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 \(E=.....\,{m} J.\)

\(\phi=\frac{4}{\pi} \times 10^{-3}\left(1-\frac{ t }{100}\right) \cdot \pi R ^{2}\)

\(\phi=4 \times 10^{-3} \times(1)^{2}\left(1-\frac{t}{100}\right)\)

\(\varepsilon=\frac{- d \phi}{ dt }\)

\(\varepsilon=\frac{- d }{ dt }\left(4 \times 10^{-3}\left(1-\frac{ t }{100}\right)\right)\)

\(\varepsilon=4 \times 10^{-3}\left(\frac{1}{100}\right)=4 \times 10^{-5} V\)

When \(B=0\)

\(1-\frac{t}{100}=0\)

\(t=100 sec\)

\(E=\frac{\varepsilon^{2}}{R} t\)

\(E=\frac{\left(4 \times 10^{-5}\right)^{2}}{2 \times 10^{-6}} \times 100 J\)

\(E=\frac{16 \times 10^{-10} \times 100}{2 \times 10^{-6}} J\)

\(E=0.08 J\)