#### Charge Conservation and the Continuity Equation
ems.ms.ce

As an aside, now that we have defined our volume current density vcurd, we can obtain one of the most important and useful equations in electrodynamics.

Consider a generic surface \({\cal S}\). The total current \(I\) going through it is \[ I = \int_{\cal S} {\bf J} \cdot d{\bf a} \] Specializing to a closed surface, we can use the divergence theorem to rewrite this as \[ \oint_{\cal S} d{\bf a} \cdot {\bf J} = \int_{\cal V} d\tau ~{\boldsymbol \nabla} \cdot {\bf J} \] Let us now invoke the principle of charge conservation. By its definition, the total current going through the surface is (minus) the change of charge contained inside the surface per unit time, \[ \int_{\cal V} d\tau ~{\boldsymbol \nabla} \cdot {\bf J} = -\frac{d}{dt} \int_{\cal V} d\tau ~\rho = - \int_{\cal V} d\tau ~\frac{\partial \rho}{\partial t}. \] Since this is valid for any volume, we get the

Continuity Equation

- FLS II (13.8)
- Gr (5.29)
- PM (4.10)

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

Although you might think that we've made assumptions (e.g. about the current being steady), it turns out (as we will see later after studying the Maxwell equations) that the continuity equation is a fundamental consequence of the basic laws of electrodynamics, and is always obeyed. It is a simple consequence of charge conservation and locality.

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Author: Jean-Sébastien Caux

Created: 2022-03-24 Thu 08:42