Electromagnetic Force and the Maxwell Stress Tensor in Condensed Systems

TL;DR

This paper provides a clear, pedagogical derivation of the Maxwell stress tensor and electromagnetic force in condensed media, clarifying their validity, addressing misconceptions, and illustrating their application to strongly polarizable systems like ferrofluids.

Contribution

It offers a comprehensive, accessible derivation of key electromagnetic quantities in condensed media, clarifying their scope and addressing common misconceptions.

Findings

Derived the Maxwell stress tensor from fundamental principles.

Clarified the range of validity for various electromagnetic force expressions.

Demonstrated applications to ferrofluid systems.

Abstract

While the electromagnetic force is microscopically simply the Lorentz force, its macroscopic form is more complicated, and given by expressions such as the Maxwell stress tensor and the Kelvin force. Their derivation is fairly opaque, at times even confusing, and their range of validity all but a well kept secret. These circumstances unnecessarily reduce the usefulness and trustworthiness of some key quantities in macroscopic electrodynamics. This article presents a thorough yet pedagogical derivation of the Maxwell stress tensor and electromagnetic force in condensed media. It starts from universally accepted inputs: conservation laws, thermodynamics and the Maxwell equations. Simplifications are considered for various limits, especially the equilibrium, with a range of validity assigned to each expression. Some widespread misconceptions are scrutinized, and hidden ambiguities in popular notations revealed. A number of phenomena typical of strongly polarizable systems, especially ferrofluid, are then considered. In addition to enhancing the appreciation of these systems, it helps to solidify the grasp of the introduced concepts and derived formulas, and it demonstrates the ease with which the Maxwell stress tensor can be handled, inviting theorists and experimentalists alike to embrace this useful quantity.