The correct form of the interaction energy between two current carrying circuits was much debated in the 1870s by Maxwell and the German theoretical physicists Carl Neumann, Wilhelm Weber, and Hermann Helmholtz. If ds₁ is a line element of circuit 1 and ds₂ is a line element of circuit 2, all of their proposals can be described as different choices for a constant k in an expression for the mutual inductance of the two line elements:

Not obviously, the value of k cannot be determined by force measurements on real circuits because the integration of d²M₁₂ over two closed loops produces a formula which does not depend on k. Show this is true by applying Stokes’ theorem (twice) to the second term in the curly brackets.

dM12 = po 1+ k ds === ((+^) ds - $ + ( =^)= 47 2 |r - 1- k ds (r r)(r r). ds2 |r r