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It is known that if we take into account the magnetic charges, Maxwell's equations acquire a symmetrical form (Jackson, 3rd edition, eq. 6.150):

\begin{align} \begin{aligned} -\nabla \times \mathbf{E} &=\frac{\partial \mathbf{B}}{\partial t} + \mathbf J_m,& \nabla \cdot \mathbf{B} &=\rho_m, \\ \nabla \times \mathbf H &=\frac{\partial \mathbf D}{\partial t} +\mathbf J_e,& \nabla \cdot \mathbf{D} &=\rho_e. \end{aligned} \end{align}

But what about "magnetic Lorentz" force? Should it be $q_m(\mathbf H-\mathbf v \times \mathbf D)$, or $q_m(\mathbf B-\mathbf v \times \mathbf E)$, or something else?

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  • $\begingroup$ See en.wikipedia.org/wiki/Magnetic_monopole. The force is $q_m(\vec{B} - \vec{v} \times \vec{E})$ (in natural units). $\endgroup$ – gj255 Feb 5 at 12:53
  • $\begingroup$ In vacuum, indeed. But shouldn't we use $\vec H$ instead of $\vec B$ and/or $\vec D$ instead of $\vec E$ in case of field in matter? $\endgroup$ – warlock Feb 5 at 13:32
  • $\begingroup$ The force on a charge is always governed by $E$ and $B$. The fields $H$ and $D$ are merely convenient objects for solving Maxwell's equations in the presence of both free and bound charges and currents. Magnetic monopoles aside, you know that at the end of the day, it's the $E$ and $B$ fields that determine how electric charges move. The same is true of magnetic charges. $\endgroup$ – gj255 Feb 27 at 23:14

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