Question

Magnetic flux linked with a stationary loop of resistance R varies with respect to time during the time period T as follows: $\phi=at(T-t)$ the amount of heat generated in the loop during that time (inductance of the coil is negligible) is

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

Answer 1

Answer: (D)

$\frac{a^2T^3}{3R}$

Answer 2

Given that $\phi = at (T - t)$ induced emf, $E = \frac{d \phi}{dt} = \frac{d}{dt} [ at(T - t)]$ = $at (0-1) + a (T - t) = a(T - 2t)$ So, indeced emf is also a function of time. $\therefore$ Heat generated in time $T$ is $H = \int^T_0 \frac{E^2}{R} dt = \frac{a^2}{R} \int^T_0 (T - 2t)^2 dt = \frac{a^2 T^3}{3R}$