Name

Without magnetic monopoles

With magnetic monopoles

Gauss’s law

$\nabla \cdot E=4\text{π}{\rho }_e$

Gauss’s law for magnetism

$\nabla \cdot B=0$

$\nabla \cdot B=4\text{π}{\rho }_m$

Faraday’s law of induction

$-\nabla \times E=\frac{1}{c}\frac{\partial B}{\partial t}$

$-\nabla \times E=\frac{1}{c}\frac{\partial B}{\partial t}+\frac{4\text{π}}{c}{j}_m$

Ampere’s law (with Maxwell’s extension)

$\nabla \times B=\frac{1}{c}\frac{\partial E}{\partial t}+\frac{4\text{π}}{c}{j}_e$

Lorentz force law

$F=q_e\left(E+\frac{v}{c}\times B\right)$

$F=q_e\left(E+\frac{v}{c}\times B\right)+q_m\left(B-\frac{v}{c}\times E\right)$