Teaching Maxwell's Equations from 2D to 3D with Bivectors

Abstract

Electromagnetism is one of the few core physics topics without simple two-dimensional examples to start from: the cross product and curl require three dimensions. Previous work described magnetism as a bivector field, visualized with oriented (clockwise/counterclockwise) "tiles" rather than the traditional (pseudo)vector "arrows." Here, we express Maxwell's equations in this bivector language: magnetic flux is understood as a sum along a surface rather than through it, and the magnetic field tiles encircle the boundary of an Amperian loop or ribbon in a natural way. This allows a gentle two-dimensional starting point and makes symmetry arguments natural for magnetism.