Two insulated wires are wound on the same hollow cylinder, s as to from two solenoids shering a common air-filled core. Let `l` be the length of the core, A the cross -sectional area of the core, `N_(1)` the number of times the first wire is wound around the core, and `N_(2)` the number of times the second wire is wound around the core. Find the mutual inductance of the two solenoids, neglecting the end effects.

Explanation

If a current `I_(1)` flows around the first wire then a unfirom axial magnetic field of strength `B_(1) = (mu_(0)N_(1)I_(1))/(l)` is generated in the core. The magnetic field in the region outside the core is of negligible magnitude.The flux linked to a single turn of the second wire is `B_(1)A`. Thus, flux linked to all `N_(2)` turns of the second wire is
Now, `phi = N_(2)B_(1)A = (mu_(0)N_(1)N_(2)AI_(1))/(l) = MI_(1)`
`:. M = (mu_(0)N_(1)N_(2)A)/(l)`
As described pervisously, `M` is a genometric quantity depending on the dimensions of the core and the manner in which the two wires are wound around the core, but not no the actual actual currents flowing through the wires.