Worked exercise — loop entering and leaving a field

Problem

A rectangular loop with sides $a=3\ \text{m}$ and $b=2\ \text{m}$, and resistance $R=3\ \Omega$, moves at constant velocity $v=2\ \text{m/s}$ to the right. At $t=0$ the loop is $2\ \text{m}$ from the left edge of a region $6\ \text{m}$ wide, in which there is a uniform field $B=5\ \text{T}$ pointing out of the page. Study the magnetic flux, the induced current, and the force on the loop as functions of time.

Loop approaching the field region, moving to the right with velocity $\vv{v}$.

Solution

The useful side for the flux is $b=2\ \text{m}$ (height of the loop). The right edge of the loop starts $2\ \text{m}$ from the field region, the left edge at $2+3=5\ \text{m}$. The region is $6\ \text{m}$ wide. Let’s analyse five phases.

Phase 1 — $0\le t<1\ \text{s}$ (loop outside): no part is immersed in the field. $\Phi_B=0,\qquad \text{emf}=0,\qquad i=0$

Phase 2 — $1\le t<2{,}5\ \text{s}$ (the loop enters): the immersed portion has width $v(t-1)$, so the flux grows linearly. $\Phi_B=B\,b\,v\,(t-1),\qquad \text{emf}=-\ddt{\Phi_B}=-B\,b\,v=-5\cdot 2\cdot 2=-20\ \text{V}$ $i=\frac{\abs{\text{emf}}}{R}=\frac{20}{3}\approx 6{,}7\ \text{A}$

Phase 3 — $2{,}5\le t<3\ \text{s}$ (loop fully inside): the flux is constant and maximum. $\Phi_B=B\,b\,a=5\cdot 2\cdot 3=30\ \text{Wb},\qquad \text{emf}=0,\qquad i=0$

Phase 4 — $3\le t<4{,}5\ \text{s}$ (the loop exits): the flux decreases linearly, the emf changes sign and the current flows in the opposite direction. $\text{emf}=+B\,b\,v=+20\ \text{V},\qquad i=\frac{20}{3}\approx 6{,}7\ \text{A}\ \text{(opposite direction)}$

Phase 5 — $t\ge 4{,}5\ \text{s}$ (loop outside): $\Phi_B=0,\qquad i=0$

Force (magnetic friction). During entry and exit, the Laplace force acts on the current-carrying immersed side: $\abs{F}=i\,b\,B=\frac{20}{3}\cdot 2\cdot 5\approx 66{,}7\ \text{N}$ By Lenz’s law this force always opposes the loop’s motion: it acts as a “magnetic friction” that slows the relative motion. The work needed to overcome it is dissipated in $R$ by the Joule effect ($P=Ri^2$). The magnetic friction exists only as long as the flux changes: when the loop is stationary or fully inside/outside, it disappears.

$\ev{\abs{\text{emf}}=B\,b\,v=20\ \text{V},\quad i=\tfrac{20}{3}\approx 6{,}7\ \text{A},\quad \abs{F}=i\,b\,B\approx 66{,}7\ \text{N}}$

Links

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