Self-Force of a Dirac String: An Explicit Calculation

TL;DR

This paper explicitly calculates the self-force on a Dirac string modeled as a semi-infinite solenoid, demonstrating its divergence as the radius approaches zero, thus clarifying the string's singular nature.

Contribution

It provides a direct, elementary derivation of the self-force on a Dirac string, which was previously rarely explicitly calculated.

Findings

The self-force diverges as the solenoid radius approaches zero.

The force is given by $F=\Phi^2/(2\pi\mu_0 a^2)$, illustrating its dependence on flux and radius.

Explicit demonstration of the singular behavior of the Dirac string.

Abstract

A Dirac string can be modeled as a semi-infinite solenoid carrying a fixed magnetic flux. Dirac pointed out that such a string should experience a nonvanishing and divergent self-force, but explicit calculations are rarely shown. Motivated by a recent comment by McDonald, we present a direct and elementary derivation of this self-force. Treating the string as a stack of current loops, we compute the axial force produced by the radial magnetic field generated by the rest of the solenoid. The resulting force, $F=\Phi^2/(2\pi\mu_0 a^2)$, diverges as the solenoid radius $a\to0$ with flux $\Phi$ fixed, making explicit the singular nature of the Dirac string.