Coulomb force between two Dirac monopoles

TL;DR

The paper demonstrates that Dirac monopoles experience a Coulomb-like force even when connected by a Dirac string. It models the string as a semi-infinite line of dipoles, derives the string field, and uses a force-based calculation to show that the interaction between two strings reduces to a Coulomb-like magnetic-force between charges $g_1$ and $g_2$. The resulting force magnitude is $F = rac{\mu_0}{4\\pi} rac{g_1 g_2}{r^2}$ along $\\hat{\\mathbf r}$, matching the familiar monopole picture. This work provides an accessible, undergraduate- and early graduate-level demonstration that reinforces core electromagnetism concepts and clarifies monopole dynamics in the Dirac-string framework.

Abstract

The model of magnetic monopoles that was proposed by Paul Dirac in 1931 has long been a subject of theoretical interest in physics because of its potential to explain the quantization of electric charge. While much attention has been given to non-Dirac monopoles, Dirac's model, which involves an infinitely thin solenoid known as a Dirac string, presents subtleties in the interaction between monopoles. In this paper, we show that the force between two Dirac monopoles obeys a Coulomb-like interaction law. This derivation offers an instructive exercise in fundamental electromagnetism concepts and is appropriate for undergraduate and early graduate-level students.

Figures (2)

• Figure 1: Dirac string consisting of a one dimensional semi-infinite line $\gamma$ along the $z$-axis that terminates at the origin and carries a magnetic flux $\Phi$. The field at any point is the superposition of the fields created by infinitesimal dipoles $d\mathbf m$ that point along the direction of $\gamma$. At any point outside the string, the field is that of a monopole, $\mathbf{B}_{\rm mon}$, of magnetic charge $g=\Phi/\mu_0$, which emanates in all directions from the end of the string.
• Figure 2: Two Dirac strings $\gamma_1$ and $\gamma _2$ with end points at the origin and at $\mathbf x$ respectively.