If, on the other hand, the current in the inner solenoid is varied, the field due to it which is non-zero only within the inner solenoid. The flux enclosed by the outer solenoid is, therefore,
If is varied, the emf in the outer solenoid is giving
One can see that .
This equality can be proved quite generally from Biot-Savart's law. Consider two circuits shown in the figure.
The field at , due to current in the loop (called the primary ) is
where . We have seen that can be expressed in terms of a vector potential , where
, by Biot-Savart's law
The flux enclosed by the second loop, (called the secondary ) is
Clearly,
It can be seen that the expression is symmetric between two loops. Hence we would get an identical expression for . This expression is, however, of no significant use in obtaining the mutual inductance because of rather difficult double integral.
Thus a knowledge of mutual inductance enables us to determine, how large should be the change in the current (or voltage) in a primary circuit to obtain a desired value of current (or voltage) in the secondary circuit. Since , we represent mutual inductance by the symbol . The emf in the secondary circuit is given by , where is the variable current in the primary circuit.
Units of is that of Volt-sec/Ampere which is known as Henry (h)