Question

A conducting loop (as shown) has total resistance $R$. A uniform magnetic field $B=\gamma t$ is applied perpendicular to plane of the loop where $\gamma$ is a constant and $t$ is time. The induced current flowing through loop is:

• A. $\frac{(b^2+a^2)\gamma t}{R}$
• B. $\frac{(b^2-a^2)\gamma }{R}$
• C. $\frac{(b^2-a^2)\gamma t}{R}$
• D. $\frac{(b^2+a^2)\gamma }{R}$

Answer 1

Answer: (B)

$\frac{(b^2-a^2)\gamma }{R}$

Given,
uniform magnetic field $\left(B\right)=\gamma t$
Total flux, $\varphi =B_1{A}_1+B_2{A}_2$
$=B{b}^2\cos 0^{\circ }+B{a}^2\cos {180}^{\circ }$
$=B{b}^2-B{a}^2$
$\varphi =B(b^2-a^2)$
$\varphi =\gamma t(b^2-a^2)....\left(i\right)$
We know that,
induced current $\left(i\right)=\frac{|e|}{R}=\frac{\left|\frac{d\varphi }{dt}\right|}{R}$
From Eq, (i),
$i=\frac{\frac{d}{dt}\left[\gamma t\right(b^2-a^2\left)\right]}{R}$
$i=\frac{(b^2-a^2)\gamma }{R}$.