Question

The magnetic field in the cylindrical region shown in figure increases at a constant rate of $10.0mT$ $s^{-1}$. Each side of the square loop $abcd$ and $defa$ has a length of $20.0cm$ and a resistance of $2.00\Omega$. Correctly match the current in the wire $ad$ in four different situations as listed in column I with the values given in column II.

Answer 1

$S_1$ is closed, $S_2$ is open:
Area of the loop: $20\times 20=400$ $c{m}^2=4\times {10}^{-2}{m}^2$

Induced emf, $\epsilon =Area\times \frac{dB}{dt}=4\times {10}^{-2}\times (10\times {10}^{-3})$

Induced Current: $i=\frac{4\times {10}^{-2}\times 10\times {10}^{-3}}{4\times 2}=5\times {10}^{-5}A$

The induced current will flow from a to d as the magnetic field due to induced current opposes the external magnetic field.
$S_2$ is closed, $S_1$ is open:
Area of the loop: $20\times 20=400$ $c{m}^2=4\times {10}^{-2}{m}^2$

Induced emf, $\epsilon =Area\times \frac{dB}{dt}=4\times {10}^{-2}\times (10\times {10}^{-3})$

Induced current, $i=\frac{4\times {10}^{-2}\times 10\times {10}^{-3}}{4\times 2}=5\times {10}^{-5}A$

The induced current will flow from d to a as the magnetic field due to induced current opposes the external magnetic field.
Both switches are open:
No current flows since there is closed loop.
Both switches are closed.
No current flows.