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

Answer

D

Explanation

Given that `phi = at (T-t)`
induced emf, `E = (d phi)/(dt) = (d)/(dt)[at(T-t)]`
`= at (0-1)+a(T-t) , = a(T-2t)`
So, induced emf is alos 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)`