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Relaxation Time

If a charge is distributed within a conducting body, the charge will distributed itself in such a way that a zero electric field exists within and tangent to the conductor surface. The time required for this process is called the relaxation time.
Consider a homogeneous conduction region with a conductivity σ and permittivity ε. We have the equation of continuity,
            ∇ . j = - ∂ρ / dt.                                … (1)
But            j = σ E and ∇. E = ρ/ε.
. :            ∇. (σE- = σ ∇. E = σρ/ε = - ∂ρ/∂t.
Thus            σ/ε ∫ dt = - ∫ dρ/ρ
Or            ρ = ρ0 e–σt/ε                                 … (2)
Here, ρ0 is the initial charge density. The relaxation time tre is defined as
            tre = ε /σ,                                    … (3)
i.e., the time required for the initial charge distribution to decay to 1/e of its initial value.

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