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Substituting the wire potential from part (b) into the flux formula from part (a) gives Neumann's mutual-inductance formulaInterchanging the two curves proves $L_{12}=L_{21}$.

Parametrize the coaxial circles byThen, with $\theta =\varphi -\psi$ and $R=\sqrt{a^2+b^2+c^2}$,where $q=2ab/R^2$. One angular integration contributes $2\pi$, soSince $ab=q{R}^2/2$, this is

The symmetry is electromagnetic reciprocity applied to mutual inductance. For thin closed circuits in vacuum, the magnetic vector potential generated by circuit B isStokes theorem converts the magnetic flux through circuit A into its magnetic vector potential circulation. Dividing by $I_B$ gives Neumann's mutual-inductance formula,The dot product and separation are unchanged by interchanging A and B, soEach has units $\mathrm{Wb}{\mathrm{A}}^{-1}=\mathrm{H}$. The winding orientations must be used consistently to determine the common sign. The same reciprocity holds in ordinary linear reciprocal magnetic media; it is not an unrestricted statement about biased nonreciprocal materials. Equivalently, a reversible magnetic energy function has equal mixed derivatives with respect to the two currents.