The whole Electromagnetics formula sheet, to print

Force on $Q_2$ due to $Q_1$; ${\hat{a}}_{12}$ is the unit vector from $Q_1$ toward $Q_2$ and $R=|{\mathbf{r}}_2-{\mathbf{r}}_1|$. A positive product $Q_1{Q}_2$ means repulsion. Valid for point charges (or bodies small compared with $R$). In a homogeneous dielectric filling all space replace ${\epsilon }_0$ by $\epsilon$: the force falls by the factor ${\epsilon }_r$.

Field at distance $R$ from a point charge $Q$: away from a positive charge, toward a negative one. A charge $q$ placed in the field feels $\mathbf{F}=q\mathbf{E}$. Units: V/m, the same as N/C.

$\rho$ is the perpendicular distance from the line and ${\hat{a}}_{\rho }$ points perpendicularly away from it (for ${\rho }_L>0$). For a line not on the $z$-axis, use the perpendicular vector from the line to the field point.

Ring of radius $a$ in the plane $z=0$, centred on the origin, total charge $Q=2\pi a{\rho }_L$. Zero at the centre; $|\mathbf{E}|$ is maximum at $z=\pm a/\sqrt{2}$, where it equals $Q/(6\sqrt{3}\pi {\epsilon }_0{a}^2)$; for $|z|\gg a$ it looks like a point charge.

Disc of radius $a$ in the plane $z=0$. Below the disc ($z<0$) the field has the same magnitude with $|z|$ in place of $z$ and points along $-{\hat{a}}_z$. As $a\to \infty$ it becomes the infinite sheet.

${\hat{a}}_n$ is the unit normal pointing away from the sheet, on each side. The magnitude does not depend on the distance from the sheet.

Radius $a$, total charge $Q$ (for the solid sphere $Q=\frac{4}{3}\pi a^3{\rho }_v$). The shell is a thin surface carrying a uniform ${\rho }_S$; an isolated charged conducting sphere, solid or hollow, gives the same field because its charge spreads uniformly over its surface. A shell whose charge is not uniform can have a field inside. Outside, any spherically symmetric charge acts as a point charge at the centre. Inside the solid sphere $E$ grows linearly with $r$ and peaks at the surface.

Units C/m². For a point charge $\mathbf{D}=\frac{Q}{4\pi r^2}{\hat{a}}_r$, whatever the homogeneous medium around it.

$S$ is any closed surface, $d\mathbf{S}$ points outward and $Q_{\text{enc}}$ is the total free charge inside (the last integral is written for a volume charge; enclosed point, line and surface charges count too). Always true, but it yields $\mathbf{D}$ only when symmetry makes $\mathbf{D}$ normal to the surface with constant magnitude (or tangential) on every part of $S$.

Follows from the integral form and the divergence theorem. To get the charge density from a given field, differentiate; do not integrate.

The factors $\rho$, $r^2$ and $\sin \theta$ go INSIDE the derivative. A purely radial field with $D_r\propto 1/r^2$ has zero divergence everywhere except $r=0$.

Work done per unit charge by an external agent in moving a positive test charge from $B$ to $A$. Integrate along any convenient path; the lower limit is the reference point $B$.

Work done by an external agent moving $Q$ slowly (no change in kinetic energy) from $A$ to $B$; the field itself does $-W_{\text{ext}}$. Positive when a positive charge is moved toward higher potential; zero along an equipotential and around any closed path.

Reference at infinity. $V$ is a scalar, so add the contributions with their signs and no components; this is often easier than integrating $\mathbf{E}$ directly, and $-\nabla V$ then gives the field.

The minus sign makes $\mathbf{E}$ point toward decreasing $V$. $|\nabla V|$ is the largest rate of change of $V$ with distance at that point.

${\rho }_A$ and ${\rho }_B$ are the perpendicular distances of $A$ and $B$ from the line. The reference cannot be at infinity (the logarithm diverges); only differences are meaningful.

Radius $a$, total charge $Q$, reference at infinity; the shell result also holds for an isolated charged conducting sphere. Inside the shell $V$ is constant and equal to its surface value, although $\mathbf{E}=0$ there. At the centre of the solid sphere $V=3Q/(8\pi {\epsilon }_{0}a)$, 1.5 times its surface value. Stored energy: $Q^2/(8\pi {\epsilon }_{0}a)$ for the shell or conducting sphere, $3Q^2/(20\pi {\epsilon }_{0}a)$ for the uniformly charged solid sphere.

Charges $+Q$ and $-Q$ a distance $d$ apart, centred on the origin along the $z$-axis; the dipole moment $\mathbf{p}=Qd{\hat{a}}_z$ points from $-Q$ to $+Q$, and $\theta$ is measured from $+z$. Valid for $r\gg d$. $V$ falls as $1/r^2$ and $E$ as $1/r^3$; $V=0$ on the whole plane $\theta =90^{\circ }$. In a uniform external field ${\mathbf{E}}_0$ a dipole feels no net force, but a torque $\mathbf{T}=\mathbf{p}\times {\mathbf{E}}_0$, and has potential energy $U=-\mathbf{p}\cdot {\mathbf{E}}_0$, lowest when $\mathbf{p}$ is aligned with ${\mathbf{E}}_0$.

In the sum, $V_k$ is the potential at $Q_k$ due to all the OTHER charges; for two point charges $W_E=\frac{Q_1{Q}_2}{4\pi {\epsilon }_{0}R}$. The first integral runs over the charge, the second over all space where the field exists. Applied to point charges, the integral forms would include each charge's infinite self-energy, which the sum leaves out. $w_E$ is the energy density, to be integrated over the whole region where the field exists.

From $\nabla \cdot (\epsilon \mathbf{E})={\rho }_v$ and $\mathbf{E}=-\nabla V$ in a homogeneous medium. Cartesian: ${\nabla }^{2}V=\frac{{\partial }^{2}V}{\partial x^2}+\frac{{\partial }^{2}V}{\partial y^2}+\frac{{\partial }^{2}V}{\partial z^2}$. When $V$ depends only on the radius: $\frac{1}{\rho }\frac{d}{d\rho }\left(\rho \frac{dV}{d\rho }\right)$ (cylindrical) and $\frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{dV}{dr}\right)$ (spherical). When $V$ depends on one coordinate only, Laplace's equation integrates directly: $V=Ax+B$ (planar), $V=A\ln \rho +B$ (cylindrical), $V=A/r+B$ (spherical), with $A$ and $B$ fixed by the two boundary potentials. So between parallel plates with one homogeneous, charge-free dielectric in the gap (fringing neglected), $V$ is linear in distance.

${\chi }_e$ is the electric susceptibility (zero in vacuum). The last two relations hold for linear isotropic media; $\epsilon ={\epsilon }_0{\epsilon }_r$ is the permittivity.

${\hat{a}}_n$ is the outward normal of the dielectric's surface. The total bound charge of a dielectric body is zero. Gauss's law in terms of total charge: $\nabla \cdot ({\epsilon }_0\mathbf{E})={\rho }_v+{\rho }_{pv}$.

${\hat{a}}_n$ is the unit normal pointing from region 2 into region 1, and ${\rho }_S$ is the FREE surface charge on the interface. Between two perfect dielectrics ${\rho }_S=0$, so $D_n$ is continuous and ${\epsilon }_1{E}_{n1}={\epsilon }_2{E}_{n2}$. Tangential $\mathbf{E}$ is always continuous; tangential $\mathbf{D}$ is not (unless ${\epsilon }_1={\epsilon }_2$): $D_{t1}/{\epsilon }_1=D_{t2}/{\epsilon }_2$.

Angles measured from the NORMAL to the interface; valid when ${\rho }_S=0$. The field line lies farther from the normal in the medium with the larger permittivity.

Equal to $\frac{1}{2}\int \epsilon E^{2}dv$ over the region between the conductors. Capacitances in parallel add; in series their reciprocals add.

Plate area $A$, gap $d$, one homogeneous dielectric; uniform field, fringing neglected.

Layers parallel to the plates act as capacitors in series. $D$ is the same in every layer, so the field is larger in the layer of lower permittivity. For any number of layers, $C=A/{\sum }_k(d_k/{\epsilon }_k)$. Special cases in an air gap $d$: a dielectric slab of thickness $t$ and relative permittivity ${\epsilon }_r$, parallel to the plates, gives $C={\epsilon }_{0}A/(d-t+t/{\epsilon }_r)$; an uncharged metal slab of thickness $t$ gives $C={\epsilon }_{0}A/(d-t)$. Neither depends on where the slab sits in the gap.

Each dielectric fills the whole gap $d$ over part of the plate area ($A_1+A_2=A$): capacitors in parallel. $E=V/d$ is the same in both; $D$ differs.

Inner radius $a$, outer radius $b$, length $L$. The field is largest at the surface of the inner conductor, $\rho =a$.

Concentric spheres of radii $a<b$; letting $b\to \infty$ gives the isolated sphere. With two layers (${\epsilon }_1$ for $a<r<c$, ${\epsilon }_2$ for $c<r<b$): $C=4\pi /\left[\frac{1}{{\epsilon }_1}\left(\frac{1}{a}-\frac{1}{c}\right)+\frac{1}{{\epsilon }_2}\left(\frac{1}{c}-\frac{1}{b}\right)\right]$.

In F/m. Wire radius $a$, centre-to-centre spacing $d$, height $h$ of the wire axis above a grounded plane; the approximations need $d\gg a$ and $h\gg a$. The wire-plane result follows from the image method: the plane lies midway between the wire and its image, at half the voltage, so the capacitance doubles.

${\rho }_S={\epsilon }_0{E}_z$ at $z=0^{+}$. Its magnitude is largest, $Q/(2\pi h^2)$, directly below the charge, and it integrates to $-Q$ over the whole plane. The force on $Q$ is the Coulomb attraction of its image, a distance $2h$ away.

Linear, isotropic medium. $\mathbf{M}$ is the magnetization (magnetic dipole moment per unit volume, A/m) and ${\chi }_m$ the magnetic susceptibility. In a ferromagnetic material $\mu$ is not a constant: it depends on $H$ and on the magnetic history (see the B-H curve below).

$d\mathbf{S}$ is normal to the surface. For a uniform $B$ normal to a flat area $A$, $\Phi =BA$.

Steady current. $\mathbf{R}$ points FROM the current element TO the field point. For a surface current density $\mathbf{K}$ (A/m) replace $Id\mathbf{l}$ by $\mathbf{K}dS$; for a volume current density $\mathbf{J}$ (A/m²) by $\mathbf{J}dv$. A point on the line of a straight segment (or its extension) gets no field from that segment, since $d\mathbf{l}\times {\hat{a}}_R=0$.

$\rho$ is the perpendicular distance from the point to the line of the wire. ${\alpha }_1,{\alpha }_2$ are the angles, at the point, between that perpendicular and the lines to the two ends, when the ends lie on opposite sides of the foot of the perpendicular. If both ends are on the same side, use $\sin {\alpha }_2-\sin {\alpha }_1$ with ${\alpha }_2$ the larger angle. Direction by the right-hand rule.

Infinite wire: ${\alpha }_1={\alpha }_2=90^{\circ }$. Semi-infinite wire with the point on the perpendicular through its finite end: ${\alpha }_1=0$, ${\alpha }_2=90^{\circ }$.

Loop of radius $a$ in the $xy$-plane centred at the origin, current counter-clockwise seen from $+z$; $z$ is the distance along the axis. Far away ($z\gg a$), $H\approx I{a}^2/(2z^3)$, the on-axis field of a magnetic dipole.

Arc of radius $a$ subtending angle $\theta$ at its centre (radial leads to the centre add nothing). Rectangle of sides $s_1$ and $s_2$: four finite wires, all giving fields in the same direction at the centre; each side of length $s_1$ is at $\rho =s_2/2$ with $\sin {\alpha }_1=\sin {\alpha }_2=s_1/\sqrt{s_1^2+s_2^2}$, and likewise for the other pair. Square of side $s$ ($s_1=s_2=s$): each side has $\rho =s/2$ and ${\alpha }_1={\alpha }_2=45^{\circ }$.

$n=N/l$ turns per metre, radius $a$, point on the axis inside the solenoid at axial distances $d_1$ and $d_2$ from the two ends (for a point on the axis beyond one end, subtract the near-end term from the far-end term). A long, closely wound solenoid has a uniform $H=nI$ inside and $H\approx 0$ outside.

$N$ closely wound turns; $\rho$ is the distance from the toroid's axis. $H$ does not depend on the core material; $B=\mu NI/(2\pi \rho )$ does.

Uniform surface current density $\mathbf{K}$ (A/m); ${\hat{a}}_n$ is the unit normal from the sheet towards the field point. $|\mathbf{H}|=K/2$ at any distance. Two parallel sheets carrying $+\mathbf{K}$ and $-\mathbf{K}$: $H=K$ between them and $0$ outside.

Steady currents. $S$ is any surface bounded by the closed path $C$; the positive sense of $I_{\text{enc}}$ follows the right-hand rule on the direction of travel round $C$. For $N$ turns threading the path, $I_{\text{enc}}=NI$. Always true for steady currents, but it yields $H$ directly only when symmetry lets a path be chosen along which $H$ is tangential and constant in magnitude, and elsewhere perpendicular to the path or zero: infinite wire, round conductor, coaxial cable, current sheet, solenoid, toroid.

Magnetostatics only (time-varying fields add the displacement current density $\partial \mathbf{D}/\partial t$). In a current-free region $\nabla \times \mathbf{H}=0$.

No isolated magnetic poles: flux lines close on themselves and the net flux out of any closed surface is zero. Holds for time-varying fields too. It is the flux-junction rule of magnetic circuits.

The curl is circulation per unit area: $(\nabla \times \mathbf{H})\cdot {\hat{a}}_n={\lim }_{\Delta S\to 0}\oint \mathbf{H}\cdot d\mathbf{l}/\Delta S$, with the small path and ${\hat{a}}_n$ related by the right-hand rule.

Links the two forms of Ampere's law. $S$ is any open surface bounded by $C$, with $d\mathbf{S}$ oriented by the right-hand rule on the direction of travel round $C$.

Radius $a$, current $I$ spread uniformly ($J=I/\pi a^2$, DC). $H$ rises linearly from zero on the axis to its maximum $I/(2\pi a)$ at the surface, then falls as $1/\rho$. Inside a hollow tube whose current flows only in the wall, $H=0$.

The energy lost per cycle per unit volume equals the area enclosed by the B-H loop; multiply by the frequency $f$ and the core volume $V$ for the power.

$B_m$ = peak flux density, $t$ = lamination thickness, $k_h$ and $k_e$ material constants, $x$ an empirical exponent that depends on the material (1.6 is the classical value). At a fixed $B_m$, hysteresis loss is proportional to $f$ and eddy-current loss to $f^2$. Building the core from thin insulated laminations cuts eddy loss through the $t^2$ factor; it does not reduce hysteresis loss.

$q$ carries its sign (an electron has $q=-e=-1.602\times 10^{-19}$ C). The magnetic part $q\mathbf{u}\times \mathbf{B}$ is perpendicular to the velocity, so it does no work: it changes the direction of motion, never the speed or the kinetic energy.

$u_{\perp }$ and $u_{\parallel }$ are the velocity components perpendicular and parallel to $\mathbf{B}$; with $\mathbf{E}=0$ the path is a circle (if $u_{\parallel }=0$) or a helix. The period and angular frequency do not depend on the speed (non-relativistic motion). A charge accelerated from rest through a potential difference $V$ has $u=\sqrt{2|q|V/m}$, so entering perpendicular to $\mathbf{B}$ it moves on a circle of radius $r=\sqrt{2mV/|q|}/B$. In crossed $\mathbf{E}$ and $\mathbf{B}$ (both perpendicular to $\mathbf{u}$ and to each other, forces opposed), a charge passes undeflected when $u=E/B$.

$\mathbf{L}$ points along the current and $\theta$ is the angle between the conductor and $\mathbf{B}$; the second and third forms need a straight conductor in a uniform $\mathbf{B}$. A closed loop of any shape in a UNIFORM field feels zero net force (it can still feel a torque).

A slab carries a current $I$ perpendicular to a uniform $\mathbf{B}$; $t$ is the slab's thickness measured along $\mathbf{B}$, $n$ the number density of the charge carriers and $q$ their charge (one type of carrier). The magnetic force pushes the carriers, drifting at speed $u_d$, towards one side until the transverse Hall field $E_H$ balances it. $V_H$ does not depend on the slab's width. The Hall coefficient $R_H$ (m³/C) is negative for electrons and positive for holes, so the polarity of $V_H$ identifies the carrier type; because $V_H\propto 1/n$, it is far larger in a semiconductor than in a metal.

Infinitely long parallel wires with separation $d$, in air. Currents in the same direction attract; in opposite directions they repel. It follows from ${\mathbf{F}}_2=I_2\mathbf{L}\times {\mathbf{B}}_1$ with $B_1={\mu }_0{I}_1/(2\pi d)$.

Planar loop of area $S$ with $N$ turns carrying $I$; ${\hat{a}}_n$ is the normal given by the right-hand rule on the current; $\mathbf{m}$ in A·m². Uniform $\mathbf{B}$; $\theta$ is the angle between ${\hat{a}}_n$ and $\mathbf{B}$, so the torque is largest when the PLANE of the loop is parallel to $\mathbf{B}$. Stable equilibrium (minimum $U$, zero torque) is $\mathbf{m}$ parallel to $\mathbf{B}$.

Small current loop at the origin with $\mathbf{m}=m{\hat{a}}_z$, spherical coordinates, $r$ much larger than the loop. On the axis ($\theta =0$), $B={\mu }_{0}m/(2\pi r^3)$, which matches the loop-axis formula for $z\gg a$.

Charges $+Q$ and $-Q$ separated by $d$, with $\mathbf{p}=Q\mathbf{d}$ directed from $-Q$ to $+Q$ along ${\hat{a}}_z$; $r\gg d$; ${\epsilon }_0=8.854\times 10^{-12}$ F/m. In a uniform $\mathbf{E}$: torque $\mathbf{T}=\mathbf{p}\times \mathbf{E}$, energy $U=-\mathbf{p}\cdot \mathbf{E}$.

Always, from $\nabla \cdot \mathbf{B}=0$. The normal component of $\mathbf{H}$ is therefore discontinuous whenever ${\mu }_1\ne {\mu }_2$.

${\hat{a}}_{n12}$ points from medium 2 into medium 1. The tangential $\mathbf{H}$ jumps by $|\mathbf{K}|$ in the tangential component perpendicular to $\mathbf{K}$. With no surface current the tangential $\mathbf{H}$ is continuous and the tangential $\mathbf{B}$ changes in the ratio of the permeabilities.

Linear isotropic media, no surface current; ${\theta }_1,{\theta }_2$ are measured from the NORMAL. Going from air into iron (${\mu }_2\gg {\mu }_1$) the lines bend towards the surface inside the iron; coming out of iron into air they leave almost normal to the iron surface, unless they run exactly along it inside.

$l$ = mean length of the flux path in that part, $A$ = its cross-sectional area, $\mathcal{P}$ = permeance (H). Valid while the material is linear, $B$ is uniform over the cross-section and leakage flux is neglected.

The loop law is Ampere's circuital law; the junction law is $\oint \mathbf{B}\cdot d\mathbf{S}=0$. Parts in series carry one flux and $\mathcal{R}={\mathcal{R}}_1+{\mathcal{R}}_2+\dots$. Parallel limbs between the same two junctions have equal mmf drops; their combined reluctance follows $1/\mathcal{R}=1/{\mathcal{R}}_1+1/{\mathcal{R}}_2$, and the flux divides between them in inverse ratio of their reluctances.

Air gap of length $l_g$. With fringing neglected, $A_g$ equals the core area. Fringing spreads the flux over a larger area, lowering the gap's reluctance and its flux density; the $(a+l_g)(b+l_g)$ rule is an approximate correction for a short gap between rectangular faces. The gap's reluctance exceeds that of an iron path of length $l_c$ and the same area whenever $l_g>l_c/{\mu }_r$, which for a core with a large ${\mu }_r$ is a very short gap.

Linear core, every turn linked by the same flux. $L\propto N^2$. When the gap dominates, ${\mathcal{R}}_{\text{total}}\approx {\mathcal{R}}_g$ and $L\approx {\mu }_0{N}^2{A}_g/l_g$, nearly independent of the core's (variable) permeability, which is one reason inductor cores are gapped. Two coils of $N_1$ and $N_2$ turns on the same core with no leakage: $L_1=N_1^2/\mathcal{R}$, $L_2=N_2^2/\mathcal{R}$ and $M=N_1{N}_2/\mathcal{R}$, so $M=\sqrt{L_1{L}_2}$ and $k=1$.

Linear circuit. At the same $B$, the energy density $B^2/(2{\mu }_0)$ in air is ${\mu }_r$ times that in iron, so the gap holds much of the energy. The pull on a pole face of area $A$ across a gap with uniform $B$ (no fringing) follows from virtual work at constant flux; a U-shaped magnet lifting a bar across two gaps exerts twice this.

Linear medium. For a nonlinear core the incremental inductance $d\lambda /di$ depends on the operating point.

${\Phi }_{21}$ is the flux produced by coil 1 that links each turn of coil 2. Reciprocity ($M_{12}=M_{21}$) holds in any linear medium, so compute $M$ from whichever coil's field is easier to find. The double line integral is the general form for two thin single-turn loops in a homogeneous non-magnetic medium, $R$ being the distance between the two elements.

Plus sign when the two fluxes aid (the current enters both dotted ends), minus when they oppose; hence $M=(L_{\text{aid}}-L_{\text{opp}})/4$. $k=1$ only if every flux line of each coil links every turn of the other. The polarity of the induced voltage $v_2$ follows the dot convention.

Linear medium. For two coupled coils, $W_m=\frac{1}{2}{L}_1{I}_1^2+\frac{1}{2}{L}_2{I}_2^2\pm M{I}_1{I}_2$, plus when the fluxes aid.

Length $l$ much greater than the radius, uniform core of permeability $\mu$ filling it, $A$ = cross-sectional area, $n=N/l$. End effects make the $L$ of a real, finite solenoid somewhat smaller.

Rectangular cross-section of height $h$, inner radius $a$ and outer radius $b$, from integrating $B=\mu NI/(2\pi \rho )$ across the section. Thin toroid: mean radius $R$ much larger than the section's dimensions, $A$ = section area; for a circular section of radius $r$, $L\approx \mu N^2{r}^2/(2R)$.

Inner conductor radius $a$, outer conductor inner radius $b$. External part: flux in the dielectric of permeability $\mu$. Internal part: solid inner conductor of permeability ${\mu }_c$ carrying a uniform (DC) current, found by the energy route. The last form is for non-magnetic conductors and dielectric with a thin outer conductor. At high frequency the current crowds to the surface (skin effect) and $L_{\text{int}}^{\prime }\to 0$.

Two parallel wires of radius $a$ with centre-to-centre spacing $d$, carrying equal and opposite currents. The second form ($d\gg a$) adds the internal inductance ${\mu }_0/(8\pi )$ of EACH wire (non-magnetic, uniform current); $0.7788a=a{e}^{-1/4}$ is the geometric mean radius of a solid round wire.

First: a coil of $N_2$ turns coaxial with a long air-cored solenoid of $n_1$ turns/m and placed well inside its length, away from the ends; $A_{\text{inner}}$ is the cross-section of whichever winding is inside, because the solenoid's field is confined to its own cross-section. Second: an infinitely long straight wire and a coplanar rectangular loop with sides $h$ parallel and $w$ perpendicular to the wire, the nearer side at distance $d$, in air.

$\Psi =N\Phi$ 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.

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 $v=Ldi/dt$.

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 $L$.

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: ${\mathbf{J}}_{ds}=j\omega \epsilon {\mathbf{E}}_s$.