PHYS 122 Electromagnetism
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 | $c = 3.00\times 10^{8} \ \mathrm{m/s}$ |