**Inductance** is a physical characteristic of an inductor, which is an electrical device that produces at any time a voltage proportional to the instantaneous rate of change in current flowing through it. The symbol *L* is used for inductance in honour of the physicist Heinrich Lenz. The SI unit of inductance is the henry (H).

In a typical inductor, whose geometry and physical properties are fixed, the voltage generated is as follows:

where

*v* is the voltage generated, measured in volts

*L* is the *inductance* of the device, measured in henry.

*di/dt* is the rate of change of current, measured in ampere/second

Strictly speaking, the quantity just defined is called *self-inductance*, because the voltage is induced in the same conductor that carries the current. If the voltage is induced in another nearby conductor, the property is called *mutual inductance*, which has the symbol *M*. The above equation, with either *L* or *M* as the constant, applies to both cases.

The operation of an inductor can be understood using a simple loop of wire as an example. The current flowing through the loop of wire produces a magnetic field by Ampere's law. A change in current (*di/dt*) results in a change in this magnetic field. This changing magnetic field causes an electromotive force, that some refer to as a counter-electromotive force because it runs against the current that induces it, in the conductor under Faraday's law of induction, which results in a voltage (*v*) forming in such a direction as to oppose the change in current (see Lenz's law). The constant of proportionality *L*, which tells us for a particular device how big a voltage should be expected for a given change in current, is called the inductance.

The self-inductance *L* of a solenoid (an idealization of a coil) can be calculated from

- ,

where

*μ* is the permeability of the core, measured in henrys per metre

*N* is the number of turns

*A* is the cross sectional area of the coil, measured in square metres

*l* is the length, measured in metres

This, and the inductance of more complicated shapes, can be derived from Maxwell's equations.

## Mutual inductance

### Final expression

The mutual inductance (in SI) by circuit i on circuit j is given by the double integral

### Derivation

where

is the magnetic flux through the i^{th} surface by the electrical circuit outlined by C_{j}

*C*_{i} is the enclosing curve of S_{i}

*B* is the magnetic field vector

*A* is the vector potential

Stokes' theorem has been used.

so that the inductance is a purely geometrical quantity independent of the current in the circuits.

## Self-inductance

Self-inductance, denoted L, is a special case of mutual inductance where, in the above equation, *i* =*j*. Thus,

Physically, the self-inductance of a circuit represents the back-emf described by Faraday's law of induction.

## Usage

The flux through the ith circuit in a set is obviously given by:

so that the induced emf, , of a specific circuit, *i*, in any given set can be given directly by:

## SI electricity units

## See also

## References

Wangsness, Roald K. (1986). Electromagnetic Fields (2nd Ed.). Wiley Text Books. ISBN 0471811866.

Last updated: 09-12-2005 02:39:13