# Mutual Inductance

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## mutual inductance

[′myü·chə·wəl in′dək·təns]## Mutual Inductance

a quantity that characterizes the magnetic coupling between two or more electric circuits. If there are two conducting circuits, (1) and (2), some of the lines of magnetic induction generated by the current flowing through the first circuit will penetrate the area within the second circuit (see Figure 1)—that is, they will be coupled with the second circuit. The magnetic flux Φ_{12} through circuit (2) generated by the current *I* in circuit (1) is directly proportional to the current:

Φ_{12} = *M*_{12}*I*_{1}

The proportionality coefficient *M*_{2} depends on the dimensions and form of circuits (1) and (2), on the distance between the circuits, on their relative positions, and on the magnetic permeability of the surrounding medium. This coefficient is called mutual inductance or the coefficient of mutual induction. In the SI system of units, mutual inductance is measured in henrys.

If current *I*_{2} flows through circuit (2), the magnetic flux Φ_{12} through the area of circuit (1) is also proportional to the current:

Φ_{21} = *M*_{21}*I*_{2}

where *M*_{21} = *M*_{12}.

The existence of a magnetic coupling between the circuits is manifested by the fact that, upon a change of current in one circuit, an induced electromotive force appears in the adjacent circuit. According to the law of electromagnetic induction,

where *E*_{2} and *E*_{1} are induced electromotive forces arising in circuits (2) and (1) and ΔΦ_{]2} and ΔΦ_{21} are the change in magnetic fluxes through the respective circuits over time Δ*t*.

The mutual energy *W*_{12} of the magnetic field is expressed with the aid of mutual inductance for the currents *I*_{1} and *I*_{2} as follows:

*W*_{l2} = ±*M*_{12}*I*_{1}*I*_{2}

The sign in (4) depends on the direction of the currents.

### REFERENCE

Kalashnikov, S.G.*Elektrichestvo*. Moscow, 1970. Chapter 10. (

*Obshchiikurs fiziki*, vol. 2.)

G. IA. MIAKISHEV