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I always thought that Maxwell equations in their differential form for B and E may be reformulated/updated to include a magnetic charge, even a hypothetical one. On the other hand, when a vector potential is introduced, and the Maxwell equations are reformulated in its terms, it looks like that only allows for no magnetic charge as B is defined as a rotor of the vector field and such a rotor may have only zero divergence.

Is that right? Or can a vector potential of electromagnetic field be used or updated to incorporate a magnetic charge?