## Mutual inductance problem

A few days ago I was browsing through my notes from the first year of study and I stumbled upon this interesting problem I've never managed to solve. I've discussed this a couple of times with the late professor Valenčič and we couldn't find a flaw in my line of reasoning. So, if you know what's wrong, please drop me a mail.

In professor's book, there is an introduction to the principles of mutual inductance that goes like this:

Imagine two coils (designated 1 and 2) in some arbitrary relative position to each other. Current i_{1} that flows through coil 1 will cause a magnetic flux through coil 1 Φ_{11} = i_{1} ⋅ L_{1} (according to the definition of inductance). However some of the flux will also flow through coil 2, designated Φ_{21}.

Mutual inductance between coils 1 and 2 is then by definition:

M_{21} = Φ_{21} / i_{1}

Obviously, the magnetic flux Φ_{21} is less or equal to Φ_{11}, so we define a coupling coefficient k ≤ 1 so that:

Φ_{21} = k ⋅ Φ_{11}

Now due to principle of reciprocity, the same holds true if the current flows through coil 2 and we calculate the flux through coil 1. Coupling coefficient stays the same:

Φ_{12} = k ⋅ Φ_{22}

Now multiply both mutual inductances:

M_{21} M_{12} = k^{2} Φ_{11} / i_{1} Φ_{22} / i_{2}

Use definition of inductance:

M^{2} = k^{2} L_{1} L_{2}

M = k sqrt(L_{1} L_{2})

Now this final formula is present in a lot of literature and it's certainly correct. It's also certainly true that from the principle of reciprocity M_{21} = M_{12}. However this way of deriving the formula seems dubious - from steps above you can also see that:

M_{21} = k Φ_{11} / i_{1}

M_{12} = k Φ_{22} / i_{2}

And since M_{12} = M_{21}:

M = k L_{1} = k L_{2}

L_{1} = L_{2}

Which would mean all pairs of inductances are equal, which is certainly false since we didn't impose any restriction on the geometry of the two coils.

If I would have to guess, I would say something is wrong with the first application of the reciprocity to the fluxes through the coils, however from what I know this should be correct.

So yeah, if you are fluent in electromagnetic theory, I would love to hear your opinion.

*Update: see this follow-up post for a solution to this problem*

I don't know if someone answered this already, since it was from a long time ago, but taking a look at it, I would say that the assumption that the coupling coefficient is the same is a little dubious.

Take for instance a configuration in which a smaller loop in sitting on the same plane inside a larger loop. If current is put through the larger loop, only a portion of the magnetic flux goes through the smaller loop, making k less than one. However if current is put through the smaller loop, all of this magnetic flux would be within the larger loop, making k = 1.