A conducting metal circular-wire-loop of radius r is placed perpendicular to a magnetic field which varies with time as `B = B_0e^(-t//tau)`, where `B_0 and tau` are constants, at time = 0. If the resistance of the loop is R then the heat generated in the loop after a long time `(t to oo)` is :

Options

A. `(pi^(2)r^(4)B_(0)^(4))/(2tau R)`
B. `(pi^(2)r^(4)B_(0)^(2))/(2tau R)`
C. `(pi^(2)r^(4)B_(0)^(2)R)/(tau)`
D. `(pi^(2)r^(4)B_(0)^(2))/(tau R)`

Answer

B

Explanation

Electric flux is given by
`phi = B.A`
`phi = B_(0)pi r^(2)e^(-t//tau) " " (because = B_(0)e^(-t//tau))`
Induced E.m.f. `epsilon = (d phi)/(dt)=(B_(0)pi r^(2))/(tau^(2))e^(-t//tau)`
Heat `= int_(0)^(oo)(epsilon^(2))/(R )=(pi^(2)r^(4)B_(0)^(2))/(2tau R)`