Maxwell's equations

$\Psi =N\Phi$ is the flux linkage of $N$ tightly wound turns. Holds for any closed circuit of thin wire, stationary or moving, including a bar sliding on rails. Where the current path through a moving conductor is not a definite wire (a rotating disc), compute the emf from the $\mathbf{u}\times \mathbf{B}$ integral below instead. A positive emf drives current in the sense that is right-handed about the $d\mathbf{S}$ used to compute $\Phi$.

Path fixed in space, $\mathbf{B}$ changing with time.

Conductor moving with velocity $\mathbf{u}$ through $\mathbf{B}$; only the moving segments contribute. Between the ends $a$ and $b$ of an open-circuited rod, $V_b-V_a={\int }_a^b(\mathbf{u}\times \mathbf{B})\cdot d\mathbf{l}$: the end towards which $\mathbf{u}\times \mathbf{B}$ points is positive.

For a thin-wire circuit this equals $-d\Phi /dt$ computed with both the field and the area changing; useful as a check or when only one part is asked for.

Stokes' theorem applied to the transformer emf; holds at every point. In static fields $\nabla \times \mathbf{E}=0$ (E is conservative); with a time-varying $\mathbf{B}$ it is not.

$N$-turn coil of area $A$ rotating at $\omega$ rad/s about an axis perpendicular to $\mathbf{B}$, with the coil plane perpendicular to $\mathbf{B}$ at $t=0$.

Stationary $N$-turn coil, every turn linked by the same flux of peak value ${\Phi }_m$ at frequency $f$ ($\omega =2\pi f$). The peak emf is $2\pi fN{\Phi }_m$, and the emf lags the flux by $90^{\circ }$.

Coils fixed in place in a linear magnetic medium. $\Phi$ is the flux that a coil's own current $i$ sets up through each of its $N$ turns; ${\Phi }_{21}$ is the flux produced by $i_1$ through each of the $N_2$ turns of coil 2, and $e_2$ is the emf induced in coil 2 by a changing $i_1$. The minus sign is Lenz's law. Treated as a circuit element with the passive sign convention, the voltage across an inductor is $v=Ldi/dt$.

Infinitely long straight wire carrying $I$ in a non-magnetic medium; single-turn rectangular loop in the same plane, with its two sides of length $h$ parallel to the wire at distances $d$ and $d+w$ from it. A stationary loop and a slowly varying $I$ give $|V_{\text{emf}}|=M|dI/dt|$. With a constant $I$ and the loop moving directly away at speed $u$, $|V_{\text{emf}}|=\frac{{\mu }_{0}Ihu}{2\pi }\left(\frac{1}{d}-\frac{1}{d+w}\right)$, the difference of the $BLu$ emfs of the near and far sides.

Rod of length $L$ rotating at $\omega$ in a plane perpendicular to a uniform static $\mathbf{B}$, pivoted at one end; also the emf between axle and rim of a disc of radius $L$.

$\mathbf{B}=B(t){\hat{a}}_z$ uniform inside radius $a$ and zero outside (as in a long solenoid), so by symmetry $\mathbf{E}$ is purely azimuthal. With $d\mathbf{l}$ along ${\hat{a}}_{\varphi }$ and $d\mathbf{S}$ along ${\hat{a}}_z$, $\oint \mathbf{E}\cdot d\mathbf{l}=E_{\varphi }(2\pi r)=-(\text{area of the field region enclosed})dB/dt$. The induced E exists even where $\mathbf{B}=0$.

Gives the retarding force on a sliding bar carrying induced current. For a bar moving at constant velocity through a static field (self-inductance neglected), the mechanical power $Fu$ supplied equals the electrical power $I^{2}R$ dissipated.

Always true. For steady currents $\nabla \cdot \mathbf{J}=0$, the field form of KCL.

A/m²; exists wherever $\mathbf{D}$ changes with time, including in vacuum. Phasor form in a linear medium: ${\mathbf{J}}_{ds}=j\omega \epsilon {\mathbf{E}}_s$.

$\mathbf{J}$ is the free (conduction or convection) current density; in a conductor $\mathbf{J}=\sigma \mathbf{E}$.

Plate area $A$, gap $d$, uniform field $E=v/d$ (fringing neglected); equals the conduction current in the leads.

Time-harmonic field in a medium with conductivity $\sigma$ and real permittivity $\epsilon$: ${\mathbf{J}}_c=\sigma {\mathbf{E}}_s$ and ${\mathbf{J}}_d=j\omega \epsilon {\mathbf{E}}_s$ are $90^{\circ }$ apart, with ${\mathbf{J}}_d$ leading.

Homogeneous conductor; from the continuity equation with $\mathbf{J}=\sigma \mathbf{E}$ and $\nabla \cdot \mathbf{E}={\rho }_v/\epsilon$. Excess charge inside decays in a few $\tau$ and appears on the surface.

General time-varying fields; ${\rho }_v$ and $\mathbf{J}$ are free charge and free current densities.

Stationary surfaces and paths. In the last two lines $S$ is any surface bounded by $C$, with $d\mathbf{S}$ right-handed about $d\mathbf{l}$, and $I_{\text{enc}}={\int }_S\mathbf{J}\cdot d\mathbf{S}$.

Linear, isotropic, homogeneous media. In general $\mathbf{D}={\epsilon }_0\mathbf{E}+\mathbf{P}$ and $\mathbf{B}={\mu }_0(\mathbf{H}+\mathbf{M})$.

Force on a point charge $q$ moving with velocity $\mathbf{u}$; the magnetic part does no work because it is perpendicular to $\mathbf{u}$.

Fields varying as $e^{j\omega t}$ in a linear medium. The last line assumes the only current is conduction current $\sigma {\mathbf{E}}_s$ (no impressed source current).

${\hat{a}}_n$ is the unit normal pointing from medium 2 into medium 1. Valid for static and time-varying fields.

Normal components are taken along ${\hat{a}}_n$ (from 2 into 1). Only the tangential H component perpendicular to $\mathbf{K}$ jumps, and it jumps by the magnitude of $\mathbf{K}$.

Two perfect dielectrics with ${\rho }_s=0$: tangential E and normal D are continuous, while normal E and tangential D jump.

$\theta$ measured from the normal, charge-free interface. The field makes the larger angle with the normal in the medium of higher permittivity.

Any interface between media of finite conductivity, where $\mathbf{K}=0$; $\theta$ from the normal.

Steady (DC) current crossing the interface between two media of finite conductivity: $\nabla \cdot \mathbf{J}=0$ makes normal $\mathbf{J}$ continuous, and $E_{1t}=E_{2t}$ with $\mathbf{J}=\sigma \mathbf{E}$ gives the tangential condition. $J_n$ is taken along ${\hat{a}}_n$ (from 2 into 1) and $\theta$ from the normal. The surface charge ${\rho }_s=D_{1n}-D_{2n}$ is zero only if ${\epsilon }_1/{\sigma }_1={\epsilon }_2/{\sigma }_2$.

Medium 2 is a perfect conductor ($\sigma \to \infty$), inside which time-varying $\mathbf{E},\mathbf{D},\mathbf{B},\mathbf{H}$ are all zero; ${\hat{a}}_n$ points out of the conductor. At the surface E is purely normal and H purely tangential.

Source-free, linear, homogeneous, isotropic medium; $\mathbf{H}$ obeys the same equation. Lossless ($\sigma =0$): ${\nabla }^2\mathbf{E}=\mu \epsilon {\partial }^2\mathbf{E}/\partial t^2$.

Time-harmonic fields ($e^{j\omega t}$) in the same medium; the same equation holds for ${\mathbf{H}}_s$.

$\alpha$ (Np/m) is the attenuation constant and $\beta$ (rad/m) the phase constant; take the root with $\alpha ,\beta \ge 0$, and the root of $\eta$ with a non-negative real part. $\tan 2{\theta }_{\eta }=\sigma /(\omega \epsilon )$, so ${\theta }_{\eta }=0$ in a lossless medium and $0^{\circ }<{\theta }_{\eta }<45^{\circ }$ in a lossy one (approaching $45^{\circ }$ in a good conductor); H lags E in time by ${\theta }_{\eta }$.

$\sigma =0$, so $\alpha =0$ and $\eta$ is real: E and H are in phase.

$E_0$ real. A wave travelling in $-z$ has $e^{+\gamma z}$ and $\cos (\omega t+\beta z)$; its amplitude $E_0{e}^{+\alpha z}$ decreases as it travels towards $-z$.

Wave travelling along the unit vector ${\hat{a}}_k$ in a homogeneous medium: $\mathbf{E}$, $\mathbf{H}$ and ${\hat{a}}_k$ are mutually perpendicular, and $\mathbf{E}\times \mathbf{H}$ points along ${\hat{a}}_k$. Example: $\mathbf{E}$ along ${\hat{a}}_x$, travel along ${\hat{a}}_z$ gives $\mathbf{H}$ along ${\hat{a}}_y$.

Sinusoidal steady state at a single frequency; e.g. ${\mathbf{E}}_s=E_0{e}^{-j\beta z}{e}^{j\varphi }{\hat{a}}_x$ ($E_0$ real) corresponds to $E_0\cos (\omega t-\beta z+\varphi ){\hat{a}}_x$.

Uses the real time-domain fields, never the phasors.

Fixed volume $v$ bounded by the closed surface $S$ (outward $d\mathbf{S}$), linear non-dispersive medium, and the only current inside is conduction current $\sigma \mathbf{E}$. The left side is the total power leaving through $S$; so net power flowing in = rate of rise of stored energy + ohmic loss. With an impressed source current ${\mathbf{J}}_s$ inside, the power it supplies, $-{\int }_v{\mathbf{J}}_s\cdot \mathbf{E}dv$, is added to the right side.

Same statement per unit volume; $\mathbf{J}$ is the total current density, and $\mathbf{J}\cdot \mathbf{E}=\sigma E^2$ when it is only conduction current.

Linear medium: instantaneous values, then time averages for time-harmonic fields with peak phasors. For a single uniform plane wave in a lossless medium $w_e=w_m$ at every instant, because $H=E/\eta$.

Time-harmonic fields with peak-value phasors; with rms phasors drop the $\frac{1}{2}$. Since only the real part is kept, $\mathrm{Re}[{\mathbf{E}}_s\times {\mathbf{H}}_s^{\ast }]=\mathrm{Re}[{\mathbf{E}}_s^{\ast }\times {\mathbf{H}}_s]$, but one of the two must be conjugated.

$E_0$ and $H_0=E_0/\eta$ are peak amplitudes; in free space $\eta ={\eta }_0\approx 120\pi \Omega$. For a wave along $z$ of any polarisation, replace $E_0^2$ by $|{\mathbf{E}}_s{|}^2=|E_{xs}{|}^2+|E_{ys}{|}^2$. Equivalently $S_{\text{avg}}=v_p⟨w_e+w_m⟩=v_p\cdot \frac{1}{2}\epsilon E_0^2$: the stored energy is carried along at $v_p$.

Wave $E_0{e}^{-\alpha z}\cos (\omega t-\beta z){\hat{a}}_x$ with $\eta =|\eta |\angle {\theta }_{\eta }$, so $\mathbf{H}=(E_0/|\eta |)e^{-\alpha z}\cos (\omega t-\beta z-{\theta }_{\eta }){\hat{a}}_y$. Power decays as $e^{-2\alpha z}$, twice the exponent of the field.

If the area is tilted by angle $\psi$ between its normal and ${\mathbf{S}}_{\text{avg}}$, multiply by $\cos \psi$.