A reinterpretation of classical magnetism via the regular representation of displacement current

TL;DR

This paper offers a new interpretation of classical magnetism by decomposing the displacement current, leading to a reformulation of magnetic field generation as an effect of external displacement current, resolving longstanding confusion.

Contribution

It introduces a regularization method to decompose displacement current, reformulating magnetic fields via an external component and clarifying its role in classical magnetism.

Findings

Magnetic field can be expressed as an instantaneous Biot-Savart law involving external displacement current.

The internal displacement current cancels electric current, simplifying magnetic field interpretation.

The approach confirms the consistency of the Biot-Savart law with Maxwell's equations.

Abstract

The displacement current, introduced by Maxwell, has led to persistent confusion regarding its role in generating magnetic fields. To find a new way to understand classical magnetism, in this work, the displacement current is first decomposed into a localized internal part and an external field component by developing a discal regularization over the dipole distribution. Due to a surprising cancellation of the electric current by the internal displacement current, the magnetic field in the Coulomb gauge can be reformulated, in terms of the external component of the displacement current, as an instantaneous Biot-Savart law. The new expression allows for a reinterpretation of the generation of magnetic fields, viewing it as a magnetic effect related to the external displacement current, rather than directly to the electric current. The consistency of the Biot-Savart law with the Amp\`{e}re-Maxwell law is thus confirmed, eliminating a dogged dispute in history. This result offers a potential recognition of the displacement current enigma. The regularization method is also applied to the polarization model to develop and refine the Clausius-Mossotti equation within the special Wigner-Seitz primitive-cell regularization.