Physics Formula Sheet

A condensed, scroll-through revision aid for the entire NCERT Class XI and Class XII Physics syllabus. Every formula is written in LaTeX and tagged with symbols and quick meanings. Bookmark this page for the week before the exam.

Class XI - Mechanics (Ch 1–8)

Kinematics

Formula	Symbols	Meaning
$v = u + at$	$u$ initial, $v$ final, $a$ accel.	First eqn. of motion
$s = ut + \tfrac{1}{2}at^2$	$s$ disp., $t$ time	Second eqn.
$v^2 = u^2 + 2as$		Third eqn.
$s_n = u + \tfrac{a}{2}(2n-1)$	$s_n$ distance in $n$ -th sec
$R = \dfrac{u^2\sin 2\theta}{g}$	$\theta$ projection angle	Range, level ground
$H = \dfrac{u^2\sin^2\theta}{2g}$		Max height
$T = \dfrac{2u\sin\theta}{g}$		Time of flight
$\vec a_c = v^2/r$		Centripetal acceleration

Vectors

Formula	Meaning
$\vec A\cdot\vec B = AB\cos\theta$	Dot product
$\vert \vec A\times\vec B\vert = AB\sin\theta$	Cross product magnitude
$\vec R = \vec A + \vec B$ , $R=\sqrt{A^2+B^2+2AB\cos\theta}$	Resultant

Laws of Motion

Formula	Meaning
$\vec F = m\vec a$	Newton's 2nd law
$\vec F = d\vec p/dt$	More general form
$\vec F_{12} = -\vec F_{21}$	Newton's 3rd law
$f_s\le\mu_s N$ , $f_k=\mu_k N$	Static / kinetic friction
$\tan\theta_\text{rep}=\mu_s$	Angle of repose
$v_\text{max}=\sqrt{\mu_s rg}$	Friction-limited speed (flat curve)
$\tan\theta = v^2/rg$	Banking, no friction

Work, Energy, Power

Formula	Meaning
$W = \vec F\cdot\vec d = Fd\cos\theta$	Work (const. force)
$K = \tfrac{1}{2}mv^2$	Kinetic energy
$W_\text{net} = \Delta K$	Work–energy theorem
$U_\text{grav} = mgh$	Near Earth's surface
$U_\text{spring} = \tfrac{1}{2}kx^2$	Spring PE
$P = dW/dt = \vec F\cdot\vec v$	Power
Elastic collision (1D): $v_1' = \dfrac{m_1-m_2}{m_1+m_2}u_1 + \dfrac{2m_2}{m_1+m_2}u_2$	Final velocity

System of Particles & Rotational Motion

Formula	Meaning
$\vec R_\text{cm} = \dfrac{\sum m_i\vec r_i}{\sum m_i}$	Centre of mass
$\vec p_\text{tot} = M\vec V_\text{cm}$	Total momentum
$\vec\tau = \vec r\times\vec F$	Torque
$\vec L = \vec r\times\vec p$ , $\vec L = I\vec\omega$	Angular momentum
$\vec\tau = d\vec L/dt$	Rotational Newton
$K_\text{rot} = \tfrac{1}{2}I\omega^2$	Rotational KE
$I_\text{rod} = ML^2/12$ (centre), $ML^2/3$ (end)	Common moments
$I_\text{disc}=MR^2/2$ , $I_\text{ring}=MR^2$
$I_\text{sphere(solid)}=\tfrac{2}{5}MR^2$ , $I_\text{sphere(hollow)}=\tfrac{2}{3}MR^2$
Parallel-axis: $I = I_\text{cm} + Md^2$	Steiner's theorem
Perpendicular-axis (planar): $I_z = I_x + I_y$
Rolling without slipping: $v=R\omega$
$K_\text{rolling}=\tfrac{1}{2}mv^2(1+k^2/R^2)$	$k$ = radius of gyration

Gravitation

Formula	Meaning
$F = G m_1m_2/r^2$	Newton's law
$g = GM/R^2$	Surface gravity
$g_h = g(1-2h/R)$ for $h\ll R$	Variation with height
$g_d = g(1-d/R)$	Inside Earth
$g_\text{lat}=g-\omega^2 R\cos^2\lambda$	Latitude variation
$U = -GMm/r$	Gravitational PE
$v_\text{orbit}=\sqrt{GM/r}$	Circular orbit speed
$v_\text{esc}=\sqrt{2GM/R}$	Escape speed
$T^2 = (4\pi^2/GM)r^3$	Kepler's third law
Energy in orbit: $E = -GMm/(2r)$

Class XI - Properties of Matter (Ch 9–11)

Elasticity

Formula	Meaning
Stress $= F/A$ ; Strain $= \Delta\ell/\ell$	Definitions
$Y = (F/A)/(\Delta L/L)$	Young's modulus
$B = -P/(\Delta V/V)$	Bulk modulus
$\eta = (F/A)/\theta$	Shear modulus
$\sigma = -\dfrac{\Delta r/r}{\Delta L/L}$	Poisson ratio
$U/V = \tfrac{1}{2}\,\text{stress}\times\text{strain}$	Energy density

Fluid Mechanics

Formula	Meaning
$P = P_0 + \rho g h$	Hydrostatic pressure
$F_B = \rho_\text{fluid}V_\text{disp}g$	Buoyancy / Archimedes
$A_1v_1 = A_2v_2$	Continuity (incompressible)
$P + \tfrac{1}{2}\rho v^2 + \rho gh =$ const.	Bernoulli's equation
$v_\text{eff} = \sqrt{2gh}$	Torricelli's theorem
$F = 6\pi\eta r v$	Stokes' law
$v_t = \dfrac{2r^2(\rho-\sigma)g}{9\eta}$	Terminal velocity
$Re = \rho v D/\eta$	Reynolds number
$h = 2\sigma\cos\theta/(r\rho g)$	Capillary rise
$P_\text{excess} = 2\sigma/r$ (drop), $4\sigma/r$ (bubble)	Surface tension

Thermal Properties

Formula	Meaning
$\Delta L = \alpha L_0\Delta T$	Linear expansion
$\Delta V = \gamma V_0\Delta T$ , $\gamma\approx 3\alpha$	Volume expansion
$Q = mc\Delta T$	Heat absorbed
$Q = mL$	Latent heat
$\dfrac{Q}{t} = -kA\dfrac{dT}{dx}$	Fourier (conduction)
Thermal resistance $R_\text{th}=L/(kA)$	Series $R$ , parallel $1/R$
$\dfrac{Q}{t}=\sigma e A(T^4-T_s^4)$	Stefan–Boltzmann (net)
$\lambda_m T = b$	Wien's law, $b=2.9\times 10^{-3}$ m·K

Class XI - Thermodynamics & Kinetic Theory (Ch 12–13)

Thermodynamics

Formula	Meaning
$\Delta U = Q - W$	First law
$W = \int P\,dV$	Work done by gas
Isobaric: $W=P\Delta V$	$Q=nC_P\Delta T$
Isochoric: $W=0$	$Q=nC_V\Delta T$
Isothermal: $W=nRT\ln(V_f/V_i)$	$\Delta U=0$
Adiabatic: $PV^\gamma=$ const	$TV^{\gamma-1}=$ const, $T^\gamma P^{1-\gamma}=$ const
$W_\text{adia}=\dfrac{P_1V_1-P_2V_2}{\gamma-1}$	$Q=0$
$\eta_\text{Carnot}=1-T_c/T_h$	Carnot efficiency
COP refrigerator $= Q_c/W = T_c/(T_h-T_c)$
Mayer: $C_P-C_V=R$	Ideal gas
$\gamma=1+2/f$	$f$ = DOF

Kinetic Theory

Formula	Meaning
$PV = nRT = Nk_BT = (\rho/M)RT$	Ideal gas
$P = \tfrac{1}{3}\rho\langle v^2\rangle$	Pressure (kinetic)
$\langle E_k\rangle=\tfrac{3}{2}k_BT$	Per molecule, translational
$v_{\rm rms}=\sqrt{3RT/M}$ , $\bar v=\sqrt{8RT/\pi M}$ , $v_p=\sqrt{2RT/M}$	Speeds
$\langle E\rangle = (f/2)k_BT$	Equipartition
$\lambda = 1/(\sqrt 2\,n\pi d^2)$	Mean free path
$C_V=(f/2)R$ , $C_P=((f+2)/2)R$	Specific heats
Dulong–Petit: $C_\text{solid}\approx 3R$	High $T$

Class XI - Oscillations & Waves (Ch 14–15)

SHM

Formula	Meaning
$a = -\omega^2 x$	Defining SHM
$x = A\sin(\omega t+\phi)$	Displacement
$v = \omega\sqrt{A^2-x^2}$	Velocity
$v_\text{max}=A\omega$ , $a_\text{max}=A\omega^2$	Max values
$T = 2\pi\sqrt{m/k}$	Spring
Parallel: $k_\text{eq}=k_1+k_2$ ; Series: $1/k_\text{eq}=1/k_1+1/k_2$
$T = 2\pi\sqrt{L/g}$	Simple pendulum
$T=2\pi\sqrt{I/Mg\ell}$	Physical pendulum
$E = \tfrac{1}{2}kA^2$	Total energy
$K = \tfrac{1}{2}k(A^2-x^2)$ , $U = \tfrac{1}{2}kx^2$
Damped: $A(t)=A_0 e^{-bt/2m}$
Resonance: $A_\text{max}\approx F_0 Q/k$	$Q=\omega_0/(2\gamma)$

Waves

Formula	Meaning
$y = A\sin(kx-\omega t+\phi)$	Travelling wave
$v = f\lambda = \omega/k$	Speed
$v=\sqrt{T/\mu}$	String
$v=\sqrt{\gamma RT/M}=\sqrt{\gamma P/\rho}$	Sound in gas
$v=\sqrt{Y/\rho}$	Solid rod
Both ends fixed: $f_n = nv/(2L)$	All harmonics
Closed pipe: $f_n=(2n-1)v/(4L)$	Odd only
Open pipe: $f_n = nv/(2L)$	All
Beat: $f_\text{beat}=\vert f_1-f_2\vert$
Doppler: $f' = f(v+v_O)/(v-v_S)$	With sign convention

Class XII - Electrostatics (Ch 1–2)

Coulomb, Field, Flux

Formula	Meaning
$F = \dfrac{1}{4\pi\varepsilon_0}\dfrac{q_1q_2}{r^2}$	Coulomb
$k=1/(4\pi\varepsilon_0)=9\times 10^9$ N·m²/C²
$\vec E = \vec F/q$	Electric field
$E_\text{dipole, axial} = \dfrac{1}{4\pi\varepsilon_0}\dfrac{2p}{r^3}$	$p=q\cdot 2a$
$E_\text{dipole, equatorial}= \dfrac{1}{4\pi\varepsilon_0}\dfrac{p}{r^3}$
$\vec\tau=\vec p\times\vec E$	Torque on dipole
$U_\text{dipole}=-\vec p\cdot\vec E$
$\Phi_E = \oint\vec E\cdot d\vec A = q_\text{enc}/\varepsilon_0$	Gauss's law
Infinite line: $E = \lambda/(2\pi\varepsilon_0 r)$
Infinite sheet: $E = \sigma/(2\varepsilon_0)$
Conducting sheet: $E = \sigma/\varepsilon_0$
Shell, outside: $E=kQ/r^2$ ; inside: 0

Potential & Capacitance

Formula	Meaning
$V = W/q$ , $V = -\int\vec E\cdot d\vec l$	Definitions
$V_\text{point} = kQ/r$
$V_\text{dipole, axial}=kp/r^2$
$E = -dV/dr$
$U = q_1q_2/(4\pi\varepsilon_0 r)$	Two-charge PE
$C = Q/V$	Capacitance
$C_\text{parallel-plate}=\varepsilon_0 A/d$	Vacuum
With dielectric: $C = \kappa C_0$
Series: $1/C_\text{eq}=\sum 1/C_i$ ; Parallel: $C_\text{eq}=\sum C_i$
$U = \tfrac{1}{2}CV^2 = Q^2/(2C) = \tfrac{1}{2}QV$
Energy density: $u = \tfrac{1}{2}\varepsilon_0 E^2$

Class XII - Current Electricity (Ch 3)

Formula	Meaning
$I = dQ/dt$	Current
$J = I/A = nev_d$	Current density
$V = IR$	Ohm's law
$R = \rho L/A$	Resistance
$\rho = \rho_0[1+\alpha(T-T_0)]$	Temperature dependence
Drift: $v_d = eE\tau/m$
Series: $R_\text{eq}=\sum R_i$ ; Parallel: $1/R_\text{eq}=\sum 1/R_i$
$P = VI = I^2R = V^2/R$	Power
Kirchhoff: $\sum I=0$ (junction), $\sum V=0$ (loop)
EMF & internal $r$ : $V = \varepsilon - Ir$
Wheatstone: $P/Q = R/S$	Balance
Potentiometer: $\varepsilon_1/\varepsilon_2 = \ell_1/\ell_2$
$RC$ charge: $Q = Q_0(1-e^{-t/RC})$ , discharge: $Q = Q_0 e^{-t/RC}$

Class XII - Magnetism & EMI (Ch 4–7)

Magnetic Field of Currents

Formula	Meaning
Biot–Savart: $d\vec B=\dfrac{\mu_0}{4\pi}\dfrac{I\,d\vec l\times\hat r}{r^2}$
Long straight wire: $B = \mu_0 I/(2\pi r)$
Circular loop, axis: $B = \dfrac{\mu_0 I R^2}{2(R^2+x^2)^{3/2}}$
Centre of loop: $B=\mu_0 I/(2R)$
Solenoid (long): $B = \mu_0 n I$	$n$ = turns/length
Toroid: $B = \mu_0 N I/(2\pi r)$
Ampère's law: $\oint\vec B\cdot d\vec l = \mu_0 I_\text{enc}$

Force on Charges & Currents

Formula	Meaning
$\vec F = q(\vec E+\vec v\times\vec B)$	Lorentz force
$\vec F = I\vec L\times\vec B$	Wire in field
Cyclotron radius $r = mv/(qB)$ , $T = 2\pi m/(qB)$
$\vec\tau = \vec m\times\vec B$ , $\vec m = NI\vec A$	Magnetic moment
$U = -\vec m\cdot\vec B$
Galvanometer: $\theta = NIBA/k$

Magnetism

Formula	Meaning
$\vec B = \mu_0(\vec H+\vec M)$	$\vec M$ magnetisation
$\chi = M/H$	Susceptibility
$\mu_r = 1 + \chi$	Relative permeability
Curie's law: $\chi = C/T$	Paramagnet

EMI

Formula	Meaning
$\Phi_B = \int\vec B\cdot d\vec A$	Flux
$\varepsilon = -d\Phi_B/dt$	Faraday
Motional EMF: $\varepsilon = BLv$
Self-inductance: $\varepsilon = -L\,dI/dt$ , $L_\text{solenoid}=\mu_0 n^2 A\ell$
Mutual inductance: $\varepsilon_2=-M\,dI_1/dt$
Energy in $L$ : $U = \tfrac{1}{2}LI^2$
Energy density: $u = B^2/(2\mu_0)$
$LR$ rise: $I = I_0(1-e^{-Rt/L})$

AC Circuits

Formula	Meaning
$V = V_0\sin\omega t$	Source
RMS: $V_\text{rms}=V_0/\sqrt 2$
$X_L=\omega L$ , $X_C=1/\omega C$	Reactance
$Z = \sqrt{R^2+(X_L-X_C)^2}$	Impedance, series RLC
$\tan\phi=(X_L-X_C)/R$	Phase
Resonance: $\omega_0 = 1/\sqrt{LC}$	$Z_\text{min}=R$
$P_\text{avg}=V_\text{rms}I_\text{rms}\cos\phi$	Average power
Transformer: $V_s/V_p=N_s/N_p$ , $I_s/I_p=N_p/N_s$	Ideal

Class XII - EM Waves & Optics (Ch 8–10)

EM Waves

Formula	Meaning
$c = 1/\sqrt{\mu_0\varepsilon_0}=3\times 10^8$ m/s
$v = c/n$ in medium	$n$ refractive index
$E_0/B_0 = c$
$u = \tfrac{1}{2}\varepsilon_0 E_0^2 = B_0^2/(2\mu_0)$	Energy density
Intensity: $I = \tfrac{1}{2}\varepsilon_0 cE_0^2$
$\vec S = (\vec E\times\vec B)/\mu_0$	Poynting vector

Ray Optics

Formula	Meaning
Reflection: $\theta_i = \theta_r$
Snell: $n_1\sin\theta_1=n_2\sin\theta_2$
Critical angle: $\sin\theta_c = n_2/n_1$	$n_1>n_2$
Mirror: $1/v + 1/u = 2/R = 1/f$	Sign convention
Lens: $1/v - 1/u = 1/f$
Magnification (mirror): $m=-v/u$
Magnification (lens): $m=v/u$
Lensmaker: $1/f = (n-1)(1/R_1 - 1/R_2)$
Power: $P = 1/f$ (in m), in dioptres
Combination: $1/f = 1/f_1 + 1/f_2$ , $P = P_1+P_2$	Thin in contact
Prism: $\delta = (n-1)A$ (small angle)
Min. deviation: $n = \sin((A+\delta_m)/2)/\sin(A/2)$
Compound microscope: $M = -(L/f_o)(1+D/f_e)$
Astronomical telescope: $M = -f_o/f_e$ (normal adjustment)

Wave Optics

Formula	Meaning
Path diff for bright: $\Delta = n\lambda$ ; dark: $(n+\tfrac{1}{2})\lambda$
YDSE fringe width: $\beta = \lambda D/d$
Intensity: $I = I_1+I_2+2\sqrt{I_1I_2}\cos\delta$
Single slit minima: $a\sin\theta = n\lambda$	$n=1,2,\dots$
Width of central max: $2\lambda D/a$
Resolving power (telescope): $1.22\lambda/D$	Rayleigh
Brewster's angle: $\tan\theta_B = n$	Polarisation
Malus's law: $I = I_0\cos^2\theta$

Class XII - Modern Physics (Ch 11–13)

Dual Nature

Formula	Meaning
Photon energy: $E = h\nu = hc/\lambda$
Photon momentum: $p = h/\lambda = E/c$
Photoelectric: $h\nu = \phi + K_\text{max}$	Einstein
Stopping potential: $eV_0 = K_\text{max}$
Threshold: $\nu_0 = \phi/h$
de Broglie: $\lambda = h/p = h/\sqrt{2mK}$
Electron: $\lambda = 1.227/\sqrt V$ nm	$V$ in volts

Atoms

Formula	Meaning
Bohr quantisation: $mvr = n\hbar$
$r_n = n^2 a_0/Z$ , $a_0 = 0.529$ Å
$v_n = Zc\alpha/n$
$E_n = -13.6\,Z^2/n^2$ eV
Rydberg: $1/\lambda = R(1/n_1^2 - 1/n_2^2)$ , $R=1.097\times 10^7$ m⁻¹
Series: Lyman ( $n_1=1$ , UV), Balmer (2, visible), Paschen (3, IR), Brackett (4), Pfund (5)

Nuclei

Formula	Meaning
$R = R_0 A^{1/3}$ , $R_0 \approx 1.2$ fm
$E = \Delta m\,c^2$	Mass–energy
Binding energy per nucleon $\sim 8$ MeV (peak Fe)
Decay law: $N = N_0 e^{-\lambda t}$
$T_{1/2} = \ln 2/\lambda = 0.693/\lambda$
Mean life: $\tau = 1/\lambda$
Activity: $A = \lambda N$ , units Bq or Ci

Class XII - Semiconductors (Ch 14)

Formula	Meaning
Energy gap: $E_g$ (Si $\sim$ 1.1 eV, Ge $\sim$ 0.7 eV)
Intrinsic carrier density: $n_i \propto T^{3/2} e^{-E_g/2k_BT}$
Mass action: $n_e\cdot n_h = n_i^2$
Conductivity: $\sigma = e(n_e\mu_e + n_h\mu_h)$
Diode equation (Shockley): $I = I_0(e^{eV/k_BT}-1)$
Half-wave rectifier ripple frequency: $f$
Full-wave rectifier ripple frequency: $2f$
Common-emitter: $\beta_\text{dc} = I_C/I_B$
$\alpha = I_C/I_E$ , $\beta = \alpha/(1-\alpha)$
Voltage gain (CE): $A_V = \beta R_L/R_i$
Logic: NOT, AND, OR, NAND, NOR	Boolean basics
NAND = universal gate

End of formula sheet. Keep this open during practice; replace memorisation with familiarity.