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M12 = M21= M (say) (6.14) We have demonstrated this equality for long co-axial solenoids. However, the relationship is far more general. Note that if the inner solenoid was much shorter than (and placed well inside) the outer solenoid, then we could still have calculated the flux linkage N1Φ1 because the inner solenoid is effectively immersed in a uniform magnetic field due to the outer solenoid. In this case, the calculation of M12 would be easy. However, it would be extremely difficult to calculate the flux linkage with the outer solenoid as the magnetic field due to the inner solenoid would vary across the length as well as the cross-section of the outer solenoid. Therefore, the calculation of M21 would also be extremely difficult in this case. The equality M12=M21 is very useful in such a situation

here M12 and M21 are the flux linkage constnats, which is, unless I'm mistaken henry's constant.

This is what my book has to say on this topic. However, they have only derived this relation assuming both solenoids are for all practical purposes, of the same length. Why is this true even when the other solenoid is much shorter than the inner solenoid?

Does this have anything to do with the fact that the magnetic field outside a solenoid is approximately zero?

Moreover, is this also true if the solenoids intersect at say an angle?

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No, it doesn't need to assume the fields outside the solenoid are zero.

The use of long solenoids is convenient because you can compute the flux easily and do the integrals easily. But you could also take two loops separated from each other and if you calculated the mutual inductance you would also find M21 = M12, or if you had some crazy geometry for two different circuits and if you were able to analytically, or numerically calculate it you would also find M12=M21. This is called the principle reciprocity.

This set of course notes goes through the proof of the reciprocity theorem and comes up with Newmann's Formula that shows they are equal. To prove the reciprocity theorem it is helpful to use the concept of magnetic vector potential. This allows you to use Stoke's theorem replace the the flux going through a small surface area with a line integrals.

So you could have your coils at some angle or intersecting, but the geometry because it is not co-axial would make it harder to solve. In general as you change the geometry you would also change the mutual inductance.

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