Not All Uniform B-Fields Are The Same!

Set up a uniform B-field in the z-direction and place a rectangular-plate capacitor with its plates parallel to the B-field. Quasistatically turn off the B-field – what happens? We show that this seemingly simple and innocuous question which appears in a famous undergraduate E&M textbook hides behind it several things about the nature of E&M which must overturn our attitude regarding the magnetic vector potential. These insights are normally obscured by the standard formulation of electrodynamics in terms of the E and B fields, and this standard formulation hinders students and professional physicists alike from a fully physical understanding of electromagnetic induction. We resolve this problem by introducing the vector potential and showing the physical nature which it captures and that the B-field leaves out. Additionally, the physical significance of gauge transformations will be discussed. This will demonstrate by a purely classical argument that the vector potential is a real specification of the B-field, and that not all uniform B-fields are the same! This theoretical insight will have applications to the Hall effect in condensed matter physics.