Framework for solving an induction problem

Faced with any problem involving a circuit with a varying magnetic flux, it is worth always following the same five-step protocol. It turns what looks like a subtle phenomenon into a mechanical procedure.

Five-step protocol

1. Orient the surface. Choose an orientation for the surface $S$ bounded by the loop, and consequently the direction of travel around the boundary (linked by the right-hand rule).
2. Compute $\Phi_B(t)$. The flux of the field $\vv{B}$ through $S$, as a function of time. If $B$ or the loop’s geometry vary, the flux varies.
3. Differentiate with respect to time. Compute $d\Phi_B/dt$; the result, with its sign flipped, is the induced EMF $\text{EMF}_\text{ind} = -d\Phi_B/dt$.
4. Treat the induction as a fictitious generator. Insert $\text{EMF}_\text{ind}$ into the circuit’s Kirchhoff equations as if it were an ordinary battery. Solve for the current.
5. Check the direction with Lenz. The induced current always opposes the change in flux that generated it. If the sign is not consistent, you have made an orientation mistake.

The fifth step is the safety net: any sign error in the orientation choices is exposed by checking against Lenz’s law, which gives the “physical” direction of the current regardless of conventions.

Links

Topics: Electromagnetic induction Concepts: Faraday-Neumann-Lenz law Skills: Flux balance · Right-hand rule

Related exercises: Worked exercise — loop entering and leaving a field · Problem — Ranking the EMF of four square loops · Problem — Loop entering a field (qualitative)