Question

Show that the mutual inductance between a pair of coils is same (M₁₂ = M₂₁).

Answer 1

When an electric current passing through a coil changes with time, an emf is induced in the neighbouring coil. This phenomenon is known as mutual induction and the emf is called mutually induced emf.

(a)

(b)

Mutual induction

Consider two coils which are placed close to each other. If an electric current i₁ is sent through coil 1, the magnetic field produced by it is also linked with coil 2. Let Φ₂₁ be the magnetic flux linked with each turn of the coil 2 of N₂ turns due to coil 1, then the total flux linked with coil 2 (N₂Φ₂₁) is proportional to the current i₁ in the coil 1.

`"N"_2Φ_21 prop "i"_1`

`"N"_2Φ_21 = "M"_21 "i"_1`   or   `"M"_12 = ("N"_2Φ_21)/"i"_1`

The constant of proportionality M₂₁ is the mutual inductance of coil 2 with respect to coil 1. It is also called as a coefficient of mutual induction. If i₁ = 1A, then M₂₁ = N₂Φ₂₁.

Therefore, the mutual inductance M₂₁ is defined as the flux linkage of coil 2 when 1A current flows through coil 1. When the current changes with time, an emf ε₂ is induced in coil 2. From Faraday’s law of electromagnetic induction, this mutually induced emf ε₂ is given by

`epsilon_2 = - ("d"("N"_2Φ_21))/"dt" = - ("d"("M"_21"i"_1))/"dt"`

`epsilon_2 = - "M"_21 "di"_1/"dt"`  or  `"M"_21 = (- epsilon_2)/("di"_1//"dt")`

The negative sign in the above equation shows that the mutually induced emf always opposes the change in current i, with respect to time. If `"di"/"dt" = 1` As^-1, then M₂₁ = -ε₂. Mutual inductance M₂₁, is also defined as the opposing emf induced in the coil 2 when the rate of change of current through the coil 1 is 1 As^-1. Similarly, if an electric current i₂ through coil 2 changes with time, then emf ε₁ is induced in coil 1. Therefore,

`"M"_12 = ("N"_1Φ_12)/"i"_2  and  "M"_12 = (- epsilon_2)/("di"_2//"dt")`

where M₁₂ is the mutual inductance of coil 1 with respect to coil 2. It can be shown that for a given pair of coils, the mutual inductance is same, i.e., M₂₁ = M₁₂ = M.

In general, the mutual induction between two coils depends on size, shape, the number of turns of the coils, their relative orientation and permeability of the medium.