Question

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

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

Answer 1

Answer: (D)

Given that $\varphi =at(T-t)$
induced emf, $E=\frac{d\varphi }{dt}=\frac{d}{dt}[at(T-t)]$
= $at(0-1)+a(T-t)=a(T-2t)$
So, indeced emf is also a function of time.
$\therefore$ Heat generated in time $T$ is
$H={\int }_0^T\frac{E^2}{R}dt=\frac{a^2}{R}{\int }_0^T(T-2t{)}^{2}dt=\frac{a^2{T}^3}{3R}$