Problem — E(t) for a loop in motion

Problem

A rectangular loop with vertical side $\ell$ enters at constant velocity $v$ into a region where the field $B$ is uniform and perpendicular to the plane of the loop, crosses it while remaining fully immersed for a stretch, and finally exits. The graph of $E(t)$ has three phases. Draw the diagram and indicate, for each phase, the sign and relative amplitude.

The loop (red) enters at velocity $v$ into the region of uniform field $\vv{B}$ (dashed).

Solution

The linked flux changes only when a vertical side (of length $\ell$) crosses a boundary of the region. In those stretches the motional emf has magnitude $\abs{E}=B\,\ell\,v$; when the loop is entirely inside, the flux is constant and $E=0$.

Three phases.

• Entry: the immersed area increases, the flux increases $\Rightarrow$ rectangular pulse of amplitude $+B\ell v$ (Lenz: the current opposes the increase).
• Inside: constant flux $\Rightarrow E=0$.
• Exit: the immersed area decreases, the flux drops $\Rightarrow$ pulse of opposite sign, $-B\ell v$.

Trend of $E(t)$: positive pulse on entry, zero inside, negative pulse on exit, all of magnitude $B\ell v$.

$\ev{\text{entry}:+B\ell v;\quad \text{inside}:0;\quad \text{exit}:-B\ell v}$

Links

Topics: Induzione elettromagnetica Concepts: Legge di Faraday-Neumann-Lenz · Legge di Lenz Skills: Lettura dei grafici · Bilancio di flusso