Physics XII — Formula Reference

Electrostatics

Coulomb's Law & Electric Field

Coulomb's Law $F = \dfrac{1}{4\pi\varepsilon_0}\dfrac{q_1 q_2}{r^2}$

Electric field $E = \dfrac{F}{q}$

Point charge $E = \dfrac{1}{4\pi\varepsilon_0}\dfrac{q}{r^2}$

Potential $V = \dfrac{1}{4\pi\varepsilon_0}\dfrac{q}{r}$

Relation $E = -\dfrac{dV}{dr}$

Electric Dipole

Dipole moment $p = q(2a)$

Axial field $E = \dfrac{1}{4\pi\varepsilon_0}\dfrac{2p}{r^3}$

Equatorial field $E = \dfrac{1}{4\pi\varepsilon_0}\dfrac{p}{r^3}$

Torque $\tau = pE\sin\theta$

Potential energy $U = -pE\cos\theta$

Gauss's Law

Gauss's Law $\displaystyle\oint \vec{E} \cdot d\vec{A} = \dfrac{q_{\text{enc}}}{\varepsilon_0}$

Infinite line charge $E = \dfrac{\lambda}{2\pi\varepsilon_0 r}$

Infinite plane sheet $E = \dfrac{\sigma}{2\varepsilon_0}$

Inside conductor $E = 0$

Capacitors

Capacitance $C = \dfrac{Q}{V}$

Parallel plate $C = \dfrac{\varepsilon_0 A}{d}$

With dielectric $C = \dfrac{K\varepsilon_0 A}{d}$

Energy stored $U = \tfrac{1}{2}CV^2 = \tfrac{1}{2}QV = \dfrac{Q^2}{2C}$

Series $\dfrac{1}{C} = \dfrac{1}{C_1} + \dfrac{1}{C_2}$

Parallel $C = C_1 + C_2$

Current Electricity

Current $I = \dfrac{Q}{t}$

Drift velocity $I = nqAv_d$

Ohm's Law $V = IR$

Resistance $R = \rho\dfrac{l}{A}$

Temp. dependence $R = R_0(1 + \alpha\Delta T)$

Power $P = VI = I^2R = \dfrac{V^2}{R}$

Kirchhoff's Laws & Wheatstone Bridge

Junction rule $\sum I = 0$

Loop rule $\sum V = 0$

Wheatstone (balanced) $\dfrac{P}{Q} = \dfrac{R}{S}$

Magnetic Field

Biot–Savart Law $dB = \dfrac{\mu_0}{4\pi}\dfrac{I\,dl\sin\theta}{r^2}$

Long straight wire $B = \dfrac{\mu_0 I}{2\pi r}$

Circular loop (center) $B = \dfrac{\mu_0 I}{2R}$

Solenoid $B = \mu_0 n I$

Magnetic Force

On moving charge $F = qvB\sin\theta$

On conductor $F = BIL\sin\theta$

Circular motion radius $r = \dfrac{mv}{qB}$

Cyclotron frequency $f = \dfrac{qB}{2\pi m}$

Electromagnetic Induction

Magnetic flux $\Phi = BA\cos\theta$

Faraday's Law $\mathcal{E} = -\dfrac{d\Phi}{dt}$

Lenz's Law → Induced EMF opposes the change in flux that causes it.

Self inductance $\mathcal{E} = -L\dfrac{dI}{dt}$

Energy in inductor $U = \dfrac{1}{2}LI^2$

Alternating Current

EMF & RMS Values

AC EMF $\mathcal{E} = \mathcal{E}_0 \sin\omega t$

Angular freq. $\omega = 2\pi f$

RMS current $I_{\text{rms}} = \dfrac{I_0}{\sqrt{2}}$

RMS voltage $V_{\text{rms}} = \dfrac{V_0}{\sqrt{2}}$

Reactance, Impedance & Resonance

Inductive reactance $X_L = \omega L$

Capacitive reactance $X_C = \dfrac{1}{\omega C}$

Impedance $Z = \sqrt{R^2 + (X_L - X_C)^2}$

Resonance condition $X_L = X_C$

Resonant frequency $f = \dfrac{1}{2\pi\sqrt{LC}}$

Ray Optics

Mirror formula $\dfrac{1}{f} = \dfrac{1}{v} + \dfrac{1}{u}$

Magnification $m = \dfrac{v}{u}$

Lens formula $\dfrac{1}{f} = \dfrac{1}{v} - \dfrac{1}{u}$

Lens maker's eq. $\dfrac{1}{f} = (\mu - 1)\!\left(\dfrac{1}{R_1} - \dfrac{1}{R_2}\right)$

Power of lens $P = \dfrac{1}{f}$

Combined power $P = P_1 + P_2$

Wave Optics

Young's Double Slit Experiment

Fringe width $\beta = \dfrac{\lambda D}{d}$

n-th bright fringe $y_n = n\beta$

Single Slit Diffraction

Minima condition $a\sin\theta = n\lambda$

Photoelectric Effect

Einstein eq. $h\nu = W_0 + KE_{\max}$

KE max $KE_{\max} = \tfrac{1}{2}mv^2$

Stopping pot. $eV_0 = KE_{\max}$

Bohr's Atomic Model

Bohr radius $r_n = \dfrac{n^2 h^2 \varepsilon_0}{\pi m e^2}$

Energy level $E_n = -\dfrac{13.6}{n^2}\;\text{eV}$

Wavelength $\dfrac{1}{\lambda} = R\!\left(\dfrac{1}{n_1^2} - \dfrac{1}{n_2^2}\right)$

Nuclei

Binding energy $BE = \Delta m \cdot c^2$

Radioactive decay $N = N_0\, e^{-\lambda t}$

Half-life $T_{1/2} = \dfrac{0.693}{\lambda}$

Semiconductor

Diode current $I = I_0\!\left(e^{V/\eta V_T} - 1\right)$

Zener breakdown → Voltage regulation in reverse bias.

Logic Gates

AND Output 1 only if all inputs are 1

OR Output 1 if any input is 1

NOT Output is the complement of input