JEE Challenger

Physics

Master the fundamentals of mechanics, electricity, and modern physics with comprehensive study materials

Formula Reference

Access essential equations and laws in a clean, interactive format.

29 Chapters • 271 Formulas

Dimensional Formula — Force

[F]=[M L T−2][F] = [M\,L\,T^{-2}]

Newton (N) = kg·m/s²

Dimensional Formula — Energy / Work

[E]=[M L2 T−2][E] = [M\,L^2\,T^{-2}]

Joule (J)

Dimensional Formula — Power

[P]=[M L2 T−3][P] = [M\,L^2\,T^{-3}]

Watt (W) = J/s

Dimensional Formula — Pressure / Stress

[P]=[M L−1 T−2][P] = [M\,L^{-1}\,T^{-2}]

Pascal (Pa) = N/m²

Dimensional Formula — Electric Charge

[q]=[A T][q] = [A\,T]

Coulomb (C)

Dimensional Formula — Gravitational Constant

[G]=[M−1 L3 T−2][G] = [M^{-1}\,L^3\,T^{-2}]

G = 6.674 × 10⁻¹¹ N·m²/kg²

Dimensional Formula — Planck's Constant

[h]=[M L2 T−1][h] = [M\,L^2\,T^{-1}]

h = 6.626 × 10⁻³⁴ J·s

Absolute Error

Δa=∣amean−ai∣\Delta a = |a_{mean} - a_i|

Mean value minus individual measurement

Relative Error

Relative Error=Δaamean\text{Relative Error} = \frac{\Delta a}{a_{mean}}

Dimensionless ratio

Percentage Error

% Error=Δaamean×100\text{\% Error} = \frac{\Delta a}{a_{mean}} \times 100

Used to report accuracy

Error in Product / Quotient

ΔZZ=ΔAA+ΔBB\frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta B}{B}

For Z = A × B or Z = A/B

Error in Power

ΔZZ=nΔAA\frac{\Delta Z}{Z} = n\frac{\Delta A}{A}

For Z = Aⁿ

Average Velocity

vavg=ΔxΔt=x2−x1t2−t1v_{avg} = \frac{\Delta x}{\Delta t} = \frac{x_2 - x_1}{t_2 - t_1}

Displacement divided by time interval

Average Speed

Speedavg=Total distanceTotal time\text{Speed}_{avg} = \frac{\text{Total distance}}{\text{Total time}}

Always ≥ |average velocity|

Instantaneous Velocity

v=lim⁡Δt→0ΔxΔt=dxdtv = \lim_{\Delta t \to 0}\frac{\Delta x}{\Delta t} = \frac{dx}{dt}

Derivative of position

First Equation of Motion

v=u+atv = u + at

Uniform acceleration

Second Equation of Motion

s=ut+12at2s = ut + \tfrac{1}{2}at^2

Displacement in time t

Third Equation of Motion

v2=u2+2asv^2 = u^2 + 2as

Velocity–displacement relation

Displacement in nth Second

sn=u+a(2n−1)2s_n = u + \frac{a(2n-1)}{2}

Displacement only in the nth second

Free Fall (dropping from rest)

v=gt,h=12gt2,v2=2ghv = gt,\quad h = \tfrac{1}{2}gt^2,\quad v^2 = 2gh

g = 9.8 m/s² downward; u = 0

Projectile — Range

R=u2sin⁡2θgR = \frac{u^2 \sin 2\theta}{g}

Maximum when θ = 45°

Projectile — Max Height

H=u2sin⁡2θ2gH = \frac{u^2 \sin^2\theta}{2g}

Reached at t = T/2

Projectile — Time of Flight

T=2usin⁡θgT = \frac{2u\sin\theta}{g}

Total time in air

Trajectory Equation

y=xtan⁡θ−gx22u2cos⁡2θy = x\tan\theta - \frac{gx^2}{2u^2\cos^2\theta}

Parabolic path of projectile

Horizontal Velocity

vx=ucos⁡θ=constv_x = u\cos\theta = \text{const}

No acceleration in horizontal direction

Vertical Velocity at Time t

vy=usin⁡θ−gtv_y = u\sin\theta - gt

Zero at maximum height

Circular Motion — Centripetal Acceleration

ac=v2r=rω2=vωa_c = \frac{v^2}{r} = r\omega^2 = v\omega

Directed toward center

Relative Velocity

v⃗AB=v⃗A−v⃗B\vec{v}_{AB} = \vec{v}_A - \vec{v}_B

Velocity of A as seen from B

River-Boat: Minimum Drift Angle

sin⁡θ=vrvb\sin\theta = \frac{v_r}{v_b}

θ upstream to cross with minimum drift

Newton's Second Law

F⃗net=ma⃗\vec{F}_{net} = m\vec{a}

Net external force = mass × acceleration

Linear Momentum

p⃗=mv⃗,F⃗=dp⃗dt\vec{p} = m\vec{v},\quad \vec{F} = \frac{d\vec{p}}{dt}

Rate of change of momentum

Impulse

J⃗=F⃗ Δt=Δp⃗\vec{J} = \vec{F}\,\Delta t = \Delta\vec{p}

Change in momentum

Static Friction

fs≤μsNf_s \leq \mu_s N

Maximum static friction: f_s^{max} = μ_s N

Kinetic Friction

fk=μkN,μk<μsf_k = \mu_k N,\quad \mu_k < \mu_s

Acts opposite to direction of sliding

Centripetal Force (Circular)

Fc=mv2r=mrω2F_c = \frac{mv^2}{r} = mr\omega^2

Required force for uniform circular motion

Banking Angle (Frictionless)

tan⁡θ=v2rg\tan\theta = \frac{v^2}{rg}

Ideal banking for speed v, radius r

Maximum Speed on Banked Road (with friction)

vmax=rg(μ+tan⁡θ)1−μtan⁡θv_{max} = \sqrt{\frac{rg(\mu+\tan\theta)}{1-\mu\tan\theta}}

Upper speed limit for banked road with friction μ

Minimum Speed on Banked Road

vmin=rg(tan⁡θ−μ)1+μtan⁡θv_{min} = \sqrt{\frac{rg(\tan\theta-\mu)}{1+\mu\tan\theta}}

Lower speed limit

Work Done

W=F⃗⋅d⃗=Fdcos⁡θW = \vec{F}\cdot\vec{d} = Fd\cos\theta

Scalar product of force and displacement

Work by Variable Force

W=∫x1x2F dxW = \int_{x_1}^{x_2} F\,dx

Area under F–x graph

Kinetic Energy

KE=12mv2=p22mKE = \tfrac{1}{2}mv^2 = \frac{p^2}{2m}

Energy due to motion

Work-Energy Theorem

Wnet=ΔKE=12mvf2−12mvi2W_{net} = \Delta KE = \tfrac{1}{2}mv_f^2 - \tfrac{1}{2}mv_i^2

Net work = change in KE

Elastic PE (Spring)

U=12kx2U = \tfrac{1}{2}kx^2

x = compression or extension from natural length

Gravitational PE

U=mghU = mgh

h = height above reference level

Conservation of Energy

KE+PE=constant (conservative forces only)KE + PE = \text{constant (conservative forces only)}

Total mechanical energy conserved

Power

P=dWdt=F⃗⋅v⃗=Fvcos⁡θP = \frac{dW}{dt} = \vec{F}\cdot\vec{v} = Fv\cos\theta

Rate of doing work; unit: Watt (W)

Coefficient of Restitution

e=v2′−v1′u1−u2e = \frac{v_2' - v_1'}{u_1 - u_2}

e = 1 (perfectly elastic), e = 0 (perfectly inelastic)

Perfectly Inelastic Collision

m1u1+m2u2=(m1+m2)vm_1 u_1 + m_2 u_2 = (m_1+m_2)v

Momentum conserved; KE not conserved

Centre of Mass

r⃗cm=m1r⃗1+m2r⃗2+...m1+m2+...\vec{r}_{cm} = \frac{m_1\vec{r}_1 + m_2\vec{r}_2 + ...}{m_1 + m_2 + ...}

Weighted average of position vectors

Torque

τ⃗=r⃗×F⃗,∣τ∣=rFsin⁡θ\vec{\tau} = \vec{r}\times\vec{F},\quad |\tau| = rF\sin\theta

Rotational analog of force

Angular Momentum

L⃗=r⃗×p⃗=Iω⃗\vec{L} = \vec{r}\times\vec{p} = I\vec{\omega}

Conserved when net torque = 0

Newton's Second Law (Rotation)

τ⃗net=Iα⃗\vec{\tau}_{net} = I\vec{\alpha}

Rotational analog of F = ma

Moment of Inertia — Disc

I=12MR2I = \tfrac{1}{2}MR^2

About axis through center, perpendicular to plane

Moment of Inertia — Ring

I=MR2I = MR^2

About axis through center, perpendicular to plane

Moment of Inertia — Solid Sphere

I=25MR2I = \tfrac{2}{5}MR^2

About diameter

Moment of Inertia — Hollow Sphere

I=23MR2I = \tfrac{2}{3}MR^2

About diameter

Moment of Inertia — Rod (center)

I=ML212I = \frac{ML^2}{12}

About axis perpendicular to rod at midpoint

Parallel Axis Theorem

I=Icm+Md2I = I_{cm} + Md^2

d = distance between parallel axes

Perpendicular Axis Theorem

Iz=Ix+IyI_z = I_x + I_y

For laminar (flat 2D) bodies only

Rolling without Slipping

vcm=Rω,acm=Rαv_{cm} = R\omega,\quad a_{cm} = R\alpha

Condition for pure rolling

KE of Rolling Body

KE=12mvcm2 ⁣(1+k2R2)KE = \tfrac{1}{2}mv_{cm}^2\!\left(1 + \frac{k^2}{R^2}\right)

k = radius of gyration

Newton's Law of Gravitation

F=Gm1m2r2F = \frac{Gm_1 m_2}{r^2}

G = 6.674 × 10⁻¹¹ N·m²/kg²

Gravitational Field Intensity

g=GMR2g = \frac{GM}{R^2}

At surface; g ≈ 9.8 m/s² for Earth

Variation of g with Height

gh=g ⁣(RR+h)2≈g ⁣(1−2hR)g_h = g\!\left(\frac{R}{R+h}\right)^2 \approx g\!\left(1-\frac{2h}{R}\right)

Approximate for h << R

Variation of g with Depth

gd=g ⁣(1−dR)g_d = g\!\left(1-\frac{d}{R}\right)

Decreases linearly; g = 0 at Earth's center

Orbital Velocity

vo=GMR+h≈gR (low orbit)v_o = \sqrt{\frac{GM}{R+h}} \approx \sqrt{gR} \text{ (low orbit)}

Velocity for circular orbit

Escape Velocity

ve=2GMR=2gRv_e = \sqrt{\frac{2GM}{R}} = \sqrt{2gR}

≈ 11.2 km/s for Earth

Time Period of Satellite

T=2π(R+h)3GMT = 2\pi\sqrt{\frac{(R+h)^3}{GM}}

Kepler's Third Law applied

Kepler's Third Law

T2∝a3T^2 \propto a^3

a = semi-major axis of orbit

Total Energy of Satellite

E=−GMm2rE = -\frac{GMm}{2r}

r = R + h; negative means bound

Binding Energy

EB=GMm2rE_B = \frac{GMm}{2r}

Energy to free satellite from orbit

Stress

σ=FA\sigma = \frac{F}{A}

Force per unit area; unit: Pa = N/m²

Longitudinal Strain

ϵL=ΔLL\epsilon_L = \frac{\Delta L}{L}

Change in length per original length

Young's Modulus

Y=σϵL=FLA ΔLY = \frac{\sigma}{\epsilon_L} = \frac{FL}{A\,\Delta L}

Resistance to stretching/compression

Shear Stress and Modulus

η=F/AΔx/L=F/Atan⁡ϕ\eta = \frac{F/A}{\Delta x/L} = \frac{F/A}{\tan\phi}

Rigidity modulus; φ = shear angle

Bulk Modulus

K=−ΔPΔV/VK = -\frac{\Delta P}{\Delta V/V}

Resistance to uniform compression

Compressibility

β=1K\beta = \frac{1}{K}

Inverse of bulk modulus

Poisson's Ratio

ν=−ϵlateralϵlongitudinal\nu = -\frac{\epsilon_{lateral}}{\epsilon_{longitudinal}}

Ratio of lateral to longitudinal strain; −0.5 ≤ ν ≤ 0.5

Elastic Potential Energy Density

u=12×stress×strainu = \frac{1}{2}\times\text{stress}\times\text{strain}

Energy stored per unit volume

Pressure in Fluid

P=P0+ρghP = P_0 + \rho g h

Absolute pressure at depth h

Buoyancy (Archimedes)

Fb=ρfluid Vsubmerged gF_b = \rho_{fluid}\,V_{submerged}\,g

Upward force on submerged object

Continuity Equation

A1v1=A2v2=constA_1 v_1 = A_2 v_2 = \text{const}

Conservation of mass for incompressible flow

Bernoulli's Equation

P+12ρv2+ρgh=constP + \tfrac{1}{2}\rho v^2 + \rho g h = \text{const}

Energy conservation for ideal fluid flow

Torricelli's Theorem

v=2ghv = \sqrt{2gh}

Speed of efflux from an orifice at depth h

Stokes' Law

Fdrag=6πηrvF_{drag} = 6\pi\eta r v

Viscous drag on sphere; η = dynamic viscosity

Terminal Velocity

vt=2r2(ρ−σ)g9ηv_t = \frac{2r^2(\rho - \sigma)g}{9\eta}

ρ = sphere density, σ = fluid density

Surface Tension — Excess Pressure (Bubble)

ΔP=4Tr\Delta P = \frac{4T}{r}

Soap bubble: 2 surfaces → factor 4

Surface Tension — Excess Pressure (Drop)

ΔP=2Tr\Delta P = \frac{2T}{r}

Liquid drop: 1 surface → factor 2

Capillary Rise

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

θ = contact angle; negative h means depression

Linear Thermal Expansion

ΔL=L0 α ΔT,L=L0(1+αΔT)\Delta L = L_0\,\alpha\,\Delta T,\quad L = L_0(1+\alpha\Delta T)

α = linear expansion coefficient

Area Expansion

ΔA=A0 β ΔT,β=2α\Delta A = A_0\,\beta\,\Delta T,\quad \beta = 2\alpha

β = areal expansion coefficient

Volume Expansion

ΔV=V0 γ ΔT,γ=3α\Delta V = V_0\,\gamma\,\Delta T,\quad \gamma = 3\alpha

γ = volumetric expansion coefficient

Heat Capacity

Q=mcΔTQ = mc\Delta T

c = specific heat capacity (J/kg·K)

Latent Heat

Q=mLQ = mL

L = specific latent heat; no temperature change during phase transition

Fourier's Law of Conduction

dQdt=−KAdTdx\frac{dQ}{dt} = -KA\frac{dT}{dx}

K = thermal conductivity; negative sign: heat flows hot→cold

Thermal Resistance

Rth=LKAR_{th} = \frac{L}{KA}

Analog of electrical resistance

Newton's Law of Cooling

dTdt=−k(T−T0)\frac{dT}{dt} = -k(T - T_0)

Rate of cooling proportional to temperature difference

Stefan-Boltzmann Law

P=εσAT4P = \varepsilon\sigma A T^4

σ = 5.67 × 10⁻⁸ W/m²K⁴; ε = emissivity

Wien's Displacement Law

λmax T=b=2.898×10−3 m⋅K\lambda_{max}\,T = b = 2.898\times10^{-3}\text{ m·K}

Peak wavelength shifts with temperature

First Law

ΔU=Q−W\Delta U = Q - W

Q = heat added to system; W = work done by system

Work Done by Gas (Isobaric)

W=PΔV=nRΔTW = P\Delta V = nR\Delta T

Constant pressure

Work Done (Isothermal)

W=nRTln⁡V2V1W = nRT\ln\frac{V_2}{V_1}

Constant temperature

Work Done (Adiabatic)

W=nR(T1−T2)γ−1=P1V1−P2V2γ−1W = \frac{nR(T_1-T_2)}{\gamma-1} = \frac{P_1V_1 - P_2V_2}{\gamma-1}

No heat exchange with surroundings

Isochoric Process

W=0,Q=ΔU=nCVΔTW = 0,\quad Q = \Delta U = nC_V\Delta T

Constant volume; all heat raises internal energy

Adiabatic Relations

PVγ=C,TVγ−1=C,TγP1−γ=CPV^\gamma = C,\quad TV^{\gamma-1} = C,\quad T^\gamma P^{1-\gamma} = C

γ = Cp/Cv

Efficiency of Heat Engine

η=1−Q2Q1=1−TLTH\eta = 1 - \frac{Q_2}{Q_1} = 1 - \frac{T_L}{T_H}

Carnot efficiency is maximum possible

COP of Refrigerator

COP=Q2W=TLTH−TLCOP = \frac{Q_2}{W} = \frac{T_L}{T_H - T_L}

Coefficient of performance; COP of Carnot is maximum

Mayer's Relation

CP−CV=RC_P - C_V = R

For one mole of ideal gas

Ideal Gas Equation

PV=nRT=NkTPV = nRT = NkT

R = 8.314 J/mol·K; k_B = 1.38 × 10⁻²³ J/K

Pressure by Gas

P=13ρ v2‾=13mNV v2‾P = \frac{1}{3}\rho\,\overline{v^2} = \frac{1}{3}\frac{mN}{V}\,\overline{v^2}

Derived from kinetic theory

RMS Speed

vrms=3RTM=3kTmv_{rms} = \sqrt{\frac{3RT}{M}} = \sqrt{\frac{3kT}{m}}

M = molar mass in kg/mol

Mean Speed

vˉ=8RTπM\bar{v} = \sqrt{\frac{8RT}{\pi M}}

Average speed of all molecules

Most Probable Speed

vp=2RTMv_p = \sqrt{\frac{2RT}{M}}

Speed at peak of Maxwell distribution

Speed Relation

vp<vˉ<vrmsv_p < \bar{v} < v_{rms}

Ratio ≈ 1 : 1.13 : 1.22

Average KE per Molecule

KEˉ=f2kT\bar{KE} = \frac{f}{2}kT

f = degrees of freedom (mono=3, di=5, poly=6)

Internal Energy

U=f2nRTU = \frac{f}{2}nRT

For n moles; f = degrees of freedom

Mean Free Path

λ=12 n πd2\lambda = \frac{1}{\sqrt{2}\,n\,\pi d^2}

n = number density; d = diameter of molecule

SHM Equation

x=Asin⁡(ωt+ϕ)x = A\sin(\omega t + \phi)

A = amplitude, φ = initial phase

Velocity in SHM

v=ωA2−x2v = \omega\sqrt{A^2-x^2}

Maximum at x=0; zero at x=±A

Acceleration in SHM

a=−ω2xa = -\omega^2 x

Always directed toward mean position

Angular Frequency

ω=2πT=2πf\omega = \frac{2\pi}{T} = 2\pi f

Unit: rad/s

Spring-Mass Period

T=2πmkT = 2\pi\sqrt{\frac{m}{k}}

k = spring constant; independent of amplitude

Simple Pendulum Period

T=2πLgT = 2\pi\sqrt{\frac{L}{g}}

Valid for θ < 4°; independent of mass and amplitude

Total Energy in SHM

E=12kA2=12mω2A2E = \frac{1}{2}kA^2 = \frac{1}{2}m\omega^2 A^2

Constant; proportional to A²

PE in SHM

U=12kx2=12mω2x2U = \frac{1}{2}kx^2 = \frac{1}{2}m\omega^2 x^2

Maximum at extreme positions

KE in SHM

KE=12mω2(A2−x2)KE = \frac{1}{2}m\omega^2(A^2 - x^2)

Maximum at mean position

Wave Equation

y=Asin⁡(kx−ωt)y = A\sin(kx - \omega t)

Transverse progressive wave in +x direction

Wave Speed — Fundamental

v=fλ=ωkv = f\lambda = \frac{\omega}{k}

Universal wave relation

Wave Number

k=2πλk = \frac{2\pi}{\lambda}

Spatial angular frequency; unit: rad/m

Speed in String

v=Tμv = \sqrt{\frac{T}{\mu}}

T = tension (N), μ = linear mass density (kg/m)

Speed of Sound in Gas

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

Laplace's formula; γ = Cp/Cv

Beats Frequency

fbeats=∣f1−f2∣f_{beats} = |f_1 - f_2|

Beats per second

Doppler Effect

f′=fv±vov∓vsf' = f\frac{v \pm v_o}{v \mp v_s}

+ for approach, − for recession (both observer & source)

Fundamental — Open Pipe

f1=v2Lf_1 = \frac{v}{2L}

All harmonics present

Fundamental — Closed Pipe

f1=v4Lf_1 = \frac{v}{4L}

Only odd harmonics

Standing Wave — Nodes & Antinodes

y=2Acos⁡(kx)sin⁡(ωt)y = 2A\cos(kx)\sin(\omega t)

Nodes at kx = π/2, 3π/2; antinodes at kx = 0, π

Intensity of Sound

I=P4πr2I = \frac{P}{4\pi r^2}

Point source; decreases as 1/r²

Coulomb's Law

F=kq1q2r2,k=14πϵ0=9×109 N⋅m2/C2F = k\frac{q_1 q_2}{r^2},\quad k = \frac{1}{4\pi\epsilon_0} = 9\times10^9\,\text{N·m}^2\text{/C}^2

Force between two point charges

Electric Field (Point Charge)

E=kqr2E = \frac{kq}{r^2}

Away from +ve, toward −ve charge

Electric Field — Infinite Line Charge

E=λ2πϵ0rE = \frac{\lambda}{2\pi\epsilon_0 r}

λ = linear charge density; radially outward

Electric Field — Infinite Sheet

E=σ2ϵ0E = \frac{\sigma}{2\epsilon_0}

σ = surface charge density; uniform and perpendicular

Electric Dipole Moment

p⃗=qd⃗\vec{p} = q\vec{d}

Direction: negative to positive charge

Field on Dipole Axis (End-on)

E=2kpr3E = \frac{2kp}{r^3}

Far field approximation

Field on Dipole Equator (Broadside-on)

E=kpr3E = \frac{kp}{r^3}

Far field; direction opposite to p

Gauss's Law

∮E⃗⋅dA⃗=Qencϵ0\oint\vec{E}\cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0}

ε₀ = 8.85 × 10⁻¹² C²/N·m²

Electric Flux

ΦE=E⃗⋅A⃗=EAcos⁡θ\Phi_E = \vec{E}\cdot\vec{A} = EA\cos\theta

Through a surface

Electric Potential (Point Charge)

V=kqrV = \frac{kq}{r}

Work done per unit charge from ∞ to r

Potential Difference

VA−VB=−∫BAE⃗⋅dl⃗V_A - V_B = -\int_B^A\vec{E}\cdot d\vec{l}

Work done per unit charge against E

E and V Relation

E=−dVdrE = -\frac{dV}{dr}

E in direction of steepest decrease in V

Potential Due to Dipole (Axis)

V=kpcos⁡θr2V = \frac{kp\cos\theta}{r^2}

θ from dipole axis

Capacitance

C=QVC = \frac{Q}{V}

Unit: Farad (F)

Parallel Plate Capacitor

C=ϵ0AdC = \frac{\epsilon_0 A}{d}

A = plate area; d = separation

With Dielectric

C′=KC=Kϵ0AdC' = KC = \frac{K\epsilon_0 A}{d}

K = dielectric constant (K > 1)

Series Combination

1Ceq=1C1+1C2+...\frac{1}{C_{eq}} = \frac{1}{C_1}+\frac{1}{C_2}+...

Same charge on each; total V adds up

Parallel Combination

Ceq=C1+C2+...C_{eq} = C_1+C_2+...

Same voltage; charges add up

Energy Stored in Capacitor

U=Q22C=12CV2=QV2U = \frac{Q^2}{2C} = \frac{1}{2}CV^2 = \frac{QV}{2}

Stored in electric field between plates

Energy Density

u=12ϵ0E2u = \frac{1}{2}\epsilon_0 E^2

Energy per unit volume in electric field

Electric Current

I=dqdt=nAevdI = \frac{dq}{dt} = nAev_d

n = free e⁻ density; v_d = drift velocity

Ohm's Law

V=IRV = IR

Valid for ohmic conductors at constant T

Resistivity

R=ρLAR = \rho\frac{L}{A}

ρ = resistivity (Ω·m)

Temperature Coefficient of Resistance

RT=R0(1+αΔT)R_T = R_0(1+\alpha\Delta T)

α ≈ 0.004/°C for metals

Power Dissipation

P=VI=I2R=V2RP = VI = I^2R = \frac{V^2}{R}

Heat generated per second

EMF & Terminal Voltage

ε=V+Ir=I(R+r)\varepsilon = V + Ir = I(R+r)

r = internal resistance; V = terminal voltage

Kirchhoff's Current Law

∑Iin=∑Iout\sum I_{in} = \sum I_{out}

Charge conservation at junction

Kirchhoff's Voltage Law

∑Vloop=0\sum V_{loop} = 0

Energy conservation in closed loop

Series Resistors

Req=R1+R2+...R_{eq} = R_1+R_2+...

Same current through all

Parallel Resistors

1Req=1R1+1R2+...\frac{1}{R_{eq}} = \frac{1}{R_1}+\frac{1}{R_2}+...

Same voltage across all

Wheatstone Bridge (Balanced)

PQ=RS\frac{P}{Q} = \frac{R}{S}

No current through galvanometer at balance

Potentiometer Principle

ε1ε2=l1l2\frac{\varepsilon_1}{\varepsilon_2} = \frac{l_1}{l_2}

EMF proportional to balancing length

Biot-Savart Law

dB⃗=μ04πI dl⃗×r^r2d\vec{B} = \frac{\mu_0}{4\pi}\frac{I\,d\vec{l}\times\hat{r}}{r^2}

μ₀ = 4π × 10⁻⁷ T·m/A

Field at Center of Circular Loop

B=μ0I2RB = \frac{\mu_0 I}{2R}

Single loop of radius R

Field Inside Solenoid

B=μ0nIB = \mu_0 nI

n = turns per meter; uniform inside

Ampere's Law

∮B⃗⋅dl⃗=μ0Ienc\oint\vec{B}\cdot d\vec{l} = \mu_0 I_{enc}

For steady current

Lorentz Force

F⃗=q(v⃗×B⃗)\vec{F} = q(\vec{v}\times\vec{B})

Force on moving charge in B field

Force on Current Wire

F⃗=I(L⃗×B⃗)\vec{F} = I(\vec{L}\times\vec{B})

L = length vector along current direction

Radius of Circular Motion

r=mvqBr = \frac{mv}{qB}

Charged particle in uniform B field

Cyclotron Frequency

f=qB2πmf = \frac{qB}{2\pi m}

Independent of velocity (non-relativistic)

Force between Parallel Wires

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

Attractive for same-direction currents

Torque on Current Loop

τ⃗=m⃗×B⃗=NIABsin⁡θ\vec{\tau} = \vec{m}\times\vec{B} = NIAB\sin\theta

m = NIA = magnetic moment

Magnetic Dipole Moment

m⃗=IAn^\vec{m} = IA\hat{n}

Current loop; direction by right-hand rule

Bar Magnet — Field on Axis

Baxis=μ04π2Mr3B_{axis} = \frac{\mu_0}{4\pi}\frac{2M}{r^3}

M = magnetic dipole moment

Bar Magnet — Field on Equator

Beq=μ04πMr3B_{eq} = \frac{\mu_0}{4\pi}\frac{M}{r^3}

Direction opposite to M

Torque on Dipole

τ=MBsin⁡θ\tau = MB\sin\theta

Aligns magnetic moment with field

Potential Energy of Dipole

U=−M⃗⋅B⃗=−MBcos⁡θU = -\vec{M}\cdot\vec{B} = -MB\cos\theta

Minimum (stable) at θ = 0

Magnetic Susceptibility

χm=MH\chi_m = \frac{M}{H}

M = magnetisation; H = magnetic field intensity

Relative Permeability

μr=1+χm\mu_r = 1 + \chi_m

Para: μr > 1; Dia: μr < 1; Ferro: μr >> 1

Curie's Law

χm=CT\chi_m = \frac{C}{T}

Paramagnetics; C = Curie constant

Magnetic Flux

ΦB=B⃗⋅A⃗=BAcos⁡θ\Phi_B = \vec{B}\cdot\vec{A} = BA\cos\theta

Unit: Weber (Wb) = T·m²

Faraday's Law

ε=−NdΦBdt\varepsilon = -N\frac{d\Phi_B}{dt}

Induced EMF opposes flux change (Lenz's Law)

Motional EMF

ε=Blv\varepsilon = Blv

Conductor of length l moving at speed v ⊥ B

Self-Inductance

εL=−LdIdt\varepsilon_L = -L\frac{dI}{dt}

L = self-inductance in Henry (H)

Inductance of Solenoid

L=μ0n2AlL = \mu_0 n^2 Al

n = turns/m, A = area, l = length

Mutual Inductance

ε2=−MdI1dt\varepsilon_2 = -M\frac{dI_1}{dt}

M = mutual inductance between two coils

Energy in Inductor

UL=12LI2U_L = \frac{1}{2}LI^2

Stored as magnetic field energy

Energy Density (B-field)

u=B22μ0u = \frac{B^2}{2\mu_0}

Magnetic energy per unit volume

RMS Values

Vrms=V02,Irms=I02V_{rms} = \frac{V_0}{\sqrt{2}},\quad I_{rms} = \frac{I_0}{\sqrt{2}}

For sinusoidal AC; V₀, I₀ = peak values

Inductive Reactance

XL=ωL=2πfLX_L = \omega L = 2\pi fL

Opposition by inductor; increases with frequency

Capacitive Reactance

XC=1ωC=12πfCX_C = \frac{1}{\omega C} = \frac{1}{2\pi fC}

Opposition by capacitor; decreases with frequency

Impedance (Series RLC)

Z=R2+(XL−XC)2Z = \sqrt{R^2+(X_L-X_C)^2}

Total AC opposition

Phase Angle

tan⁡ϕ=XL−XCR\tan\phi = \frac{X_L - X_C}{R}

Phase of voltage w.r.t. current

Resonance Frequency

f0=12πLCf_0 = \frac{1}{2\pi\sqrt{LC}}

At resonance: Z = R (minimum), current maximum

Quality Factor

Q=ω0LR=1RLCQ = \frac{\omega_0 L}{R} = \frac{1}{R}\sqrt{\frac{L}{C}}

Sharpness of resonance

Average Power

Pavg=VrmsIrmscos⁡ϕ=Irms2RP_{avg} = V_{rms}I_{rms}\cos\phi = I_{rms}^2 R

True power; φ = phase difference

Power Factor

cos⁡ϕ=RZ\cos\phi = \frac{R}{Z}

1 for pure resistive; 0 for pure reactive

Transformer

VsVp=NsNp=IpIs\frac{V_s}{V_p} = \frac{N_s}{N_p} = \frac{I_p}{I_s}

Ideal transformer; N_s > N_p: step-up

Speed of EM Wave

c=1μ0ϵ0=3×108 m/sc = \frac{1}{\sqrt{\mu_0\epsilon_0}} = 3\times10^8\text{ m/s}

In vacuum; μ₀ = 4π × 10⁻⁷, ε₀ = 8.85 × 10⁻¹²

Wave Equation (E-field)

E=E0sin⁡(kx−ωt)E = E_0\sin(kx-\omega t)

EM wave propagating in +x direction

Relation E and B

E0B0=c\frac{E_0}{B_0} = c

E and B oscillate in phase; perpendicular to each other and to direction of propagation

EM Wave Intensity

I=12cϵ0E02=c2μ0B02I = \frac{1}{2}c\epsilon_0 E_0^2 = \frac{c}{2\mu_0}B_0^2

Average intensity

Radiation Pressure (absorbed)

Prad=IcP_{rad} = \frac{I}{c}

Force per unit area from EM radiation

Radiation Pressure (reflected)

Prad=2IcP_{rad} = \frac{2I}{c}

For perfect reflector

Snell's Law

n1sin⁡θ1=n2sin⁡θ2n_1\sin\theta_1 = n_2\sin\theta_2

Refraction at interface

Critical Angle (TIR)

sin⁡θc=n2n1(n1>n2)\sin\theta_c = \frac{n_2}{n_1}\quad (n_1 > n_2)

For total internal reflection

Mirror Formula

1v+1u=1f=2R\frac{1}{v}+\frac{1}{u} = \frac{1}{f} = \frac{2}{R}

Sign convention: distances from pole

Mirror Magnification

m=−vum = -\frac{v}{u}

+ve = erect; −ve = inverted

Lens Formula

1v−1u=1f\frac{1}{v}-\frac{1}{u} = \frac{1}{f}

Cartesian sign convention

Lens Magnification

m=vum = \frac{v}{u}

+ve = virtual erect; −ve = real inverted

Lensmaker's Formula

1f=(n−1) ⁣(1R1−1R2)\frac{1}{f} = (n-1)\!\left(\frac{1}{R_1}-\frac{1}{R_2}\right)

n = refractive index of lens material

Power of Lens

P=1f(m)(Diopter, D)P = \frac{1}{f(\text{m})}\quad\text{(Diopter, D)}

Converging lens: +ve; Diverging: −ve

Combined Power (in contact)

P=P1+P2P = P_1 + P_2

Two thin lenses in contact

Prism — Minimum Deviation

μ=sin⁡(A+δm2)sin⁡A2\mu = \frac{\sin\left(\frac{A+\delta_m}{2}\right)}{\sin\frac{A}{2}}

A = apex angle; δ_m = minimum deviation

Thin Prism Deviation

δ=(μ−1)A\delta = (\mu-1)A

Small angle approximation

Compound Microscope (M)

M=Lfo×DfeM = \frac{L}{f_o}\times\frac{D}{f_e}

L = tube length, D = 25 cm (least distance of distinct vision)

Telescope Magnification

M=−fofeM = -\frac{f_o}{f_e}

−ve = inverted image; larger f_o and smaller f_e = more magnification

YDSE — Fringe Width

β=λDd\beta = \frac{\lambda D}{d}

D = screen distance; d = slit separation

YDSE — Bright Fringe Position

yn=nλDdy_n = \frac{n\lambda D}{d}

n = 0, ±1, ±2, ... (n=0 is central bright)

YDSE — Dark Fringe Position

yn=(2n−1)λD2dy_n = \frac{(2n-1)\lambda D}{2d}

n = 1, 2, 3, ...

Constructive Interference

Δ=nλ\Delta = n\lambda

Path difference = integer multiple of λ

Destructive Interference

Δ=(2n−1)λ2\Delta = (2n-1)\frac{\lambda}{2}

Path difference = odd multiple of λ/2

Resultant Intensity

I=I1+I2+2I1I2cos⁡δI = I_1 + I_2 + 2\sqrt{I_1 I_2}\cos\delta

δ = phase difference = 2πΔ/λ

Single Slit — Diffraction Minima

asin⁡θ=nλ,n=±1,±2...a\sin\theta = n\lambda,\quad n = \pm 1, \pm 2...

a = slit width; central maximum is widest

Central Maximum Width (Single Slit)

w=2λDaw = \frac{2\lambda D}{a}

Width of central bright fringe on screen

Malus's Law

I=I0cos⁡2θI = I_0\cos^2\theta

Intensity after polariser at angle θ to the analyser

Brewster's Angle

tan⁡θp=n\tan\theta_p = n

n = refractive index; reflected light fully plane-polarised

Photoelectric Equation (Einstein)

KEmax=hν−ϕ=eV0KE_{max} = h\nu - \phi = eV_0

φ = work function; V₀ = stopping potential

Threshold Frequency

ν0=ϕh\nu_0 = \frac{\phi}{h}

Minimum frequency for photoelectric effect

Photon Energy

E=hν=hcλE = h\nu = \frac{hc}{\lambda}

h = 6.626 × 10⁻³⁴ J·s; c = 3 × 10⁸ m/s

Photon Momentum

p=hλ=Ecp = \frac{h}{\lambda} = \frac{E}{c}

Photon has momentum despite zero rest mass

de Broglie Wavelength

λ=hp=hmv\lambda = \frac{h}{p} = \frac{h}{mv}

Matter wave associated with moving particle

de Broglie (Accelerated Particle)

λ=h2mqV\lambda = \frac{h}{\sqrt{2mqV}}

Particle of charge q accelerated through V

de Broglie (Thermal Particle)

λ=h3mkT\lambda = \frac{h}{\sqrt{3mkT}}

Particle in thermal equilibrium at T

Bohr's Radius (nth orbit)

rn=a0n2Z,a0=0.529 A˚r_n = a_0\frac{n^2}{Z},\quad a_0 = 0.529\text{ Å}

Z = atomic number; hydrogen: Z=1

Velocity in nth Orbit

vn=Ze22ϵ0hnv_n = \frac{Ze^2}{2\epsilon_0 hn}

Decreases as n increases; v₁ ≈ c/137

Energy of nth Orbit (H-like)

En=−13.6 Z2n2 eVE_n = -\frac{13.6\,Z^2}{n^2}\text{ eV}

Negative = bound state; ground state (n=1, Z=1): −13.6 eV

Photon Emitted (Transition)

hν=En2−En1,n2>n1h\nu = E_{n_2} - E_{n_1},\quad n_2 > n_1

Emission: from higher to lower orbit

Rydberg Formula

1λ=RHZ2 ⁣(1n12−1n22)\frac{1}{\lambda} = R_H Z^2\!\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)

R_H = 1.097 × 10⁷ m⁻¹

Spectral Series (Hydrogen)

n1=1 (Lyman),  2 (Balmer),  3 (Paschen)n_1 = 1\text{ (Lyman)},\; 2\text{ (Balmer)},\; 3\text{ (Paschen)}

Lyman: UV; Balmer: visible; Paschen: IR

de Broglie Condition (Bohr)

nλ=2πrnn\lambda = 2\pi r_n

Standing wave condition for electron orbits

Nuclear Radius

R=R0A1/3,R0=1.2×10−15 mR = R_0 A^{1/3},\quad R_0 = 1.2\times10^{-15}\text{ m}

A = mass number; nuclear density is constant

Mass Defect

Δm=[Zmp+Nmn−Mnucleus]\Delta m = [Zm_p + Nm_n - M_{nucleus}]

Z protons + N neutrons; N = A − Z

Binding Energy

BE=Δm⋅c2=Δm×931.5 MeVBE = \Delta m\cdot c^2 = \Delta m\times 931.5\text{ MeV}

1 amu = 931.5 MeV/c²

Radioactive Decay Law

N=N0e−λtN = N_0 e^{-\lambda t}

λ = decay constant (s⁻¹)

Half-Life

T1/2=ln⁡2λ=0.693λT_{1/2} = \frac{\ln 2}{\lambda} = \frac{0.693}{\lambda}

Time for half the nuclei to decay

Mean Life

τ=1λ=T1/2ln⁡2≈1.44 T1/2\tau = \frac{1}{\lambda} = \frac{T_{1/2}}{\ln 2} \approx 1.44\,T_{1/2}

Average lifetime of a nucleus

Activity

A=λN=A0e−λtA = \lambda N = A_0 e^{-\lambda t}

Unit: Becquerel (Bq) = 1 decay/s; 1 Curie = 3.7 × 10¹⁰ Bq

Alpha Decay

ZAX→Z−2A−4Y+24He^A_Z X \rightarrow{}^{A-4}_{Z-2}Y + {}^4_2He

A decreases by 4, Z by 2

Beta (β⁻) Decay

ZAX→Z+1AY+e−+νˉe^A_Z X \rightarrow{}^{A}_{Z+1}Y + e^- + \bar{\nu}_e

Z increases by 1; antineutrino emitted

Intrinsic Carrier Concentration

ni2=ne⋅nhn_i^2 = n_e \cdot n_h

Mass action law; n_i increases with temperature

Diode Current (Ideal)

I=I0 ⁣(eV/ηVT−1)I = I_0\!\left(e^{V/\eta V_T}-1\right)

V_T = kT/e ≈ 26 mV at 300 K; η = ideality factor

Rectifier — Half Wave (Avg)

Idc=ImπI_{dc} = \frac{I_m}{\pi}

Average output current

Rectifier — Full Wave (Avg)

Idc=2ImπI_{dc} = \frac{2I_m}{\pi}

Higher DC output than half wave

Transistor CE Current Gain (β)

β=ICIB\beta = \frac{I_C}{I_B}

Common-emitter configuration; typical β: 20 – 500

Transistor CB Current Gain (α)

α=ICIE,α<1\alpha = \frac{I_C}{I_E},\quad \alpha < 1

Common-base; α and β relation: β = α/(1−α)

Relation α and β

β=α1−α,α=β1+β\beta = \frac{\alpha}{1-\alpha},\quad \alpha = \frac{\beta}{1+\beta}

Both > 0; α < 1; β can be large

Logic Gates

AND: Y=A⋅B,  OR: Y=A+B,  NOT: Y=Aˉ\text{AND: }Y=A\cdot B,\;\text{OR: }Y=A+B,\;\text{NOT: }Y=\bar{A}

NAND and NOR are universal gates

Modulation Index (AM)

ma=AmAcm_a = \frac{A_m}{A_c}

A_m = message amplitude; A_c = carrier amplitude; m_a ≤ 1 for no distortion

AM Bandwidth

BW=2fmBW = 2f_m

f_m = maximum message frequency

FM Modulation Index

mf=Δffmm_f = \frac{\Delta f}{f_m}

Δf = frequency deviation; f_m = message frequency

Range of Ground Waves

d=2Rhd = \sqrt{2Rh}

h = antenna height; R = 6400 km (Earth radius)

Line of Sight (LoS)

dT+dR=2RhT+2RhRd_T + d_R = \sqrt{2Rh_T} + \sqrt{2Rh_R}

d_T and d_R for transmitter and receiver antennas

Sampling Theorem

fs≥2fmaxf_s \geq 2f_{max}

Nyquist rate: sampling frequency ≥ 2× max signal frequency

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