Inductance

Prof. Dr. Thorsten Jungmann (last updated 2026-05-04)

Inductance (German: Induktivität) is a measure of the ‘inertia’ of the electric current in inductors: if the current changes, a counter-voltage arises in the inductor that slows down this change. The higher the inductance, the greater the ‘inertia’. It is a property of the system (geometry, material, field guidance).

1 Quantity symbol and unit

The quantity symbol of inductance is the letter $L$.
Its unit is the henry, named after Joseph Henry, with the unit symbol $\mathrm{H}$:

$$[L]=\mathrm{H}=\frac{\mathrm{Wb}}{\mathrm{A}}=\frac{\mathrm{V}\mathrm{s}}{\mathrm{A}}=\Omega \mathrm{s}$$

2 Practical and worked examples

2.1 Inductance of an inductor

$$L=\frac{{\mu }_0{\mu }_r{N}^{2}A}{l}$$

where:

• ${\mu }_0$ the vacuum permeability (permeability of the vacuum)
• ${\mu }_r$ the relative permeability of the inductor core (1 for an air-core inductor)
• $N$ the number of turns
• $A$ the effective area, here the area of a circle: $A=\pi \cdot r^2=\frac{\pi \cdot d^2}{4}$
• $l$ the effective length, in the case of a toroidal inductor: mean length $l_m=r_i+\frac{d}{2}$

Example of an air-core inductor: $N=200$, $A=1.0\cdot 10^{-4}{\mathrm{m}}^2$, $l=0.02\mathrm{m}$, ${\mu }_r=1$.

$$L={\mu }_0\frac{N^{2}A}{l}=4\pi \cdot 10^{-7}\frac{\mathrm{H}}{\mathrm{m}}\cdot \frac{(200{)}^2\cdot 1.0\cdot 10^{-4}{\mathrm{m}}^2}{0.02\mathrm{m}}=2.513\cdot 10^{-4}\mathrm{H}\approx 0.251\mathrm{mH}.$$

2.2 Self-induction

The greater the change in current, the greater the counter-voltage at the same rate of change (differential law of self-induction):

$$u_L=L\frac{\mathrm{d}i}{\mathrm{d}t}$$

2.3 Magnetic energy

The energy stored in the magnetic field of an inductor is:
$W=\frac{1}{2}L{I}^2$

• Example:
  $L=4.7\mathrm{mH},I=0.50\mathrm{A}\Rightarrow W=\frac{1}{2}L{I}^2\approx 0.59\mathrm{mJ}$

2.4 Reactance

Inductive reactance $X_L$ (reactance) when an inductor is operated on a sinusoidal alternating voltage:

$$X_L=\omega L=2\pi fL$$

• Example:
  $f=1.0\mathrm{kHz},L=10\mathrm{mH}\Rightarrow X_L\approx 62.8\mathrm{\Omega }$

3 Typical orders of magnitude

Typical range of inductances in technical systems: $L=10\mathrm{nH}\dots 100\mathrm{H}$

4 Example phrasings

• ‘The inductance is $4.7\mathrm{mH}$ (millihenry).’
• ‘At a given frequency, the inductive reactance is: $X_L=\omega L$’
• ‘The energy stored in the magnetic field is: $W=\frac{1}{2}L{I}^2$.’

5 Related quantities

• Relationship with alternating-current operation:
  $X_L=\omega L,\omega =2\pi f$

6 Related components

• Inductor
• Relay and contactor