Question

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

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

Answer 1

Answer: (D)

$|E|=\left|\frac{d\varphi }{dt}\right|=(aT-2at)$
$H=\int \frac{E^2}{R}dt=\frac{a^2}{R}{\int }_0^T{\left(T-2t\right)}^{2}dt$
$=\frac{a^2}{R}{\int }_0^T\left(T^2+4t^2-4Tt\right)dt$
$=\frac{a^2}{R}\left({T^{2}t|}_0^T+{\frac{4}{3}{t}^3|}_0^T-{2T{t}^2|}_0^T\right)=\frac{a^2{T}^3}{3R}$