Question

A conducting loop of area $5.0c{m}^2$ is placed in a magnetic field which varies sinusoidally with time as $B=B_{0}sin\omega t$ where $B_0=0.20T$ and $\omega =300s^{-1}$. The normal to the coil makes an angle of ${60}^{\circ }$ with the field. Find (a) the maximum emf induced in the coil, (b) the emf induced at $\tau =\left(\frac{\pi }{900}\right)s$ and (c)the emf induced at $t=\left(\frac{\pi }{600}\right)s$.

Answer 1

Here $A=5c{m}^2=5\times {10}^{-4}{m}^2$

$B=B_{0}sin\omega t=0.2sin\left(300t\right)$

$\theta ={60}^{\circ }$

(a)Max e.m.f induced in the coil

$e=\frac{-d\theta }{dt}=\frac{d}{t}\left(BAcos\theta \right)$

$=\frac{d}{t}(B_{0}sin\omega t\times 5\times {10}^{-4}\times \frac{1}{2})$

$=B_0\times \frac{5}{2}\times {10}^{-4}\frac{d}{dt}\left(sin\omega t\right)$

$=\frac{B_{0}5}{2}{10}^{-4}cos\omega t.\omega .$

$=\frac{0.2\times 5}{2}\times 300\times {10}^{-4}\times cos\omega t$

$=15\times {10}^{-3}cost\omega t{e}_{max}=15\times {10}^{-3}=0.015V$

(b)Induced emf at $t=\left(\frac{\pi }{900}\right)s$

$e=15\times {10}^{-3}\times cos\omega t$

$=15\times {10}^{-3}\times cos(300\times \frac{\pi }{900})$

$=15\times {10}^{-3}\times \frac{1}{2}$

$=\frac{0.015}{2}=0.0075=7.5\times {10}^{-3}V$

(c) Induced emf at $t=\frac{\pi }{600}s$

$e=15\times {10}^{-3}\times cos(300\times \frac{\pi }{600})$

$=15\times {10}^{-3}\times 0=0V$