Forces Between Conductors: The “50/50 Rule” About the Partition of Supplied Energy

We prove the 50/50 rule of Fig. 1 by a Lagrangian technique (all fields pulled back to the reference configuration at time 0) which exposes the necessity of two circumstances for the result to hold: 1) time-invariance of currents, when expressed in “material form” and 2) equality of (magnetic) coenergy and energy, which excludes nonlinear $\boldsymbol {B}$ – $\boldsymbol {H}$ laws. A dual result, which holds in case of time-invariance of “flux linkages” (but without the linearity restriction), is found along the way.