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?

Solution

## (a) The magnetic flux $$\Phi_{B}$$through the loop is given by$$\Phi_{B}=2 B\left(\pi r^{2} /2\right)\left(\cos 45^{\circ}\right)=\pi r^{2} B / \sqrt{2}$$Thus,$$\varepsilon =-\dfrac{d \Phi_{B}}{dt}=-\dfrac{d}{d t}\left(\dfrac{\pi r^{2} B}{\sqrt{2}}\right)=-\dfrac{\pir^{2}}{\sqrt{2}}\left(\dfrac{\Delta B}{\Delta t}\right)=-\dfrac{\pi\left(3.7\times 10^{-2} \mathrm{m}\right)^{2}}{\sqrt{2}}\left(\dfrac{0-76 \times 10^{-3}\mathrm{T}}{4.5 \times 10^{-3} \mathrm{s}}\right)$$$$=5.1 \times 10^{-2} \mathrm{V}$$(a) The direction of the inducedcurrent is clockwise when viewed along the direction of $$\vec{B}$$Physics

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