Generalize the laws of relativistic electrodynamics (Eqs. 12.127 and 12.128) to include magnetic charge. [Refer to Sect. 7.3.4.]
\frac{\partial F^{\mu \nu}}{\partial x^{\nu}}=\mu_{0} J^{\mu}, \quad \frac{\partial G^{\mu \nu}}{\partial x^{\nu}}=0 (12.127)
K^{\mu}=q \eta_{\nu} F^{\mu \nu} (12.128)
Define the electric current 4-vector as before: J_{e}^{\mu}=\left(c \rho_{e}, J _{e}\right) , and the magnetic current analogously: J_{m}^{\mu}=\left(c \rho_{m}, J _{m}\right) . The fundamental laws are then
\partial_{\nu} F^{\mu \nu}=\mu_{0} J_{e}^{\mu}, \quad \partial_{\nu} G^{\mu \nu}=\frac{\mu_{0}}{c} J_{m}^{\mu}, \quad K^{\mu}=\left(q_{e} F^{\mu \nu}+\frac{q_{m}}{c} G^{\mu \nu}\right) \eta_{\nu}.
The first of these reproduces \nabla \cdot E =\left(1 / \epsilon_{0}\right) \rho_{e} \text { and } \nabla \times B =\mu_{0} J _{e}+\mu_{0} \epsilon_{0} \partial E / \partial t , just as before; the second yields \nabla \cdot B =\left(\mu_{0} / c\right)\left(c \rho_{m}\right)=\mu_{0} \rho_{m} \text { and }-(1 / c)[\partial B / \partial t+ \nabla \times E ]=\left(\mu_{0} / c\right) J _{m} \text {, or } \nabla \times E =-\mu_{0} J _{m}-\partial B / \partial t (generalizing Sec. 12.3.4). These are Maxwell’s equations with magnetic charge (Eq. 7.44). The third says
\left. \begin{matrix} \text { (i) } \nabla \cdot E =\frac{1}{\epsilon_{0}} \rho_{e}, & \text { (iii) } \nabla \times E =-\mu_{0} J _{m}-\frac{\partial B }{\partial t}, \\ \text { (ii) } \nabla \cdot B =\mu_{0} \rho_{m}, & \text { (iv) } \nabla \times B =\mu_{0} J _{e}+\mu_{0} \epsilon_{0} \frac{\partial E }{\partial t} . \end{matrix} \right\} (7.44)
K^{1}=\frac{q_{e}}{\sqrt{1-u^{2} / c^{2}}}[ E +( u \times B )]_{x}+\frac{q_{m}}{c}\left[\frac{-c}{\sqrt{1-u^{2} / c^{2}}}\left(-B_{x}\right)+\frac{u_{y}}{\sqrt{1-u^{2} / c^{2}}}\left(-\frac{E_{z}}{c}\right)+\frac{u_{z}}{\sqrt{1-u^{2} / c^{2}}}\left(\frac{E_{y}}{c}\right)\right],
K =\frac{1}{\sqrt{1-u^{2} / c^{2}}}\left\{q_{e}[ E +( u \times B )]+q_{m}\left[ B -\frac{1}{c^{2}}( u \times E )\right]\right\} , or
F =q_{e}\left[ E +( u \times B ]+q_{m}\left[ B -\frac{1}{c^{2}}( u \times E )\right]\right.,
which is the generalized Lorentz force law (Eq. 7.69).
F =q_{e}( E + v \times B )+q_{m}\left( B -\frac{1}{c^{2}} v \times E \right) (7.69)