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 $B=c[(x\cos \omega t)\hat{i}+(y\sin \omega t)\hat{k}]$ 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. $\left|\frac{a{b}^{2}c}{2}\omega \sin \omega t\right|$
• D. None of the options

Answer 1

Answer: (A)

Area vector of the given loop,
$A=ab\hat{k}$
Time dependent magnetic field,
$B=c[(x\cos \omega t)\hat{i}+(y\sin \omega t)\hat{k}]$
$\therefore$ Magnetic flux through the loop,
${\varphi }_n=A\cdot B$
$=ab\hat{k}[c\{(x\cos \omega t)\hat{i}+(y\sin \omega t)\hat{k}\}]$
$=abcy\sin \omega t$...(i)
$\because y\le y\le b$
$\therefore y_{av}=\frac{0+b}{2}=\frac{b}{2}$
Putting the value of $y$ in Eq. (i), we get
$\therefore {\varphi }_n=abc\cdot \frac{b}{2}\cdot \sin \omega t=\frac{a{b}^2}{2}c\sin \omega t$
$\therefore$ Induced emf in the loop,
$\epsilon =\frac{d}{dt}{\varphi }_n=\frac{d}{dt}\cdot \frac{a{b}^{2}c}{2}\sin \omega t$
Hence, $|\epsilon |=\left|\frac{a{b}^{2}c}{2}\omega \cos \omega t\right|$