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 i1 that flows through coil 1 will cause a magnetic flux through coil 1 Φ11 = i1 ⋅ L1 (according to the definition of inductance). However some of the flux will also flow through coil 2, designated Φ21.
Mutual inductance between coils 1 and 2 is then by definition:
M21 = Φ21 / i1
Obviously, the magnetic flux Φ21 is less or equal to Φ11, so we define a coupling coefficient k ≤ 1 so that:
Φ21 = k ⋅ Φ11
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:
Φ12 = k ⋅ Φ22
Now multiply both mutual inductances:
M21 M12 = k2 Φ11 / i1 Φ22 / i2
Use definition of inductance:
M2 = k2 L1 L2
M = k sqrt(L1 L2)
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 M21 = M12. However this way of deriving the formula seems dubious - from steps above you can also see that:
M21 = k Φ11 / i1
M12 = k Φ22 / i2
And since M12 = M21:
M = k L1 = k L2
L1 = L2
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
Atom feed
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.