Maxwell's equations revisited -- mental imagery and mathematical symbols

TL;DR

This paper revisits Maxwell's equations through the lens of mental imagery, deriving them using vector calculus tools and historical context, offering a new perspective on their foundational development.

Contribution

It introduces a novel derivation of Maxwell's equations based on mental imagery and vector calculus, enriching the understanding of their conceptual origins.

Findings

Derivation of Maxwell's equations using mental imagery of fluid motion

Application of divergence, curl, and Poincare's lemma in derivation

Historical remarks on electrodynamic theory development

Abstract

Using Maxwell's mental imagery of a tube of fluid motion of an imaginary fluid, we derive his equations $\operatorname{curl} \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$, $\operatorname{curl} \mathbf{H} = \frac{\partial \mathbf{D}}{\partial t} + \mathbf{j}$, $\operatorname{div} \mathbf{D} = \varrho$, $\operatorname{div} \mathbf{B} = 0$, which together with the constituting relations $\mathbf{D} = \varepsilon_0 \mathbf{E}$, $\mathbf{B} = \mu_0 \mathbf{H}$, form what we call today Maxwell's equations. Main tools are the divergence, curl and gradient integration theorems and a version of Poincare's lemma formulated in vector calculus notation. Remarks on the history of the development of electrodynamic theory, quotations and references to original and secondary literature complement the paper.