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If we ever obtain empirical confirmation of the existence of magnetic monopoles, how would Maxwell's classical equations be re-written? I'm assuming we'd only need to set the divergence of the magnetic field equal to some nonzero density as is the case with the electric field, but would there be something else we'd need to change? Would there be some sort of term we would be missing, such as was the case with the displacement current, which was initially missing in the equations?

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The existence of magnetic monopoles and therefor magnetic charge density and magnetic current density would mean that Maxwell's equations would become more symmetric. If we use the standard $\rho$ and $\vec{J}$ for the electric charge and current density, and $\eta$ and $\vec{K}$ for the magnetic charge and current density, Maxwell's equations would be $$\nabla\cdot \vec{E}=4 \pi \rho$$ $$\nabla \cdot \vec{B}=4 \pi \eta$$ $$\nabla \times \vec{E}=-\frac{4 \pi}{c}\vec{K}-\frac{1}{c}\frac{\partial \vec{B}}{\partial t}$$ $$\nabla \times \vec{B}=+\frac{4 \pi}{c}\vec{J}+\frac{1}{c}\frac{\partial \vec{E}}{\partial t}$$

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  • $\begingroup$ Thanks for the helpful comment! Would the existence of magnetic monopoles imply that magnetism is a phenomenon essentially separated from electricity, just like mass and electric charge seem not to have anything to do with eachother? $\endgroup$ Commented May 23 at 14:10
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    $\begingroup$ Not really- a magnetic current would create an electric field in the same way an electric current creates a magnetic field. We would have a Biot-savart-like law for finding the electric field made by "steady magnetic currents", and electric currents would still create magnetic fields. The equations in vacuum would be exactly the same too, and we would still have electromagnetic radiation. $\endgroup$
    – mike1994
    Commented May 23 at 14:21
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    $\begingroup$ @Lagrangiano It also worths to mention that the two fields are kind of the same thing in general relativity. A field that would be observed as purely electric in a co-moving reference would be seen as a combination of electric and magnetic field for someone moving at relativistic speeds with respect to it. They are two effects of the same phenomenon (the electromagnetic four-tensor, if you like)- hence the word "electromagnetic". $\endgroup$
    – Neinstein
    Commented May 24 at 7:58
  • $\begingroup$ I always liked to see Maxwell's equations written this way, with a subsidiary equation stating that the magnetic charge has always been determined by experiment and measurement to be zero. This is not a fundamental part of classical electromagnetism. (There are quantum reasons to believe that if magnetic monopoles exist at all, they are incredibly massive particles). $\endgroup$
    – nigel222
    Commented May 24 at 9:05
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    $\begingroup$ @Lagrangiano As you can see from the modified Maxwell's equations, magnetic and electric phenomena are coupled to each other. There is no separation. $\endgroup$ Commented May 24 at 10:21

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