Two coaxial circular coils share the same axis. The primary coil has $N_1=200$ turns, the secondary coil has $N_2=150$ turns, and their mutual inductance is $M=0.06\text{H}$. The current in the primary varies sinusoidally as $I_1(t)=5\cos (120\pi t)\text{A}$. (i) Find the maximum magnitude of the emf induced in the secondary coil. (ii) State the direction of the induced current at an instant when the primary current is decreasing.

Step-by-step solution

1. Faraday’s law for mutual induction: ${\epsilon }_2=-M\frac{d{I}_1}{dt}$.
2. Differentiate $I_1(t)$: $\frac{d{I}_1}{dt}=-5(120\pi )\sin (120\pi t)=-600\pi \sin (120\pi t)$.
3. Substitute into the emf expression: ${\epsilon }_2=-M[-600\pi \sin (120\pi t)]=M600\pi \sin (120\pi t)$.
4. The magnitude varies as $|{\epsilon }_2|=M600\pi |\sin (120\pi t)|$; its maximum occurs when $|\sin |=1$. Hence ${\epsilon }_{\text{max}}=M600\pi$.
5. Insert $M=0.06\text{H}$: ${\epsilon }_{\text{max}}=0.06\times 600\pi =36\pi \approx 113\text{V}$.
6. For part (ii), when $I_1$ is decreasing, $d{I}_1/dt<0$; the induced emf ${\epsilon }_2=-Md{I}_1/dt$ is positive, meaning the secondary current is induced in a direction that creates a magnetic field supporting the original primary field.
7. Thus the secondary current flows in the same sense as the primary current (Lenz’s law).

Concept tested: Mutual induction and Lenz’s law

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