#### The Continuity Equation
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Very important distinction: global versus local conservation of charge.

Charge in a volume \({\cal V}\): \[ Q_{\cal V} (t) = \int_{\cal V} d\tau~ \rho ({\bf r}, t) \label{Gr(8.1)} \] Current \({\bf J}\) flowing out through boundary \({\cal S}\) of \({\cal V}\): conservation of charge means \[ \frac{dQ_{\cal V}}{dt} = -\oint_{\cal S} d{\bf a} \cdot {\bf J} \label{Gr(8.2)} \] This means that \[ \int_{\cal V} d\tau \frac{\partial \rho}{\partial t} = -\int_{\cal V} d\tau {\boldsymbol \nabla} \cdot {\bf J} \label{Gr(8.3)} \] Since this is true for any volume, we have (re)derived the

Continuity equation conteq \[ \frac{\partial \rho}{\partial t} + {\boldsymbol \nabla} \cdot {\bf J} = 0 \]

Therefore, conservation of charge is a direct consequence of Maxwell's equations.

One thing to note: we have viewed \(\rho\) and \({\boldsymbol J}\) as sources (the ''right-hand side'') of Maxwell's equations. The continuity equation thus imposes a functional constraint on these sources: not any \(\rho\) and \({\boldsymbol J}\) will do the trick, only the ones what obey it.

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Author: Jean-Sébastien Caux

Created: 2022-03-24 Thu 08:42