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Assume that the Dirac string is lying along the negative $z$-axis, and is subject to a magnetic field $B$. Assume throughout this question that we are considering a static situation. The force on the Dirac string is given by

$$F=g\int_{-\infty}^0 dz'\nabla '(e_z\cdot B)$$

Assuming a) $B$ vanishes at infinity i need to show $$F_z=gB_z(x=0,y=0,z=0)$$ and also b) $$\nabla \times B = 0 $$ and c) in general

$$F=gB(x=0,y=0,z=0)$$

But isnt the first question obvious if $e_z \cdot B = B_z$ and the integral and del operator cancels. And this will indicate the third question being valid too?

Tomy
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  • Can you please provide a bit more context? I thought that the Dirac string is an inevitable singularity of the vector potential $A$ that satisfies $\nabla \times A=B$ when one assumes that $B$ is a magnetic monopole in the origin. In particular $B$ has a singularity in $(0,0,0)$ and $A$ has it on, say, the entire negative $z$-axis. What force could act on that singularity? – Kurt G. Apr 27 '23 at 13:29
  • That's something i also dont understand. I got this from my professor. I went over some pdfs on google on the subject but couldn't find anything – Tomy Apr 27 '23 at 15:27
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    In this answer the vector potential $A$ whose curl is a Coulomb field is called $F$ and explicitly discussed. It has the singularity on the negative $z$ axis which is called the Dirac string in physics. A book where it is discussed is M. Nakahara, Geometry, Topology and Physics. Nowhere so far I have read that the Dirac string is an object a force could act on. You may want to get back to your professor and seek clarity. – Kurt G. Apr 27 '23 at 16:05

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