Inductance

Inductance is the ratio of magnetic flux linkage in a coil to the current that produces it:

$L=\frac{\Lambda }{I}\text{(H = Wb/A)}.$

A 1-henry inductor produces 1 Wb of flux linkage per amp of current.

Inductance is a geometric / material property, like Capacitance. It depends on:

• Number of turns
• Shape and size of the coil
• Permeability of the magnetic core

Not on $I$ or $\Lambda$ separately.

Self-inductance examples

Long solenoid, $N$ turns, length $l$, cross-section $S$, core permeability $\mu$:

$L=\frac{\mu N^{2}S}{l}.$

Two key features:

• Quadratic in $N$: doubling turns gives 4× inductance. ($N$ shows up once in the field $B\propto NI/l$ and once in the flux linkage $\Lambda =N\Phi$.)
• Linear in $\mu$: a ferromagnetic core (${\mu }_r\gg 1$) multiplies inductance dramatically. This is why inductors and transformers use iron cores.

Coaxial cable, inner radius $a$, outer $b$, length $l$:

$L=\frac{\mu l}{2\pi }\ln \frac{b}{a},L^{\prime }=\frac{L}{l}=\frac{\mu }{2\pi }\ln \frac{b}{a}\text{(H/m)}.$

The per-length inductance $L^{\prime }$ is one of the four parameters of a TEM Transmission line ($R^{\prime },L^{\prime },G^{\prime },C^{\prime }$).

Mutual inductance

Mutual inductance $M_{12}=N_2{\Phi }_{12}/I_1$ is the corresponding cross-coil quantity: flux linkage in coil 2 per ampere driven through coil 1. It satisfies reciprocity ($M_{12}=M_{21}$) and is bounded by $|M|\le \sqrt{L_1{L}_2}$. The dimensionless ratio $k=M/\sqrt{L_1{L}_2}$ — the coupling coefficient — is what distinguishes a tightly-coupled transformer ($k\approx 1$) from loose magnetic coupling on a PCB ($k\ll 1$). See Mutual inductance for the foundational role in transformers, wireless power, and crosstalk.

Why it shows up in circuits

The terminal voltage across an ideal inductor:

$v=L\frac{di}{dt}.$

Derivation: by Faraday’s law, the induced emf around the inductor’s loop is $-d\Lambda /dt=-Ldi/dt$. By Kirchhoff’s voltage law applied to the external circuit, the terminal voltage drives the inductor against this back-emf, so $v_{\text{terminal}}=+Ldi/dt$.

Inductors resist changes in current. At DC steady state ($di/dt=0$), an inductor looks like a wire — no voltage drop. At high frequencies, $|j\omega L|$ becomes large, and the inductor approaches an open circuit.

Energy stored

Building up current from 0 to $I$ in time $t$ requires energy:

$W_m={\int }_0^{t}vidt={\int }_0^{t}L\frac{di}{dt}idt={\int }_0^{I}Lidi=\frac{1}{2}L{I}^2\text{(J)}.$

This energy is stored in the Magnetic field the inductor produces, with density $w_m=\frac{1}{2}\mu H^2$. See Magnetic energy.

The factor of $\frac{1}{2}$ has the same origin as in the Electrostatic energy formula: the field builds gradually, so the average voltage during build-up is half the final voltage.

Compared with capacitance

Two columns of dual relationships:

Property	Capacitor	Inductor
Storage	Charge / Voltage	Flux / Current
Defining	$C=Q/V$	$L=\Lambda /I$
Energy	$\frac{1}{2}C{V}^2$	$\frac{1}{2}L{I}^2$
Energy density	$\frac{1}{2}ϵE^2$	$\frac{1}{2}\mu H^2$
$v$-$i$ relation	$i=Cdv/dt$	$v=Ldi/dt$
Resists change in	Voltage	Current
DC steady state	Open circuit	Short circuit
Material parameter	$ϵ$ (permittivity)	$\mu$ (permeability)

The duality runs deep: every theorem about capacitors has an inductor analog, and vice versa.