Charge continuity equation

The charge continuity equation is the local statement of charge conservation:

$\nabla \cdot \mathbf{J}=-\frac{\partial {\rho }_v}{\partial t}.$

In words: the net outflow of current density from a region equals the rate at which charge density inside is decreasing. Charge cannot appear or disappear, only flow.

The integral form, by Divergence theorem:

${\oint }_S\mathbf{J}\cdot d\mathbf{s}=-\frac{d}{dt}{\int }_V{\rho }_{v}dv=-\frac{d{Q}_{\text{enc}}}{dt}.$

The current out of a closed surface equals the rate of decrease of enclosed charge. If ${\partial }_t{\rho }_v=0$ (static case), $\nabla \cdot \mathbf{J}=0$: current density is solenoidal in steady-state.

Why it’s automatic from Maxwell

Take the divergence of Ampère’s law with Displacement current:

$\nabla \cdot (\nabla \times \mathbf{H})=\nabla \cdot \mathbf{J}+\nabla \cdot \frac{\partial \mathbf{D}}{\partial t}.$

The left side is identically zero ($\nabla \cdot \nabla \times =0$ always). Swap the order of $\nabla \cdot$ and ${\partial }_t$ on the right, then use Gauss’s law $\nabla \cdot \mathbf{D}={\rho }_v$:

$0=\nabla \cdot \mathbf{J}+\frac{\partial {\rho }_v}{\partial t}.$

So the continuity equation is not an extra postulate. It follows automatically from Maxwell’s equations, specifically from the inclusion of the displacement current term. Without ${\partial }_t\mathbf{D}$ in Ampère’s law, charge wouldn’t be conserved.

Charge relaxation

In a conducting medium with $\mathbf{J}=\sigma \mathbf{E}$ and $\mathbf{D}=ϵ\mathbf{E}$, substitute into continuity:

$\nabla \cdot (\sigma \mathbf{E})=-\frac{\partial {\rho }_v}{\partial t}.$

For uniform $\sigma$, this is $\sigma \nabla \cdot \mathbf{E}=-{\partial }_t{\rho }_v$, and using $\nabla \cdot \mathbf{E}={\rho }_v/ϵ$:

$\frac{\sigma }{ϵ}{\rho }_v=-\frac{\partial {\rho }_v}{\partial t}.$

A first-order ODE for ${\rho }_v(t)$ at each point:

${\rho }_v(t)={\rho }_v(0)e^{-t/\tau },\tau =\frac{ϵ}{\sigma }.$

So any “extra” volume charge placed inside a conductor decays exponentially with relaxation time $\tau$.

• For copper: $\tau \approx 10^{-19}$ s, basically instantaneous.
• For mica: $\tau \approx 15$ hours.

After $\tau$, the bulk charge has moved to the surface of the conductor, leaving the interior charge-neutral (in steady state, $\mathbf{E}=0$ inside a conductor, consistent with this picture).

This is why “no free charge inside a conductor” is a sensible boundary condition for steady-state problems: even if you put some in, it leaves on a femtosecond timescale.

In context

The charge continuity equation is one of a family of “continuity equations” in physics. Same structure shows up in:

• Fluid mass conservation: $\nabla \cdot (\rho \mathbf{v})=-{\partial }_t\rho$.
• Probability conservation in quantum mechanics: $\nabla \cdot {\mathbf{J}}_{\text{prob}}=-{\partial }_t|\psi {|}^2$.
• Energy conservation in EM: $\nabla \cdot \mathbf{S}=-{\partial }_{t}u$ (Poynting’s theorem), where $\mathbf{S}$ is the Poynting vector and $u$ is energy density.

All share the form “divergence of flux = negative rate of change of density”: a conserved scalar can only redistribute, not appear or disappear.