Question

The magnetic flux through a stationary loop with resistance $R$ varies during an interval of time $T$ as $\varphi =at(T-t)$. The heat generated during this time neglecting the inductance of the loop will be

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

Answer 1

The correct option is A $\frac{a^2{T}^3}{3R}$

Given:-
Magnetic flux through the loop,$\varphi =at(T-t)$

Emf induced in the loop is

$E=\frac{-d\varphi }{dt}=\frac{-d}{dt}\left[at\right(T-t\left)\right]$

$E=-a(T-2t)$

Heat generated in the loop from $t=0$ to $t=T$ is given by

$H=\int dH=\int \frac{E^2}{R}dt$

$H={\int }_0^T\frac{a^2(T-2t{)}^2}{R}dt$

$H=\frac{a^2}{R}{\int }_0^T(T^2+4t^2-4tT)dt$

$H=\frac{a^2}{R}[T^3+\frac{4T^3}{3}-2T^3]=\frac{a^2}{R}\times \frac{T^3}{3}$

$\therefore H=\frac{a^2{T}^3}{3R}$

Hence, option $\left(\text{A}\right)$ is correct.