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PHYS 122 Electromagnetism

2021-08-17

Contents

Electrical Charge and Electric Field

Quantity	Unit	Definition
Electrical field (point charge)	$\mathrm{N/C} \newline \mathrm{(V/m)}$	$\vec{E}_{s} = \dfrac{\vec{F}_{0}}{q_{0}} = \dfrac{1}{4\pi\varepsilon_{0}} \dfrac{q}{r^{2}} \hat{r}$
Linear charge density	$\mathrm{C/m}$	$\lambda = \dfrac{dQ}{dl}$
Surface charge density	$\mathrm{C/m^{2}}$	$\sigma = \dfrac{dQ}{dA}$
Volume charge density	$\mathrm{C/m^{3}}$	$\rho = \dfrac{dQ}{dV}$
Electric dipole moment (direction from - to +)	$\mathrm{C\cdot m}$	$\vec{p} = q\vec{d}$
Induced dipole moment (direction from - to +)	$\mathrm{C\cdot m}$	$\vec{p} = \alpha\vec{E}$

Description	Equations
Coulomb’s law	$F = \dfrac{1}{4\pi\varepsilon_{0}} \dfrac{q_{1}q_{2}}{r^{2}} \hat{r}$
Force on test charge by an electric field	$\vec{F}_0 = q_{0} \vec{E}$
Superposition of electric forces	$\vec{F} = \sum\limits_{i} \vec{F}_{i}$
Superposition of electric fields	$\vec{E} = \sum\limits_{i} \vec{E}_{i}$
Torque on an electric dipole in an uniform electrical field	$\vec{\tau} = \vec{p} \times \vec{E}$
Potential energy of an electric dipole in an uniform electric field	$U = -\vec{p}\cdot\vec{E}$
Electric field of test charge on x axis caused by dipole at origin oriented in + y direction	$E_{x} = 0 \newline E_{y} = -\dfrac{kp}{\|x\|^{3}}$
Electric field of test charge on y axis caused by dipole at origin oriented in + y direction	$E_{x} = 0 \newline E_{y} = \dfrac{2kp}{\|y\|^{3}}$

Gauss’s Law

Quantity	Unit	Definition
Electric flux through a surface	$\mathrm{N \cdot m^{2}/C}$ $\mathrm{(V\cdot m)}$	$\Phi_{E} = \displaystyle\int \vec{E} \cdot d\vec{A}$

Description	Equations
Electric flux of a uniform electric field	$\Phi_{E} = \vec{E} \cdot \vec{A}$
Electric flux of a nonuniform electric field	$\Phi_{E} = \displaystyle\int \vec{E} \cdot d\vec{A} = \displaystyle\int E \cos\theta \ dA$
Gauss’s law Electric flux through a closed surface	$\begin{aligned}\Phi_{E} &= \displaystyle\oint \vec{E} \cdot d\vec{A} \cr &= \displaystyle\oint E \cos\theta \ dA = \dfrac{Q}{\varepsilon_{0}}\end{aligned}$

Electric field of uniform spherical charge distributions

charged = uniformly charged throughout (insulating)
conducting = charge only on surface

Charge Distribution	Point in Electric Field	Electric Field Magnitude
Point charge	-	$E = \dfrac{1}{4\pi\varepsilon_{0}}\dfrac{q}{r^{2}}$
Solid conducting sphere Hollow charged sphere	Outside sphere, $r>R$	$E = \dfrac{1}{4\pi\varepsilon_{0}}\dfrac{q}{r^{2}}$
Solid conducting sphere Hollow charged sphere	Inside sphere, $r<R$	$E = 0$
Solid charged sphere	Outside sphere, $r>R$	$E = \dfrac{1}{4\pi\varepsilon_{0}}\dfrac{q}{r^{2}}$
Solid charged sphere	Inside sphere, $r<R$	$E = \dfrac{1}{4\pi\varepsilon_{0}}\dfrac{r}{R^{3}}q$

Electric field of uniform cylindrical charge distributions

Charge Distribution	Point in Electric Field	Electric Field Magnitude
$\infin$ wire/rod	-	$E = \dfrac{1}{2\pi\varepsilon_{0}}\dfrac{\lambda}{r} = \dfrac{2k\lambda}{r}$
$\infin$ solid conducting cylinder $\infin$ hallow charged cylinder	Outside cylinder, $r>R$	$E = \dfrac{1}{2\pi\varepsilon_{0}}\dfrac{\lambda}{r} = \dfrac{2k\lambda}{r}$
$\infin$ solid conducting cylinder $\infin$ hallow charged cylinder	Inside cylinder, $r<R$	$E = 0$
$\infin$ solid charged cylinder	Outside cylinder, $r>R$	$E = \dfrac{1}{2\pi\varepsilon_{0}}\dfrac{\lambda}{r} = \dfrac{2k\lambda}{r}$
$\infin$ solid charged cylinder	Inside cylinder, $r<R$	$E = \dfrac{1}{2\pi\varepsilon_{0}}\dfrac{r}{R^{2}} \lambda = \dfrac{2k\lambda r}{R^{2}}$

Electric field of uniform planar charge distributions

Charge Distribution	Point in Electric Field	Electric Field Magnitude
$\infin$ charged sheet/plate	-	$E = \dfrac{\sigma}{2\varepsilon_{0}}$
$\infin$ conducting sheet/plate	-	$E = \dfrac{\sigma}{\varepsilon_{0}} = \dfrac{q}{2\varepsilon_{0}A}$ ( $q$ spreads at each surface)
Two oppositely charged conducting plates	Between plates	$E = \dfrac{\sigma}{\varepsilon_{0}}$
Charged conductor	At surface	$E = \dfrac{\sigma}{\varepsilon_{0}}$

Electric Potential

Quantity	Unit	Definition
Electric potential energy (point charge) (choose $U = 0$ at $\infty$ )	$\mathrm{J}$	$U = \dfrac{1}{4\pi\varepsilon_{0}} \dfrac{q_{s}q_{0}}{r}$
Electric potential (point charge) (choose $V = 0$ at $\infty$ )	$\mathrm{V}$ $\mathrm{(J/C)}$	$V = \dfrac{U}{q_{0}} = \dfrac{1}{4\pi\varepsilon_{0}} \dfrac{q_{s}}{r}$

Description	Equations
Electric potential energy of a test charge due to many source charges	$U = \dfrac{q_{0}}{4\pi\varepsilon_{0}} \sum\limits_{i} \dfrac{q_{i}}{r_{i}}$
Total electric potential energy of all source charges	$U = \dfrac{1}{4\pi\varepsilon_{0}} \sum\limits_{i<j} \dfrac{q_{i}q_{j}}{r_{ij}}$
Electric potential due to many source charges	$V = \dfrac{1}{4\pi\varepsilon_{0}} \sum\limits_{i} \dfrac{q_{i}}{r_{i}}$
Electric potential due to continuous distribution of charges	$V = \dfrac{1}{4\pi\varepsilon_{0}} \displaystyle\int \dfrac{dq}{r}$
Electric potential and potential energy of point charges	$U = q_{2}V_{1}$
Work by electric force and electric field	$W_{a \to b} = \displaystyle\int_{a}^{b} \vec{F} \cdot d\vec{l} = q \int_{a}^{b} \vec{E} \cdot d\vec{l}$
Work by electric force on a closed path	$W_{a \to b \to a} = q \displaystyle\oint \vec{E} \cdot d\vec{l} = 0$
Work by electric force and change in potential energy	$W_{a \to b} = -\Delta U$
Potential difference	$V_{ab} = V_{b} - V_{a}$
Potential difference between terminals of battery	$V_{\mathrm{batt}} = V_{-+} = V_{+} - V_{-}$
Potential difference and work, potential energy difference	$V_{ab} = \dfrac{\Delta U}{q_{0}} = -\dfrac{W_{a \to b}}{q_{0}}$
Potential difference and electric field	$V_{ab} = -\displaystyle\int_{a}^{b} \vec{E} \cdot d\vec{l} = -\displaystyle\int E \cos\theta \ dl$
Electric field and potential gradient	$\begin{aligned}\vec{E} &= -\vec{\nabla}V \cr &= \left<-\dfrac{\partial V}{\partial x}, -\dfrac{\partial V}{\partial y}, -\dfrac{\partial V}{\partial z} \right>\end{aligned}$

Capacitance and Dielectrics

Quantity	Unit	Definition
Capacitance (in vacuum)	$\mathrm{F} \newline \mathrm{(C/V = C^{2}/J)}$	$C = \dfrac{Q}{V_{-+}} = \dfrac{Q}{V_{+} - V_{-}}$
Electric energy density (in vacuum)	$\mathrm{J/m^{3}}$	$u = \dfrac{U}{Ad} = \dfrac{1}{2} \varepsilon_{0}E^{2}$

Description	Equations
Capacitance of a parallel-plate capacitor in vacuum	$C = \dfrac{Q}{V_{-+}} = \varepsilon_{0} \dfrac{A}{d}$
Potential energy stored in a charged capacitor (define $U_{\mathrm{uncharged}} \equiv 0)$	$U = \dfrac{Q^{2}}{2C} = \dfrac{1}{2}CV^{2} = \dfrac{1}{2}QV$
Electric energy density in vacuum	$u = \dfrac{U}{Ad} = \dfrac{1}{2} \varepsilon_{0}E^{2}$
Dielectric constant	$\kappa = \dfrac{C}{C_{0}} = \dfrac{V_{0}}{V} = \dfrac{E_{0}}{E}$
Induced surface charge density on a dielectric in an isolated capacitor	$\sigma_{\mathrm{induced}} = \sigma_{\mathrm{bound}} \newline \sigma_{0} = \sigma_{\mathrm{free}} \newline \sigma_{\mathrm{induced}} = \sigma_{0} \left(1 - \dfrac{1}{\kappa} \right)$
Permittivity of a dielectric	$\varepsilon = \kappa \varepsilon_{0}$
Capacitance of a parallel-plate capacitor with dielectric between plates	$C = \kappa C_{0} = \kappa\varepsilon_{0}\dfrac{A}{d} = \varepsilon\dfrac{A}{d}$
Electric energy density in a dielectric	$u = \dfrac{1}{2}\kappa\varepsilon_{0}E^{2} = \dfrac{1}{2}\varepsilon E^{2}$
Gauss’s law in dielectrics	$\displaystyle\oint \vec{E}\cdot d\vec{A} = \dfrac{q_{\mathrm{free, enc}}}{\kappa\varepsilon_{0}}$

Current, Resistance, and emf

Quantity	Unit	Definition
Current	$\mathrm{A} \newline \mathrm{(C/s)}$	$I = \dfrac{dQ}{dt}$
Current density (per unit cross-section area)	$\mathrm{A/m^{2}}$	$\vec{J} = nq\vec{v}_d \newline J = \dfrac{I}{A} = n \lvert q \rvert v_{d}$
Conductivity (intrinsic to a material)	$\mathrm{(\Omega\cdot m)^{-1}} \newline \mathrm{A/(V\cdot m)}$	$\sigma = \dfrac{J}{E}$
Resistivity (intrinsic to a material)	$\mathrm{\Omega\cdot m}$	$\rho = \dfrac{E}{J}$
Resistance	$\mathrm{\Omega}$	$R = \dfrac{V}{I} = \dfrac{\rho L}{A} = \dfrac{L}{\sigma A}$

Description	Equations
Drift velocity of charge carrier	$\vec{v}_{d} = -\dfrac{q\vec{E}}{m}\tau$
Current and conductor properties	$I = \dfrac{dQ}{dt} = n \lvert q \rvert v_{d}A = JA$
Current density (per unit cross-section area)	$\vec{J} = nq\vec{v}_d \newline J = \dfrac{I}{A} = n \lvert q \rvert v_{d} = \dfrac{nq^{2}\tau}{m_{q}}E$
Conductivity (intrinsic to a material)	$\sigma = \dfrac{J}{E} = \dfrac{nq^{2}\tau}{m_{q}}$
Temperature dependence of resistivity	$\rho(T) = \rho_{0} (1 + \alpha (T-T_{0}))$
Temperature dependence of resistance	$R(T) = R_{0} (1 + \alpha (T-T_{0}))$

Direct-Current (DC) Circuits

Circuit analysis

Description	Equations
Circuit elements in series $\lvert Q \rvert, I$ - Equal $V, R$ - Add $C$ - Reciprocal	$\lvert Q \rvert = \lvert Q_{1} \rvert = … = \lvert Q_{i} \rvert \newline I = I_{1} = … = I_{i} \newline V = \sum\limits_{i} V_{i} \newline R = \sum\limits_{i} R_{i} \newline \dfrac{1}{C} = \sum\limits_{i} \dfrac{1}{C_{i}}$
Circuit elements in parallel $V$ - Equal $Q, I, C$ - Add $R$ - Reciprocal	$V = V_{1} = … = V_{i} \newline Q = \sum\limits_{i} Q_{i} \newline I = \sum\limits_{i} I_{i} \newline C = \sum\limits_{i} C_{i} \newline \dfrac{1}{R} = \sum\limits_{i} \dfrac{1}{R_{i}}$
Algebra of reciprocal values of two elements	$\dfrac{1}{A} = \dfrac{1}{A_{1}} + \dfrac{1}{A_{2}} \Rightarrow A = \dfrac{A_{1}A_{2}}{A_{1} + A_{2}}$
Kirchhoff’s junction rule (conservation of charge)	$\sum I = 0$
Kirchhoff’s loop rule (conservation of energy)	$\sum V = 0$
Battery $(- \to +)$	$+\mathcal{E}$
Resistor (along reference direction)	$-IR$
Capacitor $(- \to +)$	$+\dfrac{q(t)}{C}$

Ohm’s law and power

Description	Equations
Ohm’s law	$V = IR$
Potential difference of source with internal resistance	$V_{-+} = \mathcal{E} - Ir = IR$
Current of source with internal resistance	$I = \dfrac{\mathcal{E}}{R + r}$
Power delivered to or extracted from a circuit element	$P = IV$
Power delivered to a resistor (Note: both $I$ and $V$ depend on $R$ )	$P = IV = I^{2}R = \dfrac{V^{2}}{R}$
Power output of a source	$P = I\mathcal{E} = IV + I^{2}r = I^{2}(R+r)$

R-C circuit

Description	Equations
Time constant	$\tau = RC$
Charge when charging capacitors	$\begin{aligned}q(t) &= C\mathcal{E} (1-e^{-t/RC}) \cr &= Q_{f}(1-e^{-t/RC})\end{aligned}$
Current when charging capacitors	$i(t) = \dfrac{\mathcal{E}}{R}e^{-t/RC} = I_{0}e^{-t/RC}$
Charge when discharging capacitors	$q(t) = Q_{0} e^{-t/RC}$
Current when discharging capacitors	$i(t) = -\dfrac{Q_{0}}{RC}e^{-t/RC} = I_{0}e^{-t/RC}$
Power of battery in R-C circuit	$P = i\mathcal{E} = i^{2}R + \dfrac{iq}{C}$
Total energy stored in capacitor	$U = \dfrac{1}{2}QV = \dfrac{1}{2}Q_{f}\mathcal{E}$

Magnetic Force and Motion

Quantity	Unit	Definition
Magnetic force	$\mathrm{N}$	$\begin{aligned}\vec{F} &= q\vec{v}\times\vec{B} \cr &= \lvert q \rvert v B \sin\theta\end{aligned}$
Magnetic flux through a surface	$\mathrm{Wb} \newline (\mathrm{T\cdot m^{2}})$	$\Phi_{B} = \displaystyle\int \vec{B}\cdot d\vec{A}$
Magnetic dipole moment (direction from S to N)	$\mathrm{A\cdot m^{2}} \newline \mathrm{J/T}$	$\vec{\mu} = I\vec{A}$

Magnetic interactions of charged particles

Description	Equations
Magnetic force on a charged particle	$\vec{F} = q\vec{v}\times\vec{B} = \lvert q \rvert v B \sin\theta$
Radius of a circular orbit in a magnetic field (charge where $v\perp B$ )	$R = \dfrac{mv}{\lvert q \rvert B}$
Angular speed (frequency) of circular motion	$\omega = 2\pi f = \dfrac{2\pi}{T} = \dfrac{\lvert q \rvert B}{m}$
Frequency of circular motion	$f = \dfrac{1}{T} = \dfrac{\omega}{2\pi} = \dfrac{\lvert q \rvert B}{2\pi m}$
Period of circular motion	$T = \dfrac{1}{f} = \dfrac{2\pi}{\omega} = \dfrac{2\pi m}{\lvert q \rvert B}$
Velocity selector	$v = \dfrac{E}{B}$
Thompson’s experiment	$v = \sqrt{\dfrac{2qV}{m}} \newline \dfrac{q}{m} = \dfrac{E^{2}}{2VB^{2}}$
Mass spectrometers	$m = \dfrac{\vert q \rvert B^{2}R}{E}$

Magnetic interactions of current-carrying conductor

Description	Equations
Magnetic force on a straight wire segment	$\vec{F} = I\vec{l}\times\vec{B}$
Magnetic force on an infinitesimal wire segment	$d\vec{F} = I \ d\vec{l}\times\vec{B}$
Magnetic dipole moment	$\vec{\mu} = I\vec{A}$
Magnetic torque on a current loop	$\vec{\tau} = \vec{\mu}\times\vec{B} = IAB\sin\theta$
Magnetic torque on a solenoid	$\vec{\tau} = N\vec{\mu}\times\vec{B} = NIAB\sin\theta$
Potential energy for a magnetic dipole in B field	$U = -\vec{\mu}\cdot\vec{B} = -\mu B \cos\theta$

Magnetic flux and other effects

Description	Equations
Magnetic flux through a surface	$\Phi_{B} = \displaystyle\int \vec{B}\cdot d\vec{A}$
Gauss’s law for magnetism	$\displaystyle\oint \vec{B}\cdot d\vec{A} = 0$
Electromagnetic (Lorentz) force	$\vec{F} = q(\vec{E} + \vec{v}\times\vec{B})$
Hall effect	$nq = \dfrac{-J_{x}B_{y}}{E_{z}}$

Magnetic Field

Quantity	Unit	Definition
Magnetic field	$\mathrm{T} \newline \mathrm{N/(A\cdot m)} \newline 1\mathrm{G} = 10^{-4}\mathrm{T}$	$\vec{B} = \dfrac{\mu_{0}}{4\pi}\displaystyle\int\dfrac{I d\vec{l}\times\hat{r}}{r^{2}}$

Description	Equations
Ampere’s law	$\displaystyle\oint \vec{B}\cdot d\vec{l} = \mu_{0}I$
Magnetic field of a point charge	$\vec{B} = \dfrac{\mu_{0}}{4\pi}\dfrac{q \vec{v}\times\hat{r}}{r^{2}}$
Biot-Savart law Magnetic field of infinitesimal length of wire	$d\vec{B} = \dfrac{\mu_{0}}{4\pi}\dfrac{I d\vec{l}\times\hat{r}}{r^{2}}$
Force on two $\infin$ parallel wires per unit length	$\vec{F} = q\vec{v}\times\vec{B}$ $\dfrac{F}{l} = \dfrac{\mu_{0}I_{1}I_{2}}{2\pi d}$
Force on two moving charges	$\vec{F} = I\vec{l}\times\vec{B}$ $\vec{F}_{1 \to 2} = \dfrac{\mu_{0}}{4\pi}\dfrac{q_{1}q_{2}}{r}\vec{v}_2\times\vec{v}_{1}\times\hat{r}$

Magnetic field of linear conductors

Conductor Form	Magnetic Field Magnitude
$\infin$ straight wire	$B = \dfrac{\mu_{0}I}{2\pi r}$
$\infin$ current-conducting plane	$B = \dfrac{1}{2}\mu_{0}K$

Magnetic field of circular conductors

Conductor Form	Magnetic Field Magnitude
On the axis of circular wire loop	$B_{x} = \dfrac{\mu_{0}IR^{2}}{2(x^{2}+R^{2})^{3/2}}$
On the axis of N circular wire loops	$B_{x} = \dfrac{N\mu_{0}IR^{2}}{2(x^{2}+R^{2})^{3/2}} = \dfrac{\mu_{0}\mu}{2\pi(x^{2}+R^{2})^{3/2}}$
At the center of N circular wire loops	$B_{x} = \dfrac{N\mu_{0}I}{2a}$
At the center of a circular arc	$B = \dfrac{\mu_{0}I\theta}{4\pi r}$
Inside cylindrical conductor	$B = \dfrac{\mu_{0}I}{2\pi}\dfrac{r}{R^{2}} \ \ (r<R)$
Outside cylindrical conductor	$B = \dfrac{\mu_{0}I}{2\pi r} \ \ (r>R)$
Inside $\infin$ solenoid	$B = N\mu_{0}I$
Inside finite length solenoid	$B = \dfrac{N\mu_{0}I}{l}$
Inside toroid	$B = \dfrac{N\mu_{0}I}{2\pi r}$

Changing Magnetic Field (Induction)

Quantity	Unit	Definition
Inductance	$\mathrm{H} \newline \mathrm{V \cdot s/A}$	$L = \dfrac{\Phi_{B}}{i}$

Description	Equations
Faraday’s law	$\mathcal{E} = -\dfrac{d\Phi_{B}}{dt}$
Motional emf	$\mathcal{E} = \displaystyle\oint (\vec{v}\times\vec{B})\cdot d\vec{l} \newline \mathcal{E}= vBl$
Faraday’s law for stationary integration path (Induced electric field and magnetic flux)	$\displaystyle\oint\vec{E}\cdot d\vec{l} = -\dfrac{d\Phi_{B}}{dt}$
Inductance of a solenoid	$L = \dfrac{\mu_{0}N^{2}A}{l}$
Inductance as amount of change in current associated with change in magnetic flux	$\begin{aligned}\mathcal{E} &= -L\dfrac{di}{dt} \cr \dfrac{d\Phi_{B}}{dt} &= L\dfrac{di}{dt}\end{aligned}$
Magnetic potential energy	$U = \dfrac{1}{2}LI^{2}$
Magnetic energy density	$u = \dfrac{1}{2}\dfrac{B^{2}}{\mu_{0}}$

Changing Electric Field

Description	Equations
Conduction current	$i_{C} = \dfrac{dq}{dt} = \varepsilon_{0}\dfrac{d\Phi_{E}}{dt}$
Displacement current	$i_{D} = \varepsilon_{0}\dfrac{d\Phi_{E}}{dt}$
Maxwell-Ampere’s law	$\begin{aligned}\displaystyle\oint\vec{B}\cdot d\vec{l} &= \mu_{0}(i_{C}+i_{D}) \cr &= \mu_{0}i_{C} + \mu_{0}\varepsilon_{0}\dfrac{d\Phi_{E}}{dt}\end{aligned}$
Magnetic field inside a circular capacitor	$B=\dfrac{\mu_{0}Ir}{2\pi R^{2}} \ (r<R) \newline B=\dfrac{\mu_{0}I}{2\pi R} \ (r \ge R)$
Maxwell’s Equations	$\displaystyle\oint\vec{E}\cdot d\vec{A} = \dfrac{Q}{\varepsilon_{0}} \newline \oint\vec{B}\cdot d\vec{A} = 0 \newline \oint\vec{E}\cdot d\vec{l} = -\dfrac{d\Phi_{B}}{dt} \newline \oint\vec{B}\cdot d\vec{l} = \mu_{0}\left( i_{C} + \varepsilon_{0}\dfrac{d\Phi_{E}}{dt} \right)$
Maxwell’s equation in empty free space	$\oint\vec{E}\cdot d\vec{A} = 0 \newline \oint\vec{B}\cdot d\vec{A} = 0 \newline \oint\vec{E}\cdot d\vec{l} = -\frac{d\Phi_{B}}{dt} \newline \oint\vec{B}\cdot d\vec{l} = \mu_{0}\varepsilon_{0}\frac{d\Phi_{E}}{dt}$

Alternating-Current (AC) Circuits

Quantity	Unit	Definition
Capacitive reactance	$\mathrm{\Omega}$	$X_{C} = \dfrac{1}{\omega C}$
Inductive reactance	$\mathrm{\Omega}$	$X_{L} = \omega L$
Impedance	$\mathrm{\Omega}$	$Z = \dfrac{\mathcal{E}_{\mathrm{max}}}{I}$

Description	Equations
AC source in AC circuit	$\mathcal{E} = \mathcal{E}_{\mathrm{max}}\sin(\omega t)$
Angular frequency of oscillation	$\omega = 2\pi f$
Resistor in AC circuit (i and v in phase)	$v_{R} = \mathcal{E}_{\mathrm{max}}\sin(\omega t) \newline i = I\sin(\omega t) \newline V_{R} = IR$
Capacitor in AC circuit (i leads v by 90 deg)	$v_{C} = \mathcal{E}_{\mathrm{max}}\sin(\omega t) \newline i = I\sin(\omega t + 90^{\circ}) \newline V_{C} = IX_{C} = \dfrac{I}{\omega C}$
Inductor in AC circuit (i lags v by 90 deg)	$v_{L} = \mathcal{E}_{\mathrm{max}}\sin(\omega t) \newline i = I\sin(\omega t - 90^{\circ}) \newline V_{L} = IX_{L} = I\omega L$
RC series AC circuit	$Z_{RC} = \sqrt{R^{2} + 1/(\omega C)^{2}} \newline \tan\phi = -\dfrac{V_{C}}{V_{R}} = -\dfrac{1}{\omega RC}$
RLC series AC circuit	$Z_{RLC} = \sqrt{R^{2} + (\omega L-1/\omega C)^{2}} \newline \tan\phi = \dfrac{V_{L} - V_{C}}{V_{R}} = \dfrac{\omega L - 1/\omega C}{R}$
RC filters	$V_{C} = V_{R} \newline \omega_{\mathrm{cutoff}} = \dfrac{1}{RC}$ High pass measures R Low pass measures C
RL filters	$V_{R} = V_{L} \newline \omega_{\mathrm{cutoff}} = \dfrac{R}{L}$ High pass measures L Low pass measures R
Trigonometric identities	$\sin\theta = \cos(\frac{\pi}{2} - \theta) \newline \cos\theta = \sin(\frac{\pi}{2} - \theta) \newline \sin(-\theta) = -\sin\theta \newline \cos(-\theta) = \cos\theta$

Special Relativity

Description	Equations
Lorentz factor	$\gamma = \dfrac{1}{\sqrt{1-v^{2}/c^{2}}}$
Time dilation	$\Delta t_{v} = \gamma\Delta t_{\mathrm{proper}}$
Length contraction	$l_{v} = \dfrac{l_{\mathrm{proper}}}{\gamma}$
Space-time interval	$s^{2} = (c\Delta t)^{2} - (\Delta x)^{2}$
Lorentz transformation	$x' = \gamma(x - ut) \newline y' = y \newline z' = z \newline t' = \gamma (t-ux/c^{2}) \newline v_{x}' = \dfrac{v_{x} - u}{1-uv_{x}/c^{2}}$
Relativistic inertia	$m_{v} = \gamma m$
Relativistic momentum	$p = \gamma mv$
Relativistic kinetic energy	$K = (\gamma - 1)mc^{2}$
Internal (rest) energy	$E_{\mathrm{int}} = mc^{2}$
Total energy	$E = K + E_{\mathrm{int}} \newline E^{2} = (mc^{2})^{2} + (pc)^{2}$

Appendix: List of Constants

Quantity	Value
Coulomb constant	$k = 8.99\times 10^{9} \ \mathrm{N\cdot m^{2}/C^{2}}$
Electric constant	$\varepsilon_{0} = 8.85\times 10^{-12} \ \mathrm{F/m}$
Magnetic constant	$\mu_{0} = 1.26\times 10^{-6} \ \mathrm{H/m}$
Elementary charge	$q_{e} = 1.60\times 10^{-19} \ \mathrm{C}$
Mass of electron	$m_{e} = 9.11\times 10^{-31} \ \mathrm{kg}$
Speed of light in vacuum	$c = 3.00\times 10^{8} \ \mathrm{m/s}$