Current continuity

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This fundamental law states that

The current flowing into or out of a closed surface in space must result in an increase or decrease in the amount of charge contained in the volume enclosed by the surface.

That is to say:

\oint_{\partial S}\textbf{J}\cdot d\textbf{S} = \frac{\partial}{\partial t}\int_S \rho\, dV.

The differential for of this is easy to derive from Maxwell's Equations if we consider

\nabla\cdot\nabla\times\textbf{H} = \nabla\cdot\textbf{J} + \nabla\cdot\frac{\partial\textbf{D}}{\partial t}

The divergence of a curl is always zero, so

0 = \nabla\cdot\textbf{J} + \frac{\partial}{\partial t} (\nabla\cdot\textbf{D}).

(Note: we have changed the order of divegence and time differentiation.) But, from Maxwell's Equations, we have

\nabla\cdot\textbf{D} = \rho

(the charge density), so we can write the charge continuity relationship as

\nabla\cdot\textbf{J} = -\frac{\partial\rho}{\partial t}.

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