📢How One Elegant 4D Equation Unifies Gauss's Law and the Ampère-Maxwell Law

The unification of Maxwell's equations into a single, elegant fourdimensional relativistic tensor equation: $\partial_\mu F^{\mu \nu}=K^\nu$. This equation unifies the two inhomogeneous Maxwell's equations-Gauss's Law ( $\nabla \cdot F =\rho / \epsilon_0$ ) and the Ampère-Maxwell Law ( $\nabla \times B =\mu_0 J+\mu_0 \epsilon_0 \frac{\partial F}{\partial t}$ )-which are collectively called "inhomogeneous" because they are directly sourced by charge density ( $\rho$ ) and current density ( J ), embedded in the four-current density $K^\nu$. Specifically, the $\nu=0$ (time) component of the tensor equation yields Gauss's Law, while the $\nu=j$ (spatial) components yield the Ampère-Maxwell Law, which relates the curl of the magnetic field to both current and the time-changing electric field (displacement current). This relativistic formulation, which embeds the electric (E) and magnetic (B) fields into the electromagnetic field tensor $F^{\mu \nu}$ and uses the four-dimensional spacetime coordinate $x^\mu$, reveals that electromagnetism is inherently consistent with Special Relativity.

How One Elegant 4D Equation Unifies Gauss's Law and the Ampère-Maxwell Law | Audiosviadean on Notion