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My goal is to show that a magnetic field generated by a magnetic monopole is $$-\frac{1}{2r^2}\hat{r}$$ but I'm having a little trouble doing so. I know that $$\vec{\nabla}\cdot\vec{B} = 4\pi \rho_M $$ where $\rho_M$ is the density of "magnetic charge". How do I go about deriving what the magnetic monopole field is? Furthermore, what is the value of $\rho_M$ here?

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$$\newcommand{\V}[1]{\mathbf{#1}} \rho_M(\V{r}) = e_M \delta(\V{r}) $$ where $e_M$ would be the charge of the monopole. The method is then exactly the same as for a point electric charge:

  1. use the integral form of the law $\V{\nabla}.\V{B}=4\pi\rho_M$, i.e. that

$$\iint_{S_r} \V{B}\ .\V{dS} = e_M$$

where $S_r$ is the sphere of radius $r$ centred on the origin.

  1. use the symmetry of the sources to argue that $\V{B}$ is radial and depends only on the radius $r$.
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I would start from the definition of a point magnetic dipole, which is a point that generates the magnetic field of a closed circular current at the limit r->0. In this limit, the dipole moment of the current must stay the same no matter the radius.

Next, consider that a magnetic monopole would violate Gauss's law for magnetism. The flux through a closed surface of the magnetic field generated by internal currents would stop being zero, and the divergence of the magnetic field would also stop being zero.

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