A circular conducting coil of radius 1 m is being heated by the change of magnetic field $\overrightarrow B$ 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

$B = {4 \over \pi } \times {10^{ - 3}}T\left( {1 - {t \over {100}}} \right)$

The energy dissipated by the coil before the magnetic field is switched off completely is E = ___________ mJ.

$\phi = \overrightarrow B .\overrightarrow S$<br><br>$$\phi = {4 \over \pi } \times {10^{ - 3}}\left( {1 - {t \over {100}}} \right).\pi {R^2}$$<br><br>$\phi = 4 \times {10^{ - 3}} \times {(1)^2}\left( {1 - {t \over {100}}} \right)$<br><br>$\varepsilon = {{ - d\phi } \over {dt}}$<br><br>$$\varepsilon = {{ - d} \over {dt}}\left( {4 \times {{10}^{ - 3}}\left( {1 - {t \over {100}}} \right)} \right)$$<br><br>$$\varepsilon = 4 \times {10^{ - 3}}\left( {{1 \over {100}}} \right) = 4 \times {10^{ - 5}}V$$<br><br>When B = 0<br><br>$1 - {t \over {100}} = 0$<br><br>t = 100 sec<br><br>Heat $= {{{\varepsilon ^2}} \over R}t$<br><br>Heat $= {{{{(4 \times {{10}^{ - 5}})}^2}} \over {2 \times {{10}^{ - 6}}}} \times 100$ J<br><br>Heat $= {{16 \times {{10}^{ - 10}} \times 100} \over {2 \times {{10}^{ - 6}}}}$ J<br><br>Heat = 0.08 J<br><br>Heat = 80 mJ