Mutual Inductance

One book every physics or engineering student must have in his collection is Aptitude Problems in Physics which is a collection of Moscow Physics Olympiad problems edited by S.S Krotov. The problems in the book are absolute gems. I will post a few questions from the Electricity and Magnetism section.

Problem:

Two long cylindrical coils with uniform winding of the same length and nearly the same radius have inductance $L_{1}$ and $L_{2}$. The coils are co-axially inserted into each other and connected to a current source. The directions of the current in the coils is such that the fluxes add . Determine the inductance L of such a composite coil.

Solution:

From the definition of inductance we have $\displaystyle L=\frac{N\phi}{I}$

$\displaystyle\phi =\frac{MMF}{Reluctance}=\displaystyle\frac{NIA}{\mu_{o}l}\implies L_{1}=kN_{1}^2$  and $L_{2}=kN_{2}^2$

When the coils are coaxially combined with flux additive composite inductance

$L_{c}=k(N_{1}+N_{2})^2$

$L_{c}=\displaystyle k\left(\sqrt{\frac{L_{1}}{k}}+\sqrt{\frac{L_{2}}{k}}\right)^2=L_{1}+L_{2}+2\sqrt{L_{1}L_{2}}$