A New Approach to Two-Dimensional Boundary Value Problems in Magnetostatics and Current Distributions

We present a new technique for solving two-dimensional boundary value problems (BVPs) in magnetostatics and current distributions. First, we prove a hitherto undiscovered analogy between magnetostatic fields in ferromagnetic media and the current density in a conductor. Next, we introduce a new quantity [Formula: see text], the stream function, and show that it is governed by a Poisson equation similar to those encountered in electrostatics while having a very intuitive physical meaning. This technique allows us to discover analytical expressions for various BVPs that previously required numerical methods. In particular, we calculate the actual resistance of a rectangular sheet and show that it is greater than the naive resistance predicted by the formula [Formula: see text]. This technique not only enables analytical solutions to previously intractable problems but also strengthens conceptual understanding for students by linking electromagnetic phenomena to visual and physical analogies, offering new pathways for teaching electrodynamics. Our technique also further highlights the similarities between electromagnetic fields and velocity fields in fluid mechanics, which has pedagogical uses in helping students understand electrodynamics.