Calculate the inducatance of a doughunt solenoid whose inside radius is equal to `b` and cross-section has the form of a square with side `a`. The solenoid winding consists of `N` turns. The space inside the solenoid is filled up with unifrom paramagnetic having permeability `mu`.

Explanation

Within the solenoid, `H_(varphi) 2pi r = N I` or `H_(varphi) = (NI)/(2pi r), B_(varphi) = mu mu_(0) (N I)/(2pi r)`
and the flux, `Phi = N Phi_(1) = N (mu mu_(0))/(2pi) In int_(b)^(a + b) (a dr)/(r)`
Finally, `L = (mu mu_(0))/(2pi) N^(2) a In (1 + (a)/(b))`