Question

A wooden core (Fig. 4.36) supports two coils: coil 1 with inductance L1 and short-circuited coil 2 with active resistance R and inductance L2. The mutual inductance of the coils depends on the distance x between them according to the law L12 (x). Find the mean (averaged over time) value of the interaction force between the coils when coil 1 carries an alternating current I1 = I0 cos ωt.

Answer 1

We consider the force that a circuit 1 exerts on another closed circuit 2:-

Here magnetic field at the site of the current element due to the current I1 flowing in 1.

The integral involved will depend on the vector a that defines the separation of the (suitably chosen) centre of the coils. Let C1 and C2 be the centres of the two coil suitably defined.

Write

where is the distance of stands for the vector

The bracket defines the mutual inductance L12 . Thus noting the definition of x

The repulsive nature of the force is also consistent with Lenz’* law, assuming, of comes, that L12 decreases with x.