Relaxation time

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Relaxation time $(T_{r})$ :-

from the equation $\overrightarrow{J}=\sigma \overrightarrow{E}$ and from Gauss’s law $\overrightarrow{\bigtriangledown }.\overrightarrow{E} = \frac{\rho _{v}}{\epsilon }$ .

continuity equation is $\overrightarrow{\bigtriangledown }.\overrightarrow{J} = -\frac{\partial \rho _{v}}{\partial t }$ .

$\overrightarrow{\bigtriangledown }.(\sigma \overrightarrow{E}) = -\frac{\partial \rho _{v}}{\partial t }$ .

$\sigma( \overrightarrow{\bigtriangledown }. \overrightarrow{E}) = -\frac{\partial \rho _{v}}{\partial t }$ .

$\sigma(\frac{\rho _{v}}{\epsilon }) = -\frac{\partial \rho _{v}}{\partial t }$ .

$\therefore \frac{\partial \rho _{v}}{\partial t }+(\frac{\sigma}{\epsilon })\rho _{v} = 0$ .

The solution to the above equation is of the form $\rho _{v} = \rho _{vo} \ e ^{-\frac{t}{T_{r}}}$ .

where $T_{r} = \frac{\epsilon }{\sigma }$ is known as relaxation time and defined as the time it takes a charge placed in the interior of a material to drop to $e^{-1}$ = 36.8 percent of it’s initial value.

$\rho _{vo}$ is the initial charge density (i.e, $\rho _{v}$ at t=0) the equation $\rho _{v} = \rho _{vo} \ e ^{-\frac{t}{T_{r}}}$ shows that as a result of introducing charge at some interior point of the material there is a decay of volume charge density $\rho _{v}$ this decay is associated with the charge movement from the interior point at which it was introduced to the surface of the material.