The Ampere-Maxwell Law is derived from the $\nu=j$ (spatial) components of the covariant inhomogeneous Maxwell's equation, $\partial_\mu F^{\mu \nu}=\mu_0 K^\nu$. Analyzing these spatial components relates the curl of the magnetic field ( $\nabla \times B$ ) to the two sources of the magnetic field: the conduction current density ( $J$ ) and Maxwell's added term, the displacement current ( $\epsilon_0 \frac{\partial \bar{F}}{\partial t}$ ), which accounts for the time rate of change of the electric field. This derivation confirms that the compact tensor equation encapsulates the complete Ampère-Maxwell Law in its spatial components.
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