A) \[\frac{{{\pi }^{2}}{{B}^{2}}{{r}^{2}}{{v}^{2}}}{R}\]
B) \[\frac{4{{\pi }^{2}}{{B}^{2}}{{r}^{2}}{{v}^{2}}}{R}\]
C) \[\frac{({{\pi }^{2}}+1){{B}^{2}}{{r}^{2}}{{v}^{2}}}{2R}\]
D) \[\frac{({{\pi }^{2}}+2){{B}^{2}}{{r}^{2}}{{v}^{2}}}{R}\]
Correct Answer: B
Solution :
[b] Flux: \[\phi =B.\pi {{r}^{2}}\] Induced emf: \[\varepsilon =\left| \frac{d\phi }{dt} \right|=2\pi Br\frac{dr}{dt}=2\pi Brv\] Induced current: \[I=\frac{\varepsilon }{R}=\frac{2\pi Brv}{R}\] Rate of work done = Rate of heat dissipated in Joule heating \[={{I}^{2}}R\] \[=\frac{4{{\pi }^{2}}{{B}^{2}}{{r}^{2}}{{v}^{2}}}{R}\]
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