Question

Magnetic flux linked with a stationary loop 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. $\frac{\alpha T}{3R}$
• B. $\frac{a^2{T}^2}{3R}$
• C. $\frac{a^2{T}^2}{R}$
• D. $\frac{a^2{T}^3}{3R}$

Answer 1

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

Given , $\varphi =at(T-t)$

The emf is, $\epsilon =-\frac{d\varphi }{dt}=-at+2at=at$

Power, $P=\epsilon I=\epsilon .\frac{\epsilon }{R}=\frac{a^2{t}^2}{R}$

Heat generated, $H={\int }_0^{T}Pdt={\int }_0^T\frac{a^2{t}^2}{R}dt=\frac{a^2{T}^3}{3R}$