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

TL;DR

The displacement current’s role in classical magnetism is reframed by decomposing it into an internal singular component and an external field component through discal regularization of the dipole distribution. The external component yields an instantaneous Biot-Savart law in the Coulomb gauge, establishing a field-based link to magnetism that remains consistent with the Ampère-Maxwell law and resolves historical ambiguities. The approach is extended to polarization, yielding refinements to the Clausius-Mossotti equation via a Wigner-Seitz cell framework and a bulk polarizability, demonstrating broad applicability to dielectric media. The results support a unified view in which magnetism is driven by the external displacement current rather than directly by conduction current, while preserving a notion of current equivalence in steady-state conditions. This regularization-based framework provides a powerful tool for field decomposition and clarifies foundational aspects of electromagnetic source causality.

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 Ampère-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.

Figures (6)

• Figure 1: (a) Schematic of a particle with charge $q$ moving normally through the center of a circular surface $S$ and (b) its electric flux (blue), displacement current (red), and (c) total current (red) with the nonsingular electric flux (blue) along the moving path denoted by the $z$ axis.
• Figure 2: Schematic of a disk-like cylinder along the moving direction of a charged particle, formed by two circular cross-sectional surfaces at two moments of motion.
• Figure 3: Schematic of a cylinder geometry in the cylindrical regularization along one transverse direction of $x_{\mathrm{p}1}$ ($n$=1) and the axial direction of $x_{\mathrm{p}3}$ ($n$=3), respectively.
• Figure 4: Schematic of a spherical volume centered at $\mathbf{x}$ with a discal region $V_{\varepsilon}^{\prime\prime}$ on the tangent plane of the spherical boundary $S(R)$ at point of the contact $\mathbf{x}^{\prime\prime}$, where the normal disk $V_{\mathrm{d}}^{\prime\prime}$ is shown for comparison.
• Figure 5: Geometry of (a) closed current loops that form (b) a closed circuit B is illustrated, with a prepared loop $L_T$, distinguished by a different color, whose central axis passes the field point $\mathbf{x}$. (c) One cylindrical tube $T_0$ is centered at $\mathbf{x}$ with the disk $V_{\mathrm{d}}$ removed, and its enlarged sectional view is presented on the ($x^{\prime}_1$, $x^{\prime}_3$) plane.