Question

Figure shows a closed loop of wire that consists of a pair of equal semicircles, of radius 3.7 cm, lying in mutually perpendicular planes.The loop was formed by folding a flat circular loop along a diameter until the two halves became perpendicular to each other. A uniform magnetic field $\overrightarrow{B}$ of magnitude 76 mT is directed perpendicular to the fold diameter and makes equal angles (of 45°) with the planes of the semicircles. The magnetic field is reduced to zero at a uniform rate during a time interval of 4.5 ms.
During this interval, what are the (a) magnitude and (b) direction (clockwise or counterclockwise when viewed along the direction of $\overrightarrow{B}$) of the emf induced in the loop?

Answer 1

(a) The magnetic flux ${\Phi }_B$
through the loop is given by

${\Phi }_B=2B\left(\pi r^2/2\right)\left(\cos {45}^{\circ }\right)=\pi r^{2}B/\sqrt{2}$

Thus,

$\epsilon =-\frac{d{\Phi }_B}{dt}=-\frac{d}{dt}\left(\frac{\pi r^{2}B}{\sqrt{2}}\right)=-\frac{\pi r^2}{\sqrt{2}}\left(\frac{\Delta B}{\Delta t}\right)=-\frac{\pi {(3.7\times {10}^{-2}m)}^2}{\sqrt{2}}\left(\frac{0-76\times {10}^{-3}T}{4.5\times {10}^{-3}s}\right)$

$=5.1\times {10}^{-2}V$

(a) The direction of the induced
current is clockwise when viewed along the direction of $\overrightarrow{B}$