Mutual Inductance of Coaxial Coils: Is M12 Always Equal to M21?

[unscientific]

Homework Statement

Not sure if I'm correct or wrong..because usually M₁₂ = M₂₁

The Attempt at a Solution

Here coil 1 and coil 2 are coaxial, with coil 2 being shorter and having smaller area.

 

[rude man]

M₂₁ always = M₁₂. That can be shown by an energy argument.

It's correct I think to say that
M₂₁ = N₂phi₁/i₁. That computes to
M₂₁ = N₂A₂μn₁
where N₂ is the number of turns in the inside coil 2. This agrees with your expression of M₁₂ (I think you reverse the conventional sequence of the subscripts, but OK).

This is because the flux inside coil 2 is clearly very nearly the same as that inside coil 1.

It's more difficult to argue in the same manner to derive M₁₂ because the question is how much of the flux generated by coil 2 really couples into coil 1.

The answer however has to be such that M₁₂ = M₂₁. I have looked at several derivations of mutual inductance of various coil configurations and in each case they pick the easier computation for M_ij and then assume (correctly) that M_ji = M_ij.