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 mutual inductance of the pair is

Options

A. `(mu_(0)a)/(2pi) In((2a + l)/(l))`
B. `(mu_(0)a)/(2pi) In((2a - l)/(l))`
C. `(mu_(0)a)/(pi) In((a + l)/(l))`
D. `(mu_(0)a)/(pi) In((a + l)/(l))`

Answer

A

Explanation

Consider a strip at a distance `x` from the wire of thikness `dx`.
Magnetic flux associted with this strip `phi = B(x) adx = (mu_(0)Ia)/(2pix)dx`
`phi = (mu_(0)Ia)/(2pi) [underset(1)overset(a + 1)int (dx)/(x) + underset(a + 1)overset(2a + l)int] = (mu_(0)Ia)/(2pi) In ((2a + l)/(l))`
`M = (phi)/(I) rArr M = (mu_(0)a)/(2pi)In ((2a + l)/(l))`
`e = - M(dI)/(dt)`
`e = - MI_(0) = - (mu_(0)I_(0)a)/(2pi)In ((2a + l)/(l))`
Heat produced `= (epsilon^(2))/(R ) t = ([(mu_(0)Ia)/(2pi) In ((2a + l)/(l))]^(2)at)/(8lambda)`