We consider a current-carrying wire loop made out of linear segments of arbitrary sizes and directions in three-dimensional space. We develop expressions to calculate its vector potential and magnetic field at all points in space. We then calculate the mutual inductance between two such (non-intersecting) piecewise-linear loops. As simple applications, we consider in detail the mutual inductance between two square wires of equal length that either lie in the same plane or lie in parallel horizontal planes with their centers on the same vertical axis. Our expressions can also be used to obtain approximations to the mutual inductance between wires of arbitrary three-dimensional shapes.
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This can lead to problems in numerical simulations that can be dealt with in one of two ways. One can program and the ensuing calculations using Eqs. (6) and (8), or, alternatively, one can use only Eq. (6), introducing a very small numerical deviation in case of trouble. For example, in the two-square problem in Sec. V, there are two wires on the same horizontal line when in Fig. 4. The numerical ambiguity can be solved by choosing one square of side L and the other of side with some very small δ (say, ).
One can recover Eq. (18) of Ref. 8 with the substitutions , , and noting that all j terms are equal in that case, meaning that the sum over j yields simply an overall factor of n. The greatly simplifying substitution appears because, in the case of two n-sided concentric polygons, symmetry implies that the integration along any side is the same. Eqs. (36)–(39) are more complicated than the corresponding Eqs. (15) in Ref. 8 precisely because here that simplification is absent.