A magnetic flux through a stationary loop with a resistance `R` varies during the time interval `tau` as `phi=at(tau-t)`. Find the amount of the generated in the loop during that time

Options

A. `(aT)/(3R)`
B. `(a^2T^2)/(3R)`
C. `(a^2T^2)/R`
D. `(a^2T^3)/(3R)`

Answer

D

Explanation

Given that `phi=at (T-t)`
Induced emf `E=(dphi)/(dt)=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 R is
`F=int_0^T(E^2)/R dt =(a^2)/R int_0^T(T-2t)^2dt=(a^2T^3)/(3R)`