Pre-Quantum Electrodynamics

• Gr 8.1.1

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

In fact, we can start directly from Maxwell's equations. Taking the divergence of AmpMax, \[ \nabla \cdot \nabla \times {\boldsymbol B} = \mu_0 \left (\nabla \cdot {\boldsymbol J} + \varepsilon_0 \nabla \cdot \frac{\partial {\boldsymbol E}}{\partial t} \right) \] and then using the fact that the divergence of a curl always vanishes, together with the differential form of Gauss's law, we directly get the continuity equation. Said otherwise: the conservation of charge is in fact 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.