A conducting circular loop of area \( 1.0 m^{2} \) is placed perpendicular to a magnetic field which varies as \( B = \sin(100 t) \) Tesla. If the resistance of the loop is 100 \( \Omega \), then the average thermal energy dissipated in the loop in one period is ___________ J.

Area of the loop \(=1 m^2\)
\(B=\sin (100 t)\)
\(\therefore \quad \phi= BA =\sin (100 t )\)
\(\therefore \quad \frac{ d \phi}{ dt }=100 \cos (100 t )\)
\(\therefore \quad P=\frac{V^2}{R}=\frac{10^4 \cos ^2(100 t)}{100}\)
\(\therefore \quad\) Thermal energy dissipated in 1 time period
\(=\int_0^{ T } Pdt =\int_0^{ T } 100 \cos ^2(100 t ) dt\)
\(T =\frac{2 \pi}{100}=\frac{\pi}{50} sec\)
\(\therefore \quad Q=100 \int_0^{\pi / 50} \cos ^2(100 t) d t\)
\(=100 \int_0^{\pi / 50} \frac{1+\cos 200 t}{2} d t\)
\(=100\left[\frac{\pi}{100}\right]=\pi\)