Question

Define mobility of a charge carrier. What is its relation with relaxation time?

Answer 1

Hint: We need to find the definition for the mobility using the physical parameters that are related to the parameter mobility. Then we can find a relation between the relaxation time and the mobility that is observed in an ohmic conductor.

Complete answer:
A conductor is said to be conducting electric current through it when a particular quantity of charge can pass through it in a given interval of time. This charge flowing per unit time is defined as the electric current. Now, let us consider the movement of these charge carriers in the conductor.
Consider a metal conductor, we know that the flow of electrons is responsible for the current flow in them. We know that the metal consists of positive nuclei which we hinder the path of electrons during its motion. The average velocity or the drift velocity of the electron is the velocity attained by the electron which is dependent on the applied electric field and the relaxation time.
The relaxation time is the time interval between two successive collisions. Now, we can write the formula for the mobility as –
\[\mu =\dfrac{{{v}_{d}}}{E}\]
Where, \[\mu \] is the mobility of the electron, \[{{v}_{d}}\] is the drift velocity and E is the electric field applied. Mobility is defined at the velocity of the electron per unit electric field. It is the ease with which the electrons can travel in a conductor.
Now, let us write the mobility in terms of the relaxation time \[\tau \]as –
\[\begin{align}
& {{v}_{d}}=\dfrac{eE}{{{m}_{e}}}\tau \\
& \Rightarrow \mu =\dfrac{{{v}_{d}}}{E} \\
& \Rightarrow \mu =\dfrac{\dfrac{eE}{{{m}_{e}}}\tau }{E} \\
& \therefore \mu =\dfrac{e\tau }{m} \\
\end{align}\]
So, the mobility of the electrons is directly proportional to the relaxation time of the electron.
This is the required solution.

Note:
We can understand from the solution that the mobility and the relaxation time are directly related. When the collision happens after a longer interval, i.e., higher relaxation time, the mobility of the electrons is higher, i.e., they can move smoothly for more distance.