I learned that ϵ1=−M12di2dt
And the book tells us directly that M12=M21 without a reason. Is there a mathematical proof for this?
Answer
An elegant and elementary derivation is given by Crawford:"Mutual inductance M12=M21", American Journal of Physics, vol 60 , February 1962, p186. The idea is the following: the total stored energy rate, "power" is dudt=L1i1di1dt+M12i1di2dt+L2i2di2dt+M21i2di1dt
In general, the integral ∫i2di1 can be anything and will depend on the history of the currents, and thus U is not necessarily single valued function of the present currents I1,I2 unless M21=M12! Now you can see that for a simple medium we must have equality of the mutual inductances.
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