Question

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 $\overrightarrow{B}=c\left[\left(xcos\omega t\right)\hat{i}+\left(ysin\omega t\right)\hat{k}\right]$ .Here, $c$ and $\omega$ are constants. The magnitude of emf induced in the loop as a function of time is

• 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

Answer 1

Answer: (A)

from diagram only $z$ component of magnetic field gives flux.
Flux through small elemental area
$d\varphi =adycysin\left(\omega t\right)$
net flux $=acsin\omega t{\int }_0^{b}ydy$
$\varphi =ac\frac{b^2}{2}sin\omega t$
$\therefore$ Induced emf in the loop.
$ϵ=\frac{d}{dt}{\varphi }_m=\frac{d}{dt}.\frac{a{b}^{2}c}{2}sin\omega t$
Hence, $\left|ϵ\right|=\left|\frac{a{b}^{2}c}{2}\omega cos\omega t\right|$