Mutual induction: When an electric current passing through a coil changes with time, an cmf is induced in the neighbouring coil. This phenomenon is known as mutual induction and the cmf is called mutually induced emf.
Consider two coils which are placed close to each other. If an electric current `i_(1)` is sent through coil 1, the magnetic field produced by it is also linked with coil 2.
Let `Phi_(21)` be the magnetic flux linked with each turn of the coil 2 of `N_(2)` turns due to coil 1, then the total flux linked with coil 2 `(N_(2) Phi_(21))` is proportional to the current `i_(1)` in the coil 1.
`N_(21) Phi_(21) propto i_(1)`
`N_(2)Phi_(21) = M_(21) i_(1)` or `M_(21) = (N_(2)Phi_(21))/i_(1)`
The constant of proportionality `M_(21)` is the mutual inductance of the coil 2 with respect to coil 1. It is also called as coefficient of mutual induction. If `i_(1) =1`, then `M_(21) = N_(2)Phi_(21)`. Therefore, the mutual inductance `M_(21)` is defined as the flux linkage of the coil 2 when 1 A current flows through coil 1.
When the current `i_(1)` changes with time, an emf `epsilon_(2)` is induced in coil 2. From Faraday.s law of electromagnetic induction, this mutually induced emf `epsilon_(2)` is given by
`epsilon_(2) = -(d(N_(2)Phi_(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 emf always opposes the change in currrent `i_(1)` with respect to time. If `(di_(1))/(dt) = 1A s^(-1)`, then `M_(21) = -epsilon_(2)`
Mutual inductance `M_(21)` is also defined as the opposing emf induced in the coil 2 when the rate 1 of change of current through the coil 1 is `1 As^(-1)`.
Similary, if an electric current `i_(2)` through coil 2 changes with time, then emfs `epsilon_(1)` is induced coil 1. Therefore,
`M_(12) = (N_(1)Phi_(12))/i_(2)` and `M_(12) = -epsilon_(1)/(di_(2)//dt)`
Where `M_(12)` is the mutual inductance of the coil 1 with respect to coil 2. It can be shown that a given pair of coils, the mutual inductance is same. i.e., `M_(21) = M_(12)` = 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.
