# Relaxation time

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

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

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

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

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

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

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

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

where $T_{r}&space;=&space;\frac{\epsilon&space;}{\sigma&space;}$  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.

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

$T_{r}$  –  is the time constant known as the relaxation time (or) rearrangement time.

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