Two coaxial circular coils each have a radius of $0.05,\text{m}$. Coil 1 has $180$ turns and Coil 2 has $120$ turns. The mutual inductance between them is $0.08,\text{H}$. If the current in Coil 1 varies as $I_1(t)=4sin(500\pi t),\text{A}$, (i) find the maximum magnitude of the induced emf in Coil 2, and (ii) state the direction of the induced current when $I_1$ is increasing through zero.

Step-by-step solution

1. Step 1: Recall Faraday’s law for mutual induction: ${\mathcal{E}}_2=-M,\frac{d{I}_1}{dt}$.
2. Step 2: Differentiate $I_1(t)$: $\frac{d{I}_1}{dt}=4\cdot 500picos(500\pi t)=2000picos(500\pi t),{\text{A s}}^{-1}$.
3. Step 3: Find the maximum magnitude of $\frac{d{I}_1}{dt}$: $2000\pi {\text{A s}}^{-1}$.
4. Step 4: Compute ${\mathcal{E}}_2^{\text{max}}=M\times 2000\pi =0.08\times 2000\pi \approx 160\pi \approx 1.6\times 10^1,\text{V}$.
5. Step 5: For the direction, use Lenz’s law: the induced current in Coil 2 will oppose the change in flux produced by Coil 1. When $I_1$ is increasing through zero, the flux in Coil 2 increases in the direction of Coil 1’s current. To oppose this, Coil 2 must produce a magnetic field opposite to that of Coil 1, which corresponds to a clockwise current when viewed from above.

Concept tested: Mutual inductance and induced emf

Published 10 September 2026 · Last updated 16 September 2026 · How this solution was prepared and checked