Not All Uniform B-Fields Are The Same!

TL;DR

This paper explores how different source configurations can produce identical uniform magnetic fields but lead to different induction dynamics when the field is turned off, emphasizing the importance of the vector potential in understanding electrodynamics.

Contribution

It demonstrates that uniform magnetic fields from different sources have distinct causal structures and highlights the utility of the vector potential in analyzing induction phenomena.

Findings

Different source configurations can produce the same uniform B-field but have different dynamics when turned off.

The vector potential provides crucial information about the source of the magnetic field.

The study clarifies the causal structure of electrodynamics in a simple capacitor setup.

Abstract

Set up a charged rectangular-plate capacitor in a uniform $\mathbf{B}$-field. Quasi-statically turn off the $\mathbf{B}$-field - what happens? We show that this seemingly simple induction problem (which appears in a famous undergraduate electrodynamics textbook) highlights the causal structure of electrodynamics. It may be (as it is in this problem) that different source configurations can produce the same uniform $\mathbf{B}$-field over some finite region, but very different dynamics when the source is turned off. We see also that the vector potential gives more information about the source of the $\mathbf{B}$-field, and is a very useful (and under-appreciated) way to understand induction and other key aspects of electrodynamics.