Problem — Induced EMF over time (graph)

Problem

A rectangular loop with vertical side $\ell$ moves at constant velocity $v$ to the right. Starting at $t=0$ it enters a region with a uniform field $B$ pointing into the page (dashed rectangle) and at instant $t_2$ it starts to leave it. (a) Sketch the induced emf $E(t)$, indicating the three intervals (entry, inside, exit) and the sign of each. (b) What is the peak value in terms of $B$, $\ell$, $v$? (c) Also sketch $i(t)$ in the loop closed on a resistance $R$, and the external force $F_\text{ext}(t)$ needed to keep $v$ constant.

The loop (red) moves to the right and enters the field region $\vv{B}$ pointing into the page (dashed).

Solution

The flux through the loop changes only while a vertical side crosses the boundary of the region: in those stretches the induced emf has magnitude $\abs{E}=B\,\ell\,v$ (motional emf of the side of length $\ell$ moving at velocity $v$).

Three phases.

• Entry ($0\le t<t_1$): the immersed area grows, the inward flux increases. Taking clockwise as positive, the induced emf opposes the increase: it is a rectangular pulse of negative sign, of magnitude $B\ell v$.
• Inside ($t_1\le t<t_2$): the loop is fully inside, the flux is constant, $E=0$.
• Exit ($t_2\le t<t_3$): the immersed area decreases, the flux drops. The emf changes sign: positive rectangular pulse, of magnitude $B\ell v$.

Time profile of $E(t)$: negative rectangular pulse on entry, zero inside, positive on exit.

(b) Peak value. $\abs{E}_\text{max}=B\,\ell\,v$

(c) Current and external force. The current follows Ohm’s law, $i=E/R$: it has the same rectangular shape as the emf, rescaled by $1/R$, with peak $\abs{i}_\text{max}=B\ell v/R$. On the current-carrying immersed side the Laplace force acts, which by Lenz’s law brakes the motion; to keep $v$ constant the external agent must apply an equal and opposite force: $F_\text{ext}=i\,\ell\,B=\frac{B^2\ell^2 v}{R}\quad(\text{during entry and exit}),\qquad F_\text{ext}=0\ (\text{inside})$ $F_\text{ext}(t)$ is thus also made of rectangular pulses, non-zero only during the entry and exit phases. The work done by $F_\text{ext}$ is dissipated in $R$ by the Joule effect.

$\ev{\abs{E}_\text{max}=B\ell v,\quad i=\frac{B\ell v}{R},\quad F_\text{ext}=\frac{B^2\ell^2 v}{R}\ \text{(entry/exit)}}$

Links

Topics: Electromagnetic induction Concepts: Faraday-Neumann-Lenz law · Lenz’s law Skills: Flux balance