🧄Verifying the Inhomogeneous Maxwell's Equations in Spacetime

The verification confirms that the single four-dimensional tensor equation $\partial_\mu F^{\mu \nu}=K^\nu$ successfully unifies two of Maxwell's equations. By analyzing the $\nu=0$ (time) component, we derive Gauss's Law ( $\nabla \cdot E =\rho / \varepsilon_0$ ), which relates the divergence of the electric field to the charge density. By analyzing the $\nu=j$ (spatial) components, we derive the Ampère-Maxwell Law ( $\nabla \times B=\mu_0 J+\mu_0 \varepsilon_0 \frac{\partial E}{\partial t}$ ), which relates the curl of the magnetic field to both the current density and the time rate of change of the electric field. These two are collectively known as the inhomogeneous Maxwell's equations because they are sourced by the charge and current densities embedded in the four-vector $K^\nu$, demonstrating the elegant and compact nature of electromagnetism within the framework of special relativity.

Verifying the Inhomogeneous Maxwell's Equations in Spacetime | Proof and Derivationviadean on Notion