Magnetostatics, magnetic circuits and inductance

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.

For the common case $\mathbf{H}=H_{\varphi }(\rho ){\hat{a}}_{\varphi }$ (the field round a straight current) this reduces to $J_z=\frac{1}{\rho }\frac{d(\rho H_{\varphi })}{d\rho }$.

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 $C$.

Homogeneous medium of permeability ${\mu }_0$ (use $\mu$ otherwise), with the choice $\nabla \cdot \mathbf{A}=0$. $\mathbf{A}$ is in Wb/m. Each current element contributes to $\mathbf{A}$ in its own direction, so the field of a straight current has $\mathbf{A}$ parallel to that current. The flux through any surface equals the circulation of $\mathbf{A}$ round its edge.

$V_m$ is in amperes. It exists only in a current-free region, because $\nabla \times (\nabla V_m)=0$ whereas $\nabla \times \mathbf{H}=\mathbf{J}$. It is not single-valued round a current: each complete circuit of a path enclosing a current $I$ changes $V_m$ by $I$ in magnitude.

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, $H=0$.

For a ferromagnetic material $\mu$ depends on the operating point: it rises from its initial value to a maximum near the knee of the curve and then falls towards ${\mu }_0$ in deep saturation. A constant ${\mu }_r$ in a problem means the core is being treated as linear (unsaturated).

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.