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Dirac famously solved Maxwell equations in the presence of a point magnetic monopole. He was able to do so in a manner which used only the standard vector potential $\vec{A}$ and gave the correct monopole field $\vec{B}$ outside the monopole. What he effectively did was to consider the monopole to be one end of a very thin, semi-infinite solenoid which, by some clever mathematics of gauge transformation, remains invisible in all observables.

My question: is the Dirac solution specific to pointlike monopoles or is it possible to produce a continuous magnetic charge distribution using his method by superposition?

If Dirac solution is specific to pointlike monopoles, is there another way to describe a continuous magnetic charge distribution via vector potential $\vec{A}$? For example, what $\vec{A}$ corresponds to a uniformly charged magnetic ball of radius $R$ and magnetic charge $g$?

If one can extend Dirac solution to an arbitrary magnetic charge distribution, what happens to the invisible solenoids/strings? Do they behave nicely or do they entangle in some complicated way?

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No, there is no way to model continuous magnetic charge densities via Dirac monopoles. Dirac argument necessarily quantizes the allowed magnetic charges, and, crucially, at the position of the monopole, the vector potential $A$ is not defined.

The condition that the magnetic four-current vanishes is essentially the necessary condition for the vector potential $A$ to exist in the first place, see e.g. this answer of mine. See also this answer of mine for other options to "add" magnetic charges to standard electromagnetism.

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