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 heat the generated in the loop during that time

Options

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

Answer

D

Explanation

(D) 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, induced emf is also a function of time.
`:.` Heat generated in time `T` is
`H = int_(0)^(T)(E^(2))/(R)dt = (a^(2))/(R)int_(0)^(T)(T - 2t)^(2)dt = (a^(2)T^(3))/(3R)`