CBSE Notes | Class 12th | Physics Part 1 | NCERT Based :Mutual inductance

Mutual inductance

two long co-axial solenoids each of length

l. We denote the radius of the inner solenoid S1 by r1 and the number of turns per unit length by n1. The corresponding quantities for the outer solenoid S2 are r2 and n2, respectively. Let N1 and N2 be the total number of turns of coils S1 and S2, respectively. When a current I2 is set up through S2, it in turn sets up a magnetic flux through S1. Let us denote it by ?1. The corresponding flux linkage with solenoid S1 is N1?1 ??M12I2 (6.9) M12 is called the mutual inductance of solenoid S1 with respect to solenoid S2. It is also referred to as the coefficient of mutual induction. For these simple co-axial solenoids it is possible to calculate M12. The magnetic field due to the current I2 in S2 is ?0n2I2. The resulting flux linkage

Note that we neglected the edge effects and considered the magnetic field ?0n2I2 to be uniform throughout the length and width of the solenoid S2. This is a good approximation keeping in mind that the solenoid is long, implying l >> r2. We now consider the reverse case. A current I1 is passed through the solenoid S1 and the flux linkage with coil S2 is, N2?2 = M21 I1 (6.12) M21 is called the mutual inductance of solenoid S2 with

respect to solenoid S1. The flux due to the current I1 in S1 can be assumed to be confined solely inside S1 since the solenoids are very long.

We have demonstrated this equality for long co-axial solenoids. However, the relation is far more general. Note that if the inner solenoid was much shorter than (and placed well inside) the outer solenoid, then we could still have calculated the flux linkage N1?1 because the inner solenoid is effectively immersed in a uniform magnetic field due to the outer solenoid. In this case, the calculation of M12 would be easy. However, it would be extremely difficult to calculate the flux linkage with the outer solenoid as the magnetic field due to the inner solenoid would vary across the length as well as cross section of the outer solenoid. Therefore, the calculation of M21 would also be extremely difficult in this case. The equality M12=M21 is very useful in such situations. We explained the above example with air as the medium within the solenoids. Instead, if a medium of relative permeability ?r had been present. It is also important to know that the mutual inductance of a pair of coils, solenoids, etc., depends on their separation as well as their relative orientation. Now, let us recollect Experiment 6.3 in Section 6.2. In that experiment, emf is induced in coil C1 wherever there was any change in current through coil C2. Let ?1 be the flux through coil C1 (say of N1 turns) when current in coil C2 is I2.

It shows that varying current in a coil can induce emf in a neighbouring coil. The magnitude of the induced emf depends upon the rate of change of current and mutual inductance of the two coils.