Question

A circular conducting coil of radius $1\, m$ 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

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