Two thin concentric wires shpaed as circles with radii `a` and `b` lie in the same plane. Allowing for `a lt lt b`, find:
(a) their mutual inductane,
(b) the magnetic flux through the surface enclosed by the outside wire, when the inside wire carries a current `I`.

Explanation

The direct calculation of the flux `Phi_(2)` is a rather complicated problem, since the configuration of the field itself is complicated. However, the application of the reciprocity theorem simplifies the solutions of the problem. Induced, let the same current `i` flow through loop 2. Then the magnetic flux created by this current through loop 1 can be easily found.
Magnetic induction at the centre of the loop, : `Phi_(12) = pi a^(2) (mu_(0) i)/(2b)`
and from reciprocity theorem,
`Phi_(12) = Phi_(21), Phi_(21) = (mu_(0) pi a^(2) i)/(2b)`
So, `L_(12) = (Phi_(21))/(i) = (1)/(2) mu_(0) pi a^(2)//b`