Relaxation time

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.

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

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Author: Lakshmi Prasanna Ponnala

Completed M.Tech in Digital Electronics and Communication Systems and currently working as a faculty.

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