Question

A circular conducting coil of radius 1m and resistance 1 $\Omega$ is placed such that half of its area is in a region of uniform magnetic field $B_1\left(\hat{k}\right)$ and the remaining half is in another region of uniform magnetic field $B_2(-\hat{k})$ as shown in figure. Column I shows values of $B_1$ and $B_2$ as a function of time t, (for t $\le$ 0), while column II predicts possible directions of current and magnitude of force.

Answer 1

${\varphi }_1\frac{B_1\pi R^2}{2};{\varphi }_2\frac{B_2\pi R^2}{2}$

For option (A) solution : $\frac{d{\varphi }_1}{dt}$ = 0 and $\frac{d{\varphi }_2}{dt}$ = 0 , induced current is zero, force is also zero.

For option (B) Solution : $\frac{d{\varphi }_1}{dt}$ = 0 and $\frac{d{\varphi }_2}{dt}=\frac{10\left(\pi R^2\right)}{2}$ , and inward field is increasing therefore current will be always anticlock wise and force on the ring will be toward left.

For option (C) Solution : $\frac{d{\varphi }_1}{dt}=\left(2t\right)\frac{\pi R^2}{2}=\left(t\right)\pi R^2;\frac{d{\varphi }_2}{dt}=\left(t\pi R^2\right)$ ; As both the induced e.m.f. are equal in magnitude therefore net current in the coil is zero. As I is zero force on ring will be zero.

For option (D) Solution : $\frac{d{\varphi }_1}{dt}=\left(t\right)\pi R^2;\frac{d{\varphi }_2}{dt}=\left(2\pi R^2\right)$ for $t<sec\frac{d{\varphi }_2}{dt}\ge 0$ current will be anticlockwise t = 2 sec.

$\frac{d{\varphi }_1}{dt}=\frac{d{\varphi }_2}{dt}$ current will be zero.

$\therefore$ force will be zero at t = 2 sec for t > 2 current will be anticlockwise