Friday, 20 July 2018

electromagnetism - When mutual inductance are occurring between two coils, why the two coils have the same mutual inductance?


I learned that ϵ1=M12di2dt

ϵ2=M21di1dt


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

This can be written as the differential du=L1d(i21/2)+M12i1di2+L2d(i22/2)+M21i2di1
Now integrate from i1(t=0)=i2(t=0)=0 to I1,I2 and get U=12L1I21+M12I1I2+12L2I22+(M21M12)i2di1
.


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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