Self-Force of a Dirac String: An Explicit Calculation

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, $F={\mathrm{\Phi }}^2\mathrm{/}(2\pi {\mu }_0{a}^2)$, diverges as the solenoid radius $a\to 0$ with flux $\mathrm{\Phi }$ fixed, making explicit the singular nature of the Dirac string.