Two paths towards circulation time derivative (Maxwell's $\mathfrak E$ revisited)

TL;DR

This paper compares two methods for computing the time derivative of circulation in vector fields over moving curves, revealing implications for Faraday's law and Maxwell's choice of electromotive intensity.

Contribution

It introduces and compares two approaches for calculating circulation derivatives, offering new insights into Maxwell's formulation of electromotive intensity.

Findings

Different methods lead to distinct conceptualizations of Faraday's law.

The analysis supports Maxwell's original choice of electromotive intensity.

Provides a new perspective on the mathematical foundations of electromagnetic induction.

Abstract

The time derivative of the circulation of a vector field $\boldsymbol{A}$ over a moving and deforming closed curve, $\frac {\mathrm{d}}{\mathrm{d} t}\oint \boldsymbol{A} \cdot \mathrm{d} \boldsymbol{r}$, is computed in two ways, with and without bringing the time derivative under the integral sign. As a by-product, the computations reveal that the conceptualization of Faraday's law of electromagnetic induction may depend on which of the two methods is employed. The discussion presented provides an unexpected argument in favor of Maxwell's mysterious choice for his electromotive intensity $\mathfrak E$, made in Article 598 of his Treatise.