The approximate formula expressing the formula of mutual inductance of two coaxial loops of the same redius `a` when their centers are separated by a distance `l` with `l gt gt a` is

Options

A. `(1)/(2) (mu_(0)pia^(4))/(l^(3))`
B. `(1)/(2) (mu_(0)a^(4))/(l^(2))`
C. `(mu_(0))/(4pi) (pia^(4))/(l^(2))`
D. `(mu_(0))/(pi) (a^(4))/(l^(3))`

Answer

A

Explanation

Let `I =` current in one loop. The magnetic field at the centre of the other co-axial loop at a distance `l` from the centre of the first loop is
`B = (mu_(0))/(4pi) (2I pia^(2))/((a^(2) + l^(2))^(3//2))`
where `p_(m) = I pia^(2)`
= magnetic moment of the loop
Flux through the other loop is
`phi_(12) = Bpi a^(2)`
`= (mu_(0))/(4pi) (2I pia^(2))/((a^(2) + l^(2))^(3//2)) pi a^(2)`
or `M = (phi_(12))/(I) = (mu_(0))/(4pi) (2I pia^(4))/((a^(2) + l^(2))^(3//2))`
`= (mu_(0) pia^(4))/(2(a^(2) + l^(2))^(3//2))= (mu_(0)pia^(4))/(2l^(3)) (alt lt l)`