A rectangular wire loop with length $a$ and width $b$ lies in the $xy$ -plane as shown. Within the loop, there is a time dependent magnetic field given by $\overrightarrow{B}= c\left[\left(x\cos \omega t\right)\widehat{i}+\left(y\sin \omega t\right)\widehat{k}\right]$.Here, $c$ and $\omega$ are constants. The magnitude of emf induced in the loop as a function of time is

• Correct answer
  A

  $\left|\frac{a{b}^{2}c}{2}\omega \cos \omega t\right|$
• B

  $\left|a{b}^{2}c\omega \cos \omega t\right|$
• C

  $\left|\frac{a^{2}bc}{2}\omega \sin \omega t\right|$
• D

  None of the options

Solution

from diagram only $z$ component of magnetic field gives flux.

Flux through small elemental area

$d\varphi =ady cy\sin \left(\omega t\right)$

net flux $=ac\sin \omega t{\int }_0^{b}y dy$

$\varphi =ac\frac{b^2}{2}\sin \omega t$

$\therefore$ Induced emf in the loop.

$\epsilon =\frac{d}{dt}{\varphi }_m=\frac{d}{dt}.\frac{a{b}^{2}c}{2}\sin \omega t$

Hence, $\left|\epsilon \right|=\left|\frac{a{b}^{2}c}{2}\omega \cos \omega t\right|$

Hence, the correct answer is option: 'A'.