Consider two loops of current with arbitrary shape. Both loops exist in planes that are parallel to each other (i.e., the loop faces are parallel to each other). Each loop makes a magnetic field that generates a flux through the other. Starting from Biot-Savart, demonstrate that the magnetic flux produced by one loop on the other is proportional to the current of the original loop (i.e., $\Phi_2 \propto I_1$). Hint: use the vector potential and Stoke’s theorem.
Find the constant of proportionality (“mutual inductance”) as pair of ciruclation integrals. What can you say about the constant of proportionality?
Use this result to find the mutual inductance in the following nested solenoid situtation: One small short solenoid with finite length, $L$, and radius, $d$, has $n_s$ turns per unit length. It is centered inside and very long solenoid with radius, $D$, and $n_L$ turns per unit length.