Two coaxial circular coils have the same radius $R=0.10\text{m}$. The first coil has $N_1=200$ turns and the second coil has $N_2=150$ turns. They are placed one directly above the other so that their planes coincide. The mutual inductance between them is $M=0.12\text{H}$. If the current in the first coil varies as $I_1(t)=I_0\sin (200\pi t)$ A, where $I_0=5\text{A}$, (i) find the maximum induced emf in the second coil, and (ii) state the direction of the induced current at the instant when $I_1$ is increasing through zero.

Step-by-step solution

1. Write Faraday’s law for mutual induction: ${\epsilon }_2=-M\frac{d{I}_1}{dt}$.
2. Differentiate $I_1(t)$: $\frac{d{I}_1}{dt}=I_{0}200\pi \cos (200\pi t)$.
3. The magnitude of the emf is $|{\epsilon }_2|=M{I}_{0}200\pi |\cos (200\pi t)|$.
4. Maximum occurs when $|\cos |=1$, so ${\epsilon }_{2\text{max}}=M{I}_{0}200\pi =0.12\times 5\times 200\pi =120\pi \text{V}$.
5. Evaluate: $120\pi \approx 3.77\times 10^2\text{V}$.
6. For direction, use Lenz’s law: when $I_1$ is increasing (positive slope), the induced emf in coil 2 opposes this increase, therefore the induced current in coil 2 is opposite to the current direction in coil 1 at that instant.

Concept tested: mutual inductance and induced emf

Published 7 September 2026 · How this solution was prepared and checked