Question

Define the coefficient of mutual inductance. Give its unit and dimensional formula.

Answer 1

Hint A magnetic field will be generated when current flows through a conductor. This magnetic field will produce an electromotive force in the second coil that happens to be in the vicinity of the magnetic field of the first coil. This is related to the coefficient of mutual inductance.

Complete Step by step solution
Let us consider two coils ${C_1}$ and ${C_2}$ . Let the magnetic flux in the second coil be ${\phi _2}$ . Let the current in the first coil ${C_1}$ be ${I_1}$.
The magnetic flux in the coil ${C_2}$ due to the current in ${C_1}$ will be
${\phi _2} \propto {I_1}$
Introducing a constant of proportionality into the above equation gives us
${\phi _2} = M{I_1}$
Here $M$ is the coefficient of mutual inductance. This constant of proportionality will depend on the number of turns of the secondary coil, area of the secondary coil, and the medium of propagation of the magnetic field.
If the current in the coil ${C_1}$ changes with time, then the magnetic flux on the second coil ${C_2}$ will change with time and according to Faraday’s law, the induced emf in the coil ${C_2}$ will be equal to the rate of change of magnetic flux. Mathematically,
${\varepsilon _2} = - \dfrac{{d{\phi _2}}}{{dt}}$
But, we have already written an equation for the magnetic flux in the second coil. Substituting that into the above equation, we get
${\varepsilon _2} = - M\dfrac{{d{I_1}}}{{dt}}$
Therefore,
$M = - \dfrac{{{\varepsilon _2}}}{{\dfrac{{d{I_1}}}{{dt}}}}$
Thus, we can define the mutual inductance as the induced emf when the rate of change of current in the primary coil is unity.
The unit of mutual inductance is Weber or Henry.
Unit of $M = \dfrac{{volt \times \sec }}{{ampere}}$
Potential is defined as the energy per unit charge. Therefore,
Dimension of volt$ = \dfrac{{M{L^2}{T^{ - 2}}}}{{IT}}$
By simplifying, we get
Dimension of volt$ = M{L^2}{T^{ - 3}}{I^{ - 1}}$
$\therefore $dimension of $M = \dfrac{{ML{T^{ - 3}}{I^{ - 1}}{T^1}}}{I}$
Hence dimension of $M = ML{T^{ - 2}}{I^{ - 2}}$

Note
The negative sign in the equation for mutual inductance is explained using Lenz’s law. The induced electromotive force in the second coil due to the changing magnetic field in the first coil will oppose the initial changing magnetic field.