B⃗⋅dA⃗=0,  ∮E⃗⋅dl⃗=−dΦBdt,  ∮B⃗⋅dl⃗=μ0(Ic+ε0dΦEdt)\oint \vec{E}\cdot d\vec{A} = \frac{Q}{\varepsilon_0}, \; \oint \vec{B}\cdot d\vec{A} = 0, \; \oint \vec{E}\cdot d\vec{l} = -\frac{d\Phi_B}{dt}, \; \oint \vec{B}\cdot d\vec{l} = \mu_0\left(I_c + \varepsilon_0\frac{d\Phi_E}{dt}\right)

Unification of electricity and magnetism describing generation of fields by charges, currents, and time-varying opposite fields.

Gauss's law for magnetism ∮ B · dA = 0 implies that magnetic monopoles do not exist in nature.

Wave Speed in Material Media & Refractive Index

v=1με=cμrεr=cn  ⟹  n≈εr=K(for μr≈1)v = \frac{1}{\sqrt{\mu \varepsilon}} = \frac{c}{\sqrt{\mu_r \varepsilon_r}} = \frac{c}{n} \implies n \approx \sqrt{\varepsilon_r} = \sqrt{K} \quad (\text{for } \mu_r \approx 1)

Maxwell's relation linking optical refractive index n to static dielectric constant K.

Both E and B amplitudes decrease by factor n in dielectric medium, while wave frequency f remains constant.
OpticsTop Priority (3-4 Qs in JEE)698 PYQs

Ray Optics & Optical Instruments

Snell's law, total internal reflection, lens maker's formula, thin lenses, prism formula, and microscopes/telescopes.

Chapter

Mirror & Thin Lens Formulas

1f=1v+1u  (Mirror),1f=1v−1u  (Lens),m=−vu  (Mirror),m=vu  (Lens)\frac{1}{f} = \frac{1}{v} + \frac{1}{u} \; (\text{Mirror}), \quad \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \; (\text{Lens}), \quad m = -\frac{v}{u} \; (\text{Mirror}), \quad m = \frac{v}{u} \; (\text{Lens})

Fundamental imaging equations with focal length and lateral magnification.

Always substitute numerical values with their proper algebraic signs.

Snell's Law & Total Internal Reflection (TIR)

n1sin⁡i=n2sin⁡r,sin⁡θc=n2n1=1μn_1 \sin i = n_2 \sin r, \quad \sin\theta_c = \frac{n_2}{n_1} = \frac{1}{\mu}

Law of refraction and critical angle condition for TIR (light traveling from denser to rarer medium).

Optical fiber core refractive index must be greater than cladding (n_core > n_cladding).

Lens Maker's Formula

1f=(μ−1)(1R1−1R2),P=1f (in meters)\frac{1}{f} = (\mu - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right), \quad P = \frac{1}{f \text{ (in meters)}}

Determines focal length from glass refractive index and surface curvature radii.

Immersing a glass lens in water increases its focal length by roughly a factor of 4.

Prism Formula (Minimum Deviation)

A+δ=i+e,μ=sin⁡(A+δm2)sin⁡(A2)A + \delta = i + e, \quad \mu = \frac{\sin\left(\frac{A + \delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)}

Connects prism angle A, angle of minimum deviation δ_m, and refractive index μ.

At minimum deviation: i = e and r₁ = r₂ = A/2.

Compound Microscope & Astronomical Telescope

mmicroscope=−vouo(1+Dfe),mtelescope=−fofe,Ltelescope=fo+fem_{\text{microscope}} = -\frac{v_o}{u_o}\left(1 + \frac{D}{f_e}\right), \quad m_{\text{telescope}} = -\frac{f_o}{f_e}, \quad L_{\text{telescope}} = f_o + f_e

Magnifying power for visual optical instruments at normal adjustment.

D = least distance of distinct vision = 25 cm.

Refraction at Spherical Curved Surface

μ2v−μ1u=μ2−μ1R,m=μ1vμ2u\frac{\mu_2}{v} - \frac{\mu_1}{u} = \frac{\mu_2 - \mu_1}{R}, \quad m = \frac{\mu_1 v}{\mu_2 u}

General Gaussian refraction formula at curved dielectric boundary separating media μ₁ and μ₂.

Apparent depth for flat boundary (R → ∞): d' = d / (μ₁/μ₂) = d / μ_rel.

Silvering of Lenses (Equivalent Curved Mirror)

1Feq=2fl+1fm  ⟹  Peq=2Pl+Pm\frac{1}{F_{\text{eq}}} = \frac{2}{f_l} + \frac{1}{f_m} \implies P_{\text{eq}} = 2P_l + P_m

A silvered thin lens acts as a concave or convex mirror. Light passes through lens twice and reflects once.

Plano-convex lens silvered on flat face acts as concave mirror of focal length F = R / (2(μ - 1)).

Normal Shift by Glass Slab & Lens Cutting Rules

Δt=t(1−1μ),Cut Along Axis: f′=f,  I′=I2,Cut Perpendicular to Axis: f′=2f\Delta t = t\left(1 - \frac{1}{\mu}\right), \quad \text{Cut Along Axis: } f' = f, \; I' = \frac{I}{2}, \quad \text{Cut Perpendicular to Axis: } f' = 2f

Apparent shift in object position caused by flat slab of thickness t, and changes in focal length and image brightness upon lens division.

Separated thin lenses: 1/F = 1/f₁ + 1/f₂ - d/(f₁ f₂).

Power of Lens Combination (In Contact & Separated)

Pcontact=P1+P2,Pseparated=P1+P2−d⋅P1P2,fseparated=f1+f2−d...  ⟹  1F=1f1+1f2−df1f2P_{\text{contact}} = P_1 + P_2, \quad P_{\text{separated}} = P_1 + P_2 - d \cdot P_1 P_2, \quad f_{\text{separated}} = f_1 + f_2 - \frac{d}{\text{...}} \implies \frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} - \frac{d}{f_1 f_2}

Equivalent focal length and power of two thin lenses separated by distance d (measured between optical centres). In contact: d = 0, so P = P₁ + P₂.

Achromatic doublet condition (no chromatic aberration): ω₁/f₁ + ω₂/f₂ = 0, giving f₁ω₂ + f₂ω₁ = 0. Both lenses must have opposite signs of focal length.

Chromatic Aberration & Dispersive Power

ω=μV−μRμY−1=fR−fVfY,Chromatic aberration=fR−fV=ωfY\omega = \frac{\mu_V - \mu_R}{\mu_Y - 1} = \frac{f_R - f_V}{f_Y}, \quad \text{Chromatic aberration} = f_R - f_V = \omega f_Y

Dispersive power ω measures how much a lens separates different colours of light. μ_V, μ_R, μ_Y are refractive indices for violet, red, and yellow light respectively.

Crown glass: ω ≈ 0.016 (low dispersion); Flint glass: ω ≈ 0.033 (high dispersion). Achromatic doublet uses crown + flint to cancel chromatic aberration.

Optical Instruments — Microscope & Telescope

mmicro=−vouo(1+Dfe),mtele=−fofe(1+feD),Ltele=fo+fe  (normal)m_{\text{micro}} = -\frac{v_o}{u_o}\left(1 + \frac{D}{f_e}\right), \quad m_{\text{tele}} = -\frac{f_o}{f_e}\left(1 + \frac{f_e}{D}\right), \quad L_{\text{tele}} = f_o + f_e \; (\text{normal})

Magnifying power and tube length for compound microscope and astronomical refracting telescope. For relaxed eye (image at infinity): m_micro ≈ -(L/f_o)(D/f_e) and m_tele = -f_o/f_e.

Resolving power of telescope: RP = D_aperture / (1.22 λ). Resolving power of microscope: RP = 2μ sin θ / (1.22 λ), where μ sin θ is numerical aperture.
OpticsHigh Yield (2-3 Qs in JEE)578 PYQs

Wave Optics & Interference

Huygens principle, Young's Double Slit Experiment (YDSE), single slit diffraction, and polarization Brewster law.

Chapter

Superposition of Two Waves (Interference)

I=I1+I2+2I1I2cos⁡ϕ,Imax=(I1+I2)2,Imin=(I1−I2)2I = I_1 + I_2 + 2\sqrt{I_1 I_2}\cos\phi, \quad I_{\text{max}} = (\sqrt{I_1} + \sqrt{I_2})^2, \quad I_{\text{min}} = (\sqrt{I_1} - \sqrt{I_2})^2

Resultant intensity of two coherent wave sources with phase difference φ.

If sources are identical (I₁ = I₂ = I₀): I = 4 I₀ cos²(φ/2).

Young's Double Slit Experiment (YDSE)

β=λDd,yn=nλDd  (Bright),yn=(n−12)λDd  (Dark)\beta = \frac{\lambda D}{d}, \quad y_n = n \frac{\lambda D}{d} \; (\text{Bright}), \quad y_n = \left(n - \frac{1}{2}\right)\frac{\lambda D}{d} \; (\text{Dark})

Fringe width β, slit separation d, and screen distance D.

Immersing YDSE in liquid of index μ compresses fringe width to β/μ.

Fringe Shift by Thin Transparent Sheet

Δy=Dd(μ−1)t=(μ−1)tλβ\Delta y = \frac{D}{d}(\mu - 1)t = \frac{(\mu - 1)t}{\lambda}\beta

Lateral displacement of the whole fringe pattern when mica/glass sheet of thickness t is inserted.

Number of fringes shifted is N = (μ - 1)t / λ.

Single Slit Diffraction

asin⁡θ=nλ  (Minima),Central Maximum Angular Width =2λa,W=2λDaa\sin\theta = n\lambda \; (\text{Minima}), \quad \text{Central Maximum Angular Width } = \frac{2\lambda}{a}, \quad W = \frac{2\lambda D}{a}

Fraunhofer diffraction minima condition and central diffraction spread.

Width of central maximum is TWICE the width of secondary maxima.

Polarization (Brewster's Law & Malus' Law)

tan⁡ip=μ,I=I0cos⁡2θ\tan i_p = \mu, \quad I = I_0 \cos^2\theta

Polarizing angle where reflected and refracted rays are perpendicular, and transmitted intensity through analyzer.

When unpolarized light passes through ideal polarizer, intensity drops to I₀/2.

Resolving Power of Telescope & Microscope (Rayleigh)

RPtelescope=a1.22λ,RPmicroscope=2μsin⁡θ1.22λ=2 NA1.22λ\text{RP}_{\text{telescope}} = \frac{a}{1.22\lambda}, \quad \text{RP}_{\text{microscope}} = \frac{2\mu\sin\theta}{1.22\lambda} = \frac{2\,\text{NA}}{1.22\lambda}

Limit of resolution based on diffraction Airy disc overlap. a is aperture diameter; NA is numerical aperture.

Oil immersion objectives increase μ, thereby increasing resolving power of microscope.

Fresnel Distance (Ray vs Wave Optics Boundary)

ZF=a2λ,Ray Optics valid for D≪ZF,  Diffraction dominates for D≫ZFZ_F = \frac{a^2}{\lambda}, \quad \text{Ray Optics valid for } D \ll Z_F, \; \text{Diffraction dominates for } D \gg Z_F

Distance beyond which ray of light spreads by an amount comparable to aperture size a.

Diffraction spread is equal to aperture size a exactly when D = Z_F.

Thin Film Interference & Phase Shift on Reflection

2μtcos⁡r=(n+12)λ  (Reflected Maxima),2μtcos⁡r=nλ  (Reflected Minima)2\mu t \cos r = \left(n + \frac{1}{2}\right)\lambda \; (\text{Reflected Maxima}), \quad 2\mu t \cos r = n\lambda \; (\text{Reflected Minima})

Interference in reflected rays from thin dielectric film of thickness t. Extra λ/2 path shift occurs upon reflection from denser boundary (Stokes' relation).

An extremely thin soap film (t << λ) appears completely dark in reflected light.
Modern PhysicsHigh Yield (2 Qs in JEE)535 PYQs

Dual Nature of Radiation & Matter

Photoelectric effect, Einstein's equation, work function, cut-off frequency, and de Broglie matter waves.

Chapter

Einstein's Photoelectric Equation

Kmax=hν−Φ0=hν−hν0=eV0K_{\text{max}} = h\nu - \Phi_0 = h\nu - h\nu_0 = e V_0

Conservation of energy in photon-electron interaction.

Graph of stopping potential V₀ versus frequency ν has slope h/e (universal constant).

De Broglie Matter Wavelength

λ=hp=hmv=h2mK\lambda = \frac{h}{p} = \frac{h}{m v} = \frac{h}{\sqrt{2mK}}

Wave-particle duality wavelength associated with moving particles of momentum p.

Wavelength of electron accelerated through potential V: λ = 1.227 / √V nm = √(150/V) Å.

Photon Momentum & Energy

E=hν=hcλ≈12400λ (A˚) eV,p=Ec=hλE = h\nu = \frac{hc}{\lambda} \approx \frac{12400}{\lambda \text{ (\AA)}} \text{ eV}, \quad p = \frac{E}{c} = \frac{h}{\lambda}

Energy and momentum of light quanta.

Useful constant: hc ≈ 1240 eV·nm ≈ 12400 eV·Å.

Radiation Pressure on Surface

P=Ic  (Complete Absorption),P=2Ic  (Complete Reflection)P = \frac{I}{c} \; (\text{Complete Absorption}), \quad P = \frac{2I}{c} \; (\text{Complete Reflection})

Mechanical force per unit area exerted by electromagnetic waves of intensity I.

Force on perfectly reflecting surface of area A is F = 2IA / c. For oblique incidence at angle θ: P = (2I/c)cos²θ.

De Broglie Wavelength of Accelerated Charged Particles

λ=h2mqV  ⟹  λe=12.27V A˚,λp=0.286V A˚,λd=0.202V A˚,λα=0.101V A˚\lambda = \frac{h}{\sqrt{2mqV}} \implies \lambda_e = \frac{12.27}{\sqrt{V}}\,\text{\AA}, \quad \lambda_p = \frac{0.286}{\sqrt{V}}\,\text{\AA}, \quad \lambda_d = \frac{0.202}{\sqrt{V}}\,\text{\AA}, \quad \lambda_\alpha = \frac{0.101}{\sqrt{V}}\,\text{\AA}

Matter wavelengths of fundamental particles accelerated from rest through potential difference V in volts.

For electron: λ = √(150/V) Å. Ratio of wavelengths for same V: λ_p : λ_d : λ_α = 1 : 1/√2 : 1/(2√2).

Thermal De Broglie Wavelength

λ=h3mkBT≈25.17T A˚(for thermal neutron)\lambda = \frac{h}{\sqrt{3m k_B T}} \approx \frac{25.17}{\sqrt{T}}\,\text{\AA} \quad (\text{for thermal neutron})

Wavelength associated with gas molecules or neutrons in thermal equilibrium at temperature T (Kelvin).

Mean translational kinetic energy is K_avg = (3/2)k_B T.

Photon Flux & Saturation Photocurrent

n=PE=Pλhc≈5×1024Pλ,isat=η⋅n⋅en = \frac{P}{E} = \frac{P\lambda}{hc} \approx 5 \times 10^{24} P\lambda, \quad i_{\text{sat}} = \eta \cdot n \cdot e

Number of photons incident per second for a source of power P, and resulting saturation current with quantum efficiency η.

In JEE questions, quantum efficiency η is typically 0.1% to 1% (i.e. only 1 out of 100 or 1000 incident photons ejects a photoelectron).
Modern PhysicsTop Priority (3 Qs in JEE)822 PYQs

Atoms & Nuclei

Bohr atomic model, energy levels, Rydberg formula, radioactive decay law, half-life, and nuclear binding energy.

Chapter

Bohr Model Radius & Velocity

rn=0.529n2Z A˚,vn=2.18×106Zn m/s=cαZnr_n = 0.529 \frac{n^2}{Z} \text{ \AA}, \quad v_n = 2.18 \times 10^6 \frac{Z}{n} \text{ m/s} = c \alpha \frac{Z}{n}

Quantized orbit radius and electron speed for hydrogen-like species.

Radius scales as n²/Z; velocity scales as Z/n.

Bohr Energy Levels & Energy Partition

En=−13.6Z2n2 eV,Kn=−En,Un=2EnE_n = -13.6 \frac{Z^2}{n^2} \text{ eV}, \quad K_n = -E_n, \quad U_n = 2E_n

Quantized energy levels in hydrogenic atoms.

Ionization energy from ground state is +13.6 Z² eV.

Rydberg Formula for Spectral Lines

1λ=RZ2(1n12−1n22)(R≈1.097×107 m−1)\frac{1}{\lambda} = R Z^2 \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right) \quad (R \approx 1.097 \times 10^7 \text{ m}^{-1})

Wavelengths of emitted photons during transitions between orbits.

Lyman (n₁=1, UV), Balmer (n₁=2, Visible), Paschen (n₁=3, IR).

Law of Radioactive Decay

N(t)=N0e−λt,T1/2=ln⁡2λ=0.693λ,τmean=1λ=1.44T1/2N(t) = N_0 e^{-\lambda t}, \quad T_{1/2} = \frac{\ln 2}{\lambda} = \frac{0.693}{\lambda}, \quad \tau_{\text{mean}} = \frac{1}{\lambda} = 1.44 T_{1/2}

Statistical exponential decay of radioactive nuclei.

Fraction remaining after n half-lives is (1/2)^n.

Nuclear Radius & Mass Defect (Binding Energy)

R=R0A1/3  (R0≈1.2 fm),ΔE=Δm×931.5 MeVR = R_0 A^{1/3} \; (R_0 \approx 1.2 \text{ fm}), \quad \Delta E = \Delta m \times 931.5 \text{ MeV}

Nuclear volume scales directly with mass number A (constant nuclear density ~ 2.3 × 10¹⁷ kg/m³).

Peak binding energy per nucleon occurs near Iron-56 (~8.8 MeV/nucleon).

Moseley's Law & Characteristic X-Rays

ν=a(Z−b),λmin=hceV=12400V (volts) A˚\sqrt{\nu} = a(Z - b), \quad \lambda_{\text{min}} = \frac{hc}{eV} = \frac{12400}{V \text{ (volts)}} \text{ \AA}

Frequency of characteristic X-ray lines where b is screening constant (b = 1 for K-alpha lines). Duane-Hunt minimum cutoff wavelength.

Plot of √ν vs atomic number Z is a straight line, which originally ordered the periodic table by atomic number.

Q-Value of Nuclear Reactions & Threshold

Q=[mreactants−mproducts]c2=∑BEproducts−∑BEreactantsQ = [m_{\text{reactants}} - m_{\text{products}}] c^2 = \sum BE_{\text{products}} - \sum BE_{\text{reactants}}

Net energy liberated or absorbed in nuclear transmutation. Exoergic if Q > 0, endoergic if Q < 0.

Threshold kinetic energy for endoergic reaction: K_{th} = |Q| (1 + m_{projectile} / m_{target}).

Activity & Radioactive Equilibrium

A(t)=λN(t)=A0e−λt,1 Curie (Ci)=3.7×1010 BqA(t) = \lambda N(t) = A_0 e^{-\lambda t}, \quad 1 \text{ Curie (Ci)} = 3.7 \times 10^{10} \text{ Bq}

Decay rate per second. In successive decay A -> B -> C with secular equilibrium: λ_A N_A = λ_B N_B.

Parallel radioactive decay: effective decay constant λ_eff = λ₁ + λ₂; 1/T_{eff} = 1/T₁ + 1/T₂.
Modern PhysicsHigh Yield (2 Qs in JEE)462 PYQs

Semiconductors & Logic Gates

P-N junction diode, Zener breakdown diode, rectifiers, and truth tables of Boolean logic gates (AND, OR, NOT, NAND, NOR).

Chapter

Intrinsic & Extrinsic Carrier Concentrations

ne⋅nh=ni2(Mass Action Law),σ=e(neμe+nhμh)n_e \cdot n_h = n_i^2 \quad (\text{Mass Action Law}), \quad \sigma = e(n_e \mu_e + n_h \mu_h)

Product of electron and hole numbers remains constant at thermal equilibrium.

In n-type, n_e ≈ N_d; in p-type, n_h ≈ N_a.

Half-Wave & Full-Wave Rectifier Efficiency

ηhalf=0.406RLrf+RL≤40.6%,ηfull=0.812RLrf+RL≤81.2%\eta_{\text{half}} = \frac{0.406 R_L}{r_f + R_L} \le 40.6\%, \quad \eta_{\text{full}} = \frac{0.812 R_L}{r_f + R_L} \le 81.2\%

AC to DC power conversion efficiency.

Output ripple frequency of full-wave rectifier is 2f_in; for half-wave it is f_in.

De Morgan's Laws & Logic Operations

A+B‾=Aˉ⋅Bˉ,A⋅B‾=Aˉ+Bˉ\overline{A + B} = \bar{A} \cdot \bar{B}, \quad \overline{A \cdot B} = \bar{A} + \bar{B}

Fundamental duality theorems for simplifying Boolean digital circuits.

NAND: Y = (A · B)' (0 only when both inputs are 1). NOR: Y = (A + B)' (1 only when both inputs are 0).

Zener Diode as Voltage Regulator

Vout=VZ,Is=IZ+IL,Rs=Vin−VZIZ+ILV_{\text{out}} = V_Z, \quad I_s = I_Z + I_L, \quad R_s = \frac{V_{\text{in}} - V_Z}{I_Z + I_L}

Maintains constant terminal load voltage V_Z regardless of unregulated DC input voltage fluctuations.

Zener operates in breakdown region in reverse bias. Minimum input voltage must exceed V_Z.

Transistor Current & Voltage Gains

IE=IB+IC,β=ICIB=α1−α,AV=βRoutRin,AP=β2RoutRinI_E = I_B + I_C, \quad \beta = \frac{I_C}{I_B} = \frac{\alpha}{1 - \alpha}, \quad A_V = \beta \frac{R_{\text{out}}}{R_{\text{in}}}, \quad A_P = \beta^2 \frac{R_{\text{out}}}{R_{\text{in}}}

Common-emitter amplifier relationships where α < 1 (typically 0.95 to 0.99) and β is large (typically 20 to 100).

In common emitter amplifier, output voltage has a 180° (π radians) phase shift relative to input signal.

Dynamic Resistance & Rectifier Ripple Factor

rd=ΔVΔI,r=(IrmsIdc)2−1  ⟹  rhalf=1.21,rfull=0.482r_d = \frac{\Delta V}{\Delta I}, \quad r = \sqrt{\left(\frac{I_{\text{rms}}}{I_{\text{dc}}}\right)^2 - 1} \implies r_{\text{half}} = 1.21, \quad r_{\text{full}} = 0.482

Differential resistance of P-N junction and effectiveness of rectifier in delivering pure DC voltage.

A smaller ripple factor indicates a purer DC output with less harmonic fluctuation.

Mass Action Law & Semiconductor Conductivity

ne⋅nh=ni2,σ=e(neμe+nhμh),σn-type≈eNDμe,σp-type≈eNAμhn_e \cdot n_h = n_i^2, \quad \sigma = e(n_e \mu_e + n_h \mu_h), \quad \sigma_{\text{n-type}} \approx e N_D \mu_e, \quad \sigma_{\text{p-type}} \approx e N_A \mu_h

Carrier concentration product in thermal equilibrium and electrical conductivity dictated by electron (μ_e) and hole (μ_h) mobilities.

Electron mobility is always higher than hole mobility (μ_e > μ_h) in semiconductors.

Optoelectronic Bandgap Wavelength (LED, Photodiode)

λthreshold=hcEg≈12400Eg (in eV) A˚\lambda_{\text{threshold}} = \frac{hc}{E_g} \approx \frac{12400}{E_g \text{ (in eV)}} \,\text{\AA}

Cut-off optical wavelength for light absorption in solar cells/photodiodes and photon emission in LEDs.

Photodiodes are operated in reverse bias because fractional change in reverse saturation current upon illumination is substantially higher than in forward bias.