Two long solenoids share the same axis. The primary solenoid has $N_1=200$ turns, the secondary has $N_2=150$ turns, and their mutual inductance is $M=0.05\text{H}$. If the current in the primary decreases uniformly from $3\text{A}$ to $0\text{A}$ in $0.010\text{s}$, (i) calculate the magnitude of the emf induced in the secondary, and (ii) state the direction of the induced current relative to the original current in the primary.

Step-by-step solution

1. Faraday’s law for mutual induction gives ${\epsilon }_2=-M\frac{d{I}_1}{dt}$.
2. Since the current falls uniformly, $\frac{d{I}_1}{dt}=\frac{\Delta I_1}{\Delta t}=\frac{0-3}{0.010}=-300{\text{A s}}^{-1}$.
3. Substitute: ${\epsilon }_2=-0.05\times (-300)=+15\text{V}$. The magnitude is $15\text{V}$.
4. Lenz’s law states that the induced emf creates a current whose magnetic field opposes the change in the original flux.
5. Because the primary current is decreasing, the secondary will try to sustain the original flux; therefore its current will be in the same sense as the original primary current.

Concept tested: Mutual inductance and Lenz’s law

Published 14 September 2026 · How this solution was prepared and checked