Self-Force of a Dirac String: An Explicit Calculation

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, $F=Φ^2/(2πμ_0 a^2)$, diverges as the solenoid radius $a\to0$ with flux $Φ$ fixed, making explicit the singular nature of the Dirac string.

Figures (1)

• Figure 1: Semi-infinite cylinder represented as a stack of current loops. Arrows indicate the direction of the circulating current of magnitude $I$. The integrals to find $A(a,z)$ and $A(a,z+dz)$ are nearly identical; they differ only due to the inclusion in the second integral of the ring located in the range $z = 0 \rightarrow dz$. Since $B_\rho=(\nabla\times\mathbf{A})_\rho=-\partial A_\phi/\partial z$, the magnitude of the radial component of the magnetic field at height $z$ on the surface of the semi-infinite cylinder, $B_\rho(a,z)$, is equal to the magnitude of the vector potential $A_\phi(a,z)$ produced by the ring at the bottom of the cylinder.