Question

A circular conducting coil of radius $1m$ is being heated by the change of magnetic field $\vec{B}$ passing perpendicular to the plane in which the coil is laid. The resistance of the coil is $2\mu \Omega$. The magnetic field is slowly switched off such that its magnitude changes in time as $B=\frac{4}{\pi }\times 10^{-3}T\left(1-\frac{t}{100}\right)$ The energy dissipated by the coil before the magnetic field is switched off completely is $E=$______ $mJ$

Answer 1

Answer: 80

$\varphi =\vec{B}\cdot \vec{S}$
$\varphi =\frac{4}{\pi }\times 10^{-3}\left(1-\frac{t}{100}\right)\cdot \pi R^2$
$\varphi =4\times 10^{-3}\times (1{)}^2\left(1-\frac{t}{100}\right)$
$\epsilon =\frac{-d\varphi }{dt}$
$\epsilon =\frac{-d}{dt}\left(4\times 10^{-3}\left(1-\frac{t}{100}\right)\right)$
$\epsilon =4\times 10^{-3}\left(\frac{1}{100}\right)$
$=4\times 10^{-5}V$
When $B=0$
$1-\frac{t}{100}=0$
$t=100\sec$
Heat $=\frac{{\epsilon }^2}{R}t$
Heat $=\frac{{\left(4\times 10^{-5}\right)}^2}{2\times 10^{-6}}\times 100J$
Heat $=\frac{16\times 10^{-10}\times 100}{2\times 10^{-6}}J$
Heat $=0.08J$
Heat $=80mJ$