Concentric rings with varying flux

Problem

Two coplanar, concentric conducting rings of different radii. In the small ring a current increases linearly with time, and an induced current appears in the large ring. If the current in the small ring now becomes constant, what happens in the large one? And if it decreases?

Solution

Increasing current (initial situation). The current $i$ in the small ring generates a magnetic field whose flux $\Phi$ is also linked with the large ring. If $i$ increases linearly, $\Phi$ increases and Faraday’s law gives a constant EMF: $\mathcal{E} = -\frac{\dd \Phi}{\dd t} \neq 0$ An induced current flows in the large ring which, by Lenz’s law, opposes the increase in flux.

Constant current. If $i$ becomes constant, the field — and hence the linked flux — no longer varies: $\frac{\dd \Phi}{\dd t} = 0 \implies \mathcal{E} = \ev{0}$ No induced current flows in the large ring. It doesn’t matter that the flux is large: only its rate of change matters.

Decreasing current. If $i$ decreases, $\Phi$ diminishes: the derivative changes sign relative to the initial case, so the EMF and induced current reverse compared with before. The large ring now, by Lenz’s law, tends to sustain the flux that is falling.

Collegamenti

Argomenti: Electromagnetic induction Concetti: Faraday-Neumann-Lenz law · Lenz’s law · Magnetic flux