Electrics
Coulomb's law: $F=\frac{1}{4 \pi \epsilon _0}\frac{q_1q_2}{r^2}$
permitivity constant ϵ0=8.85*10-12C2/N·m2
charge is quantized: $q=ne$
elementary charge e=1.6*10-14C
electric field: $\mathbf{E}=\frac{\mathbf{F}}{q_0}$
differential: $d\mathbf{E}=\frac{1}{4\pi \epsilon _0}\frac{dq}{r^3}\mathbf{r}$ (r from dq to point)
―point charge: $E=\frac{1}{4 \pi\epsilon _0}\frac{q}{r^2}$
―straight rod (perpendicular): $E=\frac{\lambda a}{2\pi \epsilon _0 r}\frac{1}{\sqrt{4r^2+a^2}}$
―arc (to centre): $E=\frac{\lambda }{4 \pi \epsilon _0r}(2\mathrm{sin}\frac{\theta }{2})$
―ring (perpendicular): $E=\frac{qz}{4 \pi \epsilon _0(z^2+R^2)^{3/2}}$
―round disk (perpendicular): $E=\frac{\sigma }{2 \epsilon _0}(1-\frac{z}{\sqrt{z^2+R^2}})$
electrostatic force in a field: $\mathbf{F}=q\mathbf{E}$ (signed)
Gauss' law: $\Phi _E=\iint \mathbf{E}\cdot d\mathbf{S}=\frac{q}{\epsilon _0}$
electric field (infinite)
―conducting surface: $E=\frac{\sigma }{\epsilon _0}$
―nonconducting surface: $E=\frac{\sigma }{2\epsilon _0}$
―straight rod: $\frac{\lambda }{2\pi r\epsilon _0}$
―two conducting plates (+ greater): $|E_L|=|E_R|=|E_{(+)}-E_{(-)}|$, $|E_{\textrm{in}}|=E_{(+)}+E_{(-)}=\frac{\sigma _1+\sigma _2}{\epsilon _0}$
―shell: $E=\frac{1}{4\pi \epsilon _0}\frac{q}{r^2}$ (outside), $E=0$ (inside)
―sphere: $E=\frac{1}{4\pi \epsilon _0}\frac{q}{r^2}$ (outside), $E=\left ( \frac{q}{4\pi \epsilon _0R^3} \right )r$ (inside)
―cylinder: $E=\frac{R^2\rho }{2\epsilon _0r}$ (outside), $E=\frac{\rho }{2\epsilon _0}r$ (inside)
work: $W=\int \mathbf{F}\cdot d\mathbf{s}=q_0\int \mathbf{E}\cdot d\mathbf{s}$
electric potential difference: $\Delta V=-\frac{W_{if}}{q_0}=\frac{\Delta U}{q_0}$
―calculate: $V_f-V_i=-\int _i^f\mathbf{E}\cdot d\mathbf{s}$
electric field of parallel plates: $E=\frac{\Delta V}{\Delta d}$
potential
―point charge: $V=\frac{1}{4\pi \epsilon _0}\frac{q}{r}$(signed)
―discrete points: $V=\frac{1}{4\pi \epsilon _0}\sum _i\frac{q_i}{r_i}$
―continuous charge: $V=\frac{1}{4\pi \epsilon _0}\int \frac{dq}{r}$
―rod (perpendicular to one end): $V=\frac{\lambda }{4\pi \epsilon _0}\, \mathrm{ln}(\frac{L+(L^2+d^2)^{1/2}}{d})$
―disk (perpendicular): $V=\frac{\sigma }{2\epsilon _0}(\sqrt{z^2+R^2}-z)$
―arc (to centre): $V=\frac{\lambda \theta }{4\pi \epsilon _0}$
E-V relation: $\mathbf{E}=-\nabla V$, $V=-\int _{i_0}^f \mathbf{E}\cdot d\mathbf{s}$
Current and circuit
current: $i=\frac{dq}{dt}$
current density: $J=i/A$, $i=\iint \mathbf{J}\cdot d\mathbf{A}$
draft speed: $\mathbf{v}_\mathrm{d}=\mathbf{J}/(ne)$, n: number of carriers per unit volume
resistance law: $R=\frac{V}{i}$
isotropic resistivity: $\rho=E/J$, $\mathbf{E}=\rho \mathbf{J}$
conductivity: $\sigma =1/\rho $
resistance: $R=\rho \frac{L}{A}$
variation with temperature: $\rho -\rho _0=\rho _0\alpha (T-T_0)$, ρ: temperature coefficient of resistivity, T0=293K, ρ0=1.69μΩ·cm
rate of electricity supply: $P=iV$
resistive dissipation: $P=i^2R=\frac{V^2}{R}$
work per unit charge to maintain ΔV/emf: $\varepsilon =\frac{dW}{dq}$
supplying current: $i=\frac{\varepsilon }{R}$
loop rule
resistance rule: $\Delta V=-iR$ (current direction), $\Delta V=+iR$ (opposite)
emf rule: $\Delta V=+\varepsilon $ (current direction), $\Delta V=-\varepsilon $ (opposite)
series charge: $q=q_1=q_2=...$
parallel charge: $q=\sum_j q_j$
series current: $i=i_1=i_2=...$
parallel current: $i=\sum_j i_j$
series voltage: $V=\sum _j V_j$
parallel voltage: $V=V_1=V_2=...$
series resistance: $R=\sum _j R_j$
parallel resistance: $\frac{1}{R}=\sum _j \frac{1}{R_j}$, two: $R=\frac{R_1R_2}{R_1+R_2}$
Capacitance
capacitance: $C=\frac{q}{V}$
capacitor
―parallel-plate: $C=\epsilon _0\frac{A}{d}$
―cylindrical: $C=2\pi \epsilon _0L\frac{1}{\mathrm{ln(b/a)}}$
―spherical: $C=4\pi \epsilon _0\frac{ab}{b-a}$
―isolated sphere: $C=4\pi \epsilon _0R$
series capacitor: $\frac{1}{C}=\sum_{j}\frac{1}{C_j}$
parallel capacitor: $C=\sum _j C_j$
potential energy: $U=\frac{q^2}{2C}=\frac{1}{2}CV^2$
volume energy density: $u=\frac{1}{2}\epsilon _0E^2$
―q unchanged: $U_f=U_i/\kappa $
―V unchanged: $U_f=\kappa U_i$
electric displacement: $\mathbf{D}=\kappa \epsilon _0\mathbf{E}=\epsilon \mathbf{E}$
RC circuit
charging equation: $R\frac{dq}{dt}+\frac{q}{C}=\varepsilon $
―solution: $q=C\varepsilon (1-e^{-t/\tau _C })$, $i=(\frac{\varepsilon }{R})e^{-t/\tau _C }$
discharging equation: $R\frac{dq}{dt}+\frac{q}{C}=0 $
―solution: $q=q_0e^{-t/\tau _C }$, $i=-i_0e^{-t/\tau _C }$
capacitive time constant $\tau _C=RC$
Magnetics
force due to moving charge: $\mathbf{F}_B=q\mathbf{v}\times \mathbf{B}$
force due to current-carrying wire: $\mathbf{F}_B=i\mathbf{L}\times \mathbf{B}$, L along direction of conventional i
Hall effect, density of carriers: $n=\frac{Bi}{Vle}$, l=A/d: thinkness of strip
circular motion: $qvB=m\frac{v^2}{r}$
period: $T=\frac{2\pi m}{qB}$
Biot-Savart law: $d\mathbf{B}=\frac{\mu _0}{4\pi }\frac{id\mathbf{s\times \mathbf{r}}}{r^3}$
permitivity constant μ0=4π*10-7T·m/A (H/m)
―arc (to centre): $B=\frac{\mu _0i\theta }{4\pi R}$
Ampere's circuital law (magnetic field due to current): $\oint \mathbf{B}\cdot d\mathbf{s}=\mu _0i$
―long straight wire: $B=\frac{\mu _0i}{2\pi r}$
―solid wire: $B=\frac{\mu _0i}{2\pi r}$ (outside), $B=(\frac{\mu _0i}{2\pi R^2})r$ (inside)
―ideal solenoid: $B=\mu _0i_0n$, n=N/L: turns per unit length
―ideal toroid: $B=\frac{\mu _0i_0N}{2\pi r}$
induced emf: $\varepsilon =-\frac{d\Phi _B}{dt}=-\frac{d}{dt} \iint \mathbf{B} \cdot d\mathbf{S}$
for coils: $\varepsilon =-N\frac{d\Phi _B}{dt}$
Maxwell-Faraday equation: $\oint \mathbf{E}\cdot d\mathbf{s}=-\frac{d}{dt} \iint \mathbf{B}\cdot d\mathbf{S}$
―induced electrodynamic field, circle: $E=\frac{R^2}{2}\frac{dB}{dt}\frac{1}{r}$ (outside), $E=\frac{1}{2}\frac{dB}{dt}r$ (inside)
Inductance
inductance: $L=\frac{N\Phi _B}{i}$
inductor
―solenoid: $L/l=\mu _0n^2A$
―toroid: $L=\frac{\mu _0N^2h}{2\pi} \, \mathrm{ln}(\frac{b}{a})$
self-induced emf: $\varepsilon _L=-L\frac{di}{dt}$
potential energy: $U_B=\frac{1}{2}Li^2$
energy density: $u_B=\frac{B^2}{2\mu _0}$
LR circuit
rise in current: $iR+L\frac{di}{dt}=\varepsilon $
―solution: $i=\frac{\varepsilon }{R}(1-e^{-t/\tau _L})$
decay in current: $iR+L\frac{di}{dt}=0$
―solution: $i=i_0e^{-t/\tau _L}$
inductive time constant: $\tau _L=L/R$
mutual induction (two coils): $\varepsilon _2=-M\frac{di_1}{dt}$, $\varepsilon _1=-M\frac{di_2}{dt}$
LC oscillation
definition: $\frac{d^2q}{dt^2}+\frac{1}{LC}q=0$
solution: $q=Q\mathrm{cos}(\omega t+\phi)$
angular frequency: $\omega =\frac{1}{\sqrt{LC}}$
electric potential energy: $U_E=\frac{Q^2}{2C}\mathrm{cos}^2(\omega t+\phi)$
magnetic potential energy: $U_B=\frac{Q^2}{2C}\mathrm{sin}^2(\omega t+\phi)$
total energy: $U=\frac{Q^2}{2C}$
series RLC oscillation
net energy dissipation: $\frac{dU}{dt}=-i^2R$
definition: $\frac{d^2q}{dt^2}+\frac{1}{LR}\frac{dq}{dt}+\frac{1}{LC}q=0$
solution: $q=Qe^{Rt/2L}\mathrm{cos}(\omega 't+\phi)$, $\omega '=\sqrt{\omega ^2-(R/2L)^2}$
electromagnetic waves
magnetic field induced by electric field: $\oint \mathbf{B}_E\cdot d\mathbf{s}=+\mu _0\epsilon _0\frac{d\Phi _E}{dt}=+\mu _0\epsilon _0\frac{d}{dt}\iint\mathbf{E}\cdot d\mathbf{S}$
―"displacement current" between parallel plates (capacitor), circle: $B=\frac{\mu _0\epsilon _0R^2}{2}\frac{dE}{dt}\frac{1}{r}$ (outside), $\frac{\mu _0\epsilon _0}{2}\frac{dE}{dt}r$ (inside)
displacement current: $i_d=\epsilon _0\frac{d\Phi _E}{dt}$
B and E are in phase: $E=E_m\mathrm{sin}(kx-\omega t)$, $B=B_m\mathrm{sin}(kx-\omega t)$
wave speed: $c=\frac{\omega }{k}$
magnitude ratio: $\frac{E_m}{B_m}=c$
speed of light: $c=\frac{1}{\sqrt{\epsilon _0\mu _0}}$
direction of wave/poynting vector: $\mathbf{S}=\frac{1}{\mu _0}\mathbf{E}\times \mathbf{B}$
plane wave's instantaneous flow rate: $S=\frac{1}{c\mu _0}E^2$ (S=P/A)
wave intensity: $I=\bar{S}=\frac{1}{c\mu _0}E_{\textrm{rms}}^2$
momentum of light: $\Delta p=\frac{\Delta U}{c}$ (total absorption), $\Delta p=\frac{2\Delta U}{c}$ (total reflection)
radiation pressure: $p_r=\frac{I}{c}$ (total absorption), $p_r=\frac{2I}{c}$ (total reflection)
law of Malus: $I=I_m \cos ^2\theta $
AC
resistive circuit: $V_R=I_RR$
capacitive circuit: $V_C=I_CX_C$
―capacitive reactance: $X_C=\frac{1}{\omega C}$
inductive circuit: $V_L=I_LX_L$
―inductive reactance: $X_L=\omega L$
series RLC circuit
current: $i=I\sin (\omega t-\phi)$
voltage: $\varepsilon =v_R+v_C+v_L$
current amplitude: $I=\frac{\varepsilon _m}{Z}$
―impedance $Z=\sqrt{R^2+(X_L-X_C)^2}$
phase constant $\tan \phi=\frac{V_L-V_C}{V_R}=\frac{X_L-X_C}{R}$
average power: $\bar{P}=I_{\textrm{rms}}^2R=\varepsilon _{\textrm{rms}}I_{\textrm{rms}}\cos \phi$
I is in phase with vR, leads vC by 90°, lags hehind vL by 90°
ideal transformer (rms) (AC supply at p end, sends to s end, R at s)
voltage: $\frac{V_s}{V_p}=\frac{N_s}{N_p}$
current: $\frac{I_s}{I_p}=\frac{N_p}{N_s}$
resistances: $R_{eq}=(\frac{N_p}{N_s})^2R$
Dipoles
electric dipole
electric field produced: $E=\frac{1}{2 \pi \epsilon _0}\frac{p}{z^3}$ (dipole axis)
electric potential: $V(\theta)=\frac{1}{4\pi \epsilon _0}\frac{p\mathrm{cos}\theta }{r^2}$
net torque: $\boldsymbol{\tau }=\mathbf{p}\times \mathbf{E}$
potential energy: $U(\theta)=-\mathbf{p}\cdot \mathbf{E}$
dipole moment: $\mathbf{p}=q\mathbf{d}$ (- to +)
magnetic dipole/current loop
magnetic field produced: $\mathbf{B}=\frac{\mu _0}{2\pi }\frac{\boldsymbol{\mu }}{z^3}$
net torque: $\boldsymbol{\tau }=\boldsymbol{\mu \times \mathbf{B}}$
potential energy: $U(\theta)=-\boldsymbol{\mu \cdot \mathbf{B}}$
magnetic dipole moment: $\boldsymbol{\mu }=Ni\mathbf{A}$, N: turns
Maxwell's equations
Gauss' law: $∯\mathbf{E}\cdot d\mathbf{S}=q/\epsilon _0$
Gauss' law for magnetism: $∯\mathbf{B}\cdot d\mathbf{S}=0$
Maxwell-Faraday equation: $\oint \mathbf{E}\cdot d\mathbf{s}=-\frac{d}{dt} \iint \mathbf{B}\cdot d\mathbf{S}$
Ampere's circuital law: $\oint \mathbf{B}\cdot d\mathbf{s}=\mu _0i+\mu _0\epsilon _0\frac{d}{dt} \iint \mathbf{E}\cdot d\mathbf{S}$
Gauss' law: $\nabla\cdot \mathbf{E}=\rho /\epsilon _0$
Gauss' law for magnetism: $\nabla\cdot \mathbf{B}=0$
Maxwell-Faraday equation: $\nabla \times \mathbf{E}=-\frac{d\mathbf{B}}{dt}$
Ampere's circuital law: $\nabla\times \mathbf{B}=\mu _0\mathbf{J}+\mu _0\epsilon _0\mathbf{E}$
D field: $\mathbf{D}=\epsilon \mathbf{E}$
H field: $\mathbf{H}=\mathbf{B}/\mu$
Magnets
magnetism due to spinning electron
spin angular momentum: $S=\frac{h}{4 \pi}=5.2729\times 10^{-35} \textrm{J/s}$
spin magnetic moment: $\mu _S=\mu _B=\frac{eh}{4 \pi m}=9.27\times 10^{-24}\textrm{J/T}$, μB: Bohr magneton
magnetism due to orbital motion
orbital angular momentum: $L_{\textrm{orb}}=mvr$
orbital magnetic moment: $\mu _{\textrm{orb}}=\frac{1}{2}evr$
―negative charge: $\boldsymbol{\mu }_{\textrm{orb}}=-\frac{e}{2m}\boldsymbol{L}_{\textrm{orb}}$