# Mutual inductances - are they always equal?

## Main Question or Discussion Point

I think not necessarily: consider two coaxial solenoids od significantly different diameters.
Practically the whole magnetic flux through the inner solenoid goes through the outer one, but not the other way around.

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mutual inductance is the same whichever coil is taken to be the primary or secondary.
The important quantity is not simply magnetic flux but magnetic flux LINKAGE.

LCS = LSC, but:

kCS does not equal kSC, for a coaxial cable, where kCS = center to shield coupling factor; & kSC = shield to center coupling factor.

So a coax has unequal flux coupling factors but I'll leave it as an exercise to the interested reader to find that the mutual inductance is simply the self inductance of the shield, LSS.

Claude

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anyone interested should first of all be clear about the (text book) definition of mutual inductance.
Without that we will get nowhere

AlephZero
Homework Helper
Practically the whole magnetic flux through the inner solenoid goes through the outer one, but not the other way around.
But if the bigger outer solenoid has a bigger self-inductance, it only takes a smaller fraction of its total magnetic field to give the same coupling effect.

The neat proofs that the two mutual inductances are equal use vector calculus - I don't know of a "simple" proof.

If the mutual inductances were different, you could construct a perpetual motion machine using that fact - but since PF doesn't discuss perpetual motion machines, I'm not going into any more details of how to do it It is possible to show equality by considering conservation of energy. The principle is central to analysing transformer action giving Vs/Vp = Ns/Np.
Too much to reproduce here !!! but available in any good A level/1st year degree text book

Page 310 to 311 of Introduction to Electrodynamics by Griffiths has complete explanation of $L_{12}=L_{21}$. I don't think you can always explain Maxwell's equation in a simple ( in easy English) way. You try too hard, you might misinterpret the equations. My suggestion is to look at the meaning of the double integral and learn to make sense of it. You need to be very good and study inside out of Vector Calculus, there is no short cut way. This is not for the weak of heart. I spent over two years studying Vector Calculus, studied and worked through the exercises in 3 different EM books before I think I get some feel of it. If there is an "ABC" way of explain EM, I have not found it yet. And I am by no means claiming I understand EM.

The mutual inductance involves a double integration respect to both loops, so both mutual inductance has the same formula.

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