In Fig. there is a conducting loop `ABCDEF` of resistance `lambda` per unit length placed near a long straight current-carrying wire. The dimension are shows in the figure. The long wire lies in the plane of the loop. The current in the long wire varies as `I = I_(0)(t)`.

The heat produced in the loop in time `t` is

Explanation

(`a`) Consider a strip at a distance `x` from the wire of thickness `dx`.
Magnetic flux associated with this strip `Deltaphi=B(x)adx=(mu_(0)Ia)/(2pix)dx`
`rArr phi=(mu_(0)Ia)/(2pi)[int_(l)^(a+l)(dx)/(x)-int_(a-l)^(2a+l)(dx)/(x)]=(mu_(0)Ia)/(2pi)ln(((a+l)^(2))/(a(2a+l)))`
`:' M=(phi)/(I_((primary)))`
`rArr M=(mu_(0)a)/(2pi)ln((a+l)^(2))/(a(2a+l))`
(`b`) `epsilon=-M(dI)/(dt)`
`rArr epsilon= -MI_(0)`
Heat produced `=(epsilon^(2))/(R )t=([(mu_(0))/(2pi)ln"(a+l)^(2)/(a(2a+l))I_(0)]^(2)at)/(8lambda)`