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Section 45.3 Conservation of Charge

The principle of conservation of charge is automatically included in Maxwell's equations. To see it explicitly we can use the point form of Maxwell's equations. First, let us take the divergence of both sides of the Ampere-Maxwell's law and then use the Gauss's law for the electric field.

\begin{align*} \vec{\nabla}\cdot(\vec{\nabla}\times \vec B) \amp = \vec{\nabla} \cdot \left(\mu_0 \vec J + \mu_0\epsilon_0 \frac{\partial \vec E}{\partial t} \right)\\ \amp = \mu_0 \vec{\nabla} \cdot \vec J + \mu_0\epsilon_0 \frac{\partial }{\partial t}(\vec{\nabla}\cdot \vec E)\\ \amp = \mu_0 \left( \vec{\nabla} \cdot \vec J + \frac{\partial \rho}{\partial t} \right) \end{align*}

Since, the divergence of a curl is zero, the left side is identically zero. Therefore, we find that

\begin{equation} \vec{\nabla} \cdot \vec J + \frac{\partial \rho}{\partial t} = 0. \label{eq-eqn-continuity-1}\tag{45.3.1} \end{equation}

This equation is called the equation of continuity, which is a statement of conservation of charge at each space point. We can see this fact by integrating =over a volume, \(V\text{,}\) which is enclosed by a close surface \(S\text{.}\)

\begin{equation*} \int_V \vec{\nabla} \cdot \vec J\, dxdydz = - \frac{\partial }{\partial t} \int_V \rho\, dxdydz. \end{equation*}

The left side is the flux of charges out of the surface and the right side is the rate at which net charge within the volume, \(Q_v\text{,}\) decreases with time.

\begin{equation*} \oint_S \vec J\cdot d\vec A = - \dfrac{d}{dt}Q_V. \end{equation*}

That is, amount of charges leaving the volume due to current through the enclosing surface in some time \(\Delta t\) must equal the net decrease in charge within the volume in that time. Thus, net electric charge can neither be created or destroyed.