Relaxation time constant from continuity

Relaxation time constant

When free charge is suddenly placed inside a conducting (or lossy dielectric) material, it does not stay. It decays exponentially as the charge moves to the surface. The relaxation time constant T_r = ε/σ is the time in which the volume charge density inside the material falls to 1/e (about 36.8 %) of its initial value.

Derivation using the continuity equation

The point form of the continuity equation expresses charge conservation:

∇ · J = −∂ρ_v/∂t

For a conducting medium, Ohm's law in point form gives J = σE, and E = D/ε. Then

∇ · J = σ ∇ · E = (σ/ε) ∇ · D

By Gauss's law, ∇ · D = ρ_v. Therefore

(σ/ε) ρ_v = −∂ρ_v/∂t

∂ρ_v/∂t + (σ/ε) ρ_v = 0

This is a first-order differential equation. Its solution is

ρ_v = ρ₀ e^(−(σ/ε) t) = ρ₀ e^(−t/T_r)

where   T_r = ε/σ     (seconds)

Here ρ₀ is the charge density at t = 0.

Interpretation

• The charge density inside a conductor decays exponentially to zero. The charge that was inside appears as surface charge on the boundary of the conductor.
• Larger σ means faster decay. At t = T_r, ρ_v = 0.368 ρ₀.
• Good conductor (copper, σ = 5.8 × 10⁷ S/m, ε ≈ ε₀): T_r = 8.854 × 10⁻¹²/5.8 × 10⁷ ≈ 1.5 × 10⁻¹⁹ s. This is extremely short, so we may say no charge remains inside a conductor.
• Good insulator (e.g. fused quartz, σ ≈ 10⁻¹⁷ S/m): T_r is of the order of days to years, so charge placed in it stays for a long time.