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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$\begingroup$ Do you know how to do the same problem for an electric charge? $\endgroup$– knzhouCommented May 22, 2017 at 20:40
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$\begingroup$ Yes I do. What differences would there be between a magnetic monopole field and say, an electric field from a point charge? $\endgroup$– Servin VashnarCommented May 22, 2017 at 20:45
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$\begingroup$ None!$\hspace{0mm}$ $\endgroup$– knzhouCommented May 22, 2017 at 20:52
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1$\begingroup$ This is mostly a duplicate of Are the Maxwell's equations enough to derive the law of Coulomb? and Deriving Coulomb's Law from Gauss's Law - the different setting makes them look different, but at heart they are identical. $\endgroup$– Emilio PisantyCommented May 22, 2017 at 21:14
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2 Answers
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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:
- 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.
- 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.
