Question

Magnetic flux $\varphi$ linked with a stationary loop of resistance $R$ vaires with time $t$ as $\varphi =at(T-t)$. Amount of heat generated in loop during 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)

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

Flux is given by,

$X=at(T-t)$

emf induced is given by,

$e=-\frac{dX}{dt}$

$e=2at-at$

Heat produced during this time perios is given by,

$H={\int }_0^T\frac{e^2}{R}dt$

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

$H=\frac{1}{3R}(4a^2{T}^3+3a^2{T}^3-6a^2{T}^3)$

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