A metal disc of radius R rotates with an angu | vedclass

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(D) Consider a small radial element of length $dr$ at a distance $r$ from the centre of the disc.As the disc rotates,this element moves with a linear

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$\frac{BR^2 \omega}{2}$

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(D) Consider a small radial element of length $dr$ at a distance $r$ from the centre of the disc.As the disc rotates,this element moves with a linear velocity $v = r\omega$ in the magnetic field $B$.The motional e.m.f. $de$ induced across this small element is given by $de = Bv dr = B(r\omega) dr$.To find the total induced e.m.f. $e$ between the centre $(r=0)$ and the rim $(r=R)$,we integrate the expression:$e = \int_{0}^{R} B\omega r dr$$e = B\omega \int_{0}^{R} r dr$$e = B\omega \left[ \frac{r^2}{2} \right]_{0}^{R}$$e = \frac{B\omega R^2}{2}$The magnitude of the induced e.m.f. is $|e| = \frac{B\omega R^2}{2}$.

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