Question

Magnetic flux $\phi$ linked with a stationary loop of resistance $R$ varies with time $t$ as $\phi=at\left(\right.T-t\left.\right)$ . The amount of heat generated in the loop, during the time interval $T$ , is

• A. $\frac{a T}{3 R}$
• B. $\frac{a^{2} T^{2}}{3 R}$
• C. $\frac{a^{2} T^{2}}{R}$
• D. $\frac{a^{2} T^{3}}{3 R}$

Answer 1

Answer: (D)

$\left|\right.E\left|\right.=\left|\frac{d \phi}{d t}\right|=\left(\right.aT-2at\left.\right)$
$H=\displaystyle \int \frac{E^{2}}{R}dt=\frac{a^{2}}{R}\displaystyle \int _{0}^{T}\left(T - 2 t\right)^{2}dt$
$=\frac{a^{2}}{R}\displaystyle \int _{0}^{T}\left(T^{2} + 4 t^{2} - 4 T t\right)dt$
$=\frac{a^{2}}{R}\left(\left.T^{2} t\right|_{0} ^{T}+\left.\frac{4}{3} t^{3}\right|_{0} ^{T}-\left.2 T t^{2}\right|_{0} ^{T}\right)=\frac{a^{2} T^{3}}{3 R}$