Charge relaxation

Charge relaxation is the exponential decay of any free volume charge density placed inside a conductor (or imperfect dielectric) to its equilibrium value of zero in the bulk. The decay is governed by the relaxation time

$\tau =\frac{ϵ}{\sigma }\text{(s)},$

where $ϵ$ is the Permittivity and $\sigma$ is the conductivity of the medium. After a few $\tau$, all volume charge has migrated to surfaces, leaving the interior electrically neutral.

Derivation

Start with the Charge continuity equation:

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

In a conductor with point-form Ohm’s law $\mathbf{J}=\sigma \mathbf{E}$ (uniform $\sigma$) and Gauss’s law $\nabla \cdot \mathbf{E}={\rho }_v/ϵ$ (uniform $ϵ$):

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

Rearrange:

$\frac{\partial {\rho }_v}{\partial t}+\frac{\sigma }{ϵ}{\rho }_v=0.$

A first-order ODE for ${\rho }_v(\mathbf{r},t)$ at each point, with solution

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

Volume charge density at every interior point decays exponentially with the same time constant — regardless of the spatial pattern of initial charge. The spatial structure is preserved; only the magnitude shrinks.

Where the charge goes

The charge doesn’t vanish — charge is conserved. It migrates to the surface of the conductor, where the conductivity drops abruptly (or where the conductor meets an insulator). Once on the surface, it sits stationary in electrostatic equilibrium.

The picture: drop free volume charges into a conductor at $t=0$. The internal field pushes them outward. They drift until they reach a boundary, at which point they accumulate as a surface charge density. Total charge is unchanged; spatial redistribution is complete after a few $\tau$.

Numerical values

The dramatic spread of $\tau$ across materials:

Material	$\sigma$ (S/m)	${ϵ}_r$	$\tau =ϵ/\sigma$
Copper	$5.8\times 10^7$	1	$1.5\times 10^{-19}$ s
Silicon (intrinsic)	$4\times 10^{-4}$	11.8	$2.6\times 10^{-7}$ s
Distilled water	$10^{-4}$	81	$7\times 10^{-6}$ s
Glass	$10^{-12}$	5	44 s
Mica	$10^{-15}$	6	$5\times 10^4$ s ≈ 15 hours
Fused quartz	$10^{-17}$	3.8	$3.4\times 10^6$ s ≈ 40 days

For copper, relaxation is sub-femtosecond — faster than any electronic process. This is why “no free volume charge in a conductor” is a vacuum-tight approximation in essentially all electrical engineering: even if you somehow inject a bulk charge, it leaves on a timescale far shorter than any signal.

For glass or mica, relaxation takes minutes to days. Glass storing electrostatic charge (think of a charged piece of plexiglass holding electrons for an hour) is not a “conductor” on engineering timescales.

Why the timescale matters

Whether a material behaves as “conductor” or “dielectric” at a given frequency depends on whether the conduction current $\sigma \mathbf{E}$ or the displacement current ${\partial }_t(ϵ\mathbf{E})$ dominates — i.e., on the size of $\omega \tau$:

• $\omega \tau \ll 1$ (low frequency, or short $\tau$): conduction current dominates. Charge relaxes within a fraction of a period; the material acts as a conductor.
• $\omega \tau \gg 1$ (high frequency, or long $\tau$): displacement current dominates. The field oscillates faster than free charge can rearrange; the material acts as a (mostly lossless) dielectric.

The conduction loss tangent $\tan {\delta }_{\sigma }=\sigma /(\omega ϵ)=1/(\omega \tau )$ measures how much of an applied AC field is dissipated as ohmic loss vs stored reactively. Small $\tan \delta$ (large $\omega \tau$) means a low-loss dielectric for the conduction channel.

For example, distilled water at 60 Hz: $\omega \tau \approx 2.6\times 10^{-3}\ll 1$ — conduction dominates, the water acts like a conductor. At 10 GHz: $\omega \tau \approx 440\gg 1$ — displacement dominates and the conduction-relaxation channel is essentially lossless.

Caveat — microwave heating of water is a different mechanism. A microwave oven at 2.45 GHz puts water deep in the "$\omega \tau \gg 1$" conduction-relaxation regime, where the $\tau =ϵ/\sigma$ framework predicts low loss. The strong absorption used for cooking comes from a separate mechanism: dipolar (Debye) relaxation of the permanent water-molecule dipole reorienting in the oscillating field. That has its own relaxation time set by rotational diffusion (loss peak around 10–20 GHz at room temperature), modelled by ${ϵ}^{\ast }(\omega )={ϵ}_{\infty }+({ϵ}_s-{ϵ}_{\infty })/(1+j\omega {\tau }_D)$. The $\tau =ϵ/\sigma$ relaxation discussed in this note is the free-charge relaxation in a conductive dielectric; dipolar relaxation in a polar molecule is a distinct, complementary loss channel that this note does not derive.

Relation to RC time

The relaxation time is mathematically identical to the RC time constant of the same geometry:

${\tau }_{RC}=RC=\frac{ϵ}{\sigma },$

where $R$ and $C$ are the resistance and capacitance of the same two-conductor structure (see Resistance and Capacitance). This is the “RC product is geometry-independent” identity. The two viewpoints — internal relaxation vs. external circuit charging — give the same time constant.

For a parallel-plate capacitor with a slightly conductive dielectric: charging it up establishes $\mathbf{E}$ in the dielectric; the conductivity allows charge to leak through, so the voltage decays as $e^{-t/RC}=e^{-t\sigma /ϵ}$ — same expression as the bulk relaxation, because they’re the same physics expressed two ways.

Why “no free volume charge in a conductor” is a good approximation

The boundary condition ${\rho }_v=0$ inside a perfect conductor is justified by:

1. Mathematically: $\sigma \to \infty$ gives $\tau \to 0$ — any volume charge relaxes instantly.
2. Practically: for real conductors, $\tau$ is so short that on any engineering timescale, the interior is charge-neutral.

This is what lets us solve Laplace’s equation ${\nabla }^{2}V=0$ inside conductors (no source term) and apply Gauss’s law to confirm $\mathbf{E}=0$ inside, with all the source charge living on the surface as ${\rho }_s$.

In context

Charge relaxation is the bridge between Maxwell’s equations (which guarantee charge conservation via the Charge continuity equation) and the static boundary conditions of conductors (which assume ${\rho }_v=0$ inside). It also justifies the frequency split between “conductor” and “dielectric” — a single material is either, depending on $\omega \tau$.