Question

Establish the relation between drift velocity and electric field. What is mobility? Describe the relation between mobility and velocity.

Answer 1

Relation between Drift Velocity and Potential Difference
Suppose PQ is a conductor of length l and a potential difference V is applied across PQ as shown in the figure 5.5. i.e., its P end remains negative and Q end positive. Therefore an electric field is developed directing from Q to P and its intensity is given by
$E=\frac{V}{l}$ .........(1)
Each free electron of the conductor experiences a force (F) towards the end Q,
$\therefore \overrightarrow{F}=-e\overrightarrow{E}$
If m be the mass of the electron, then its acceleration
$\overrightarrow{a}=\frac{\overrightarrow{F}}{m}$
or $\overrightarrow{a}=\frac{-e\overrightarrow{E}}{m}$
$\because$ The average thermal velocity of electrons is zero.
$\therefore u=0$
Therefore using first equation of motion ($v=u+\alpha t$), we have
$\overrightarrow{v_d}=0+(\frac{-e\overrightarrow{E}}{m})\tau$
or $\overrightarrow{v_d}=(\frac{-e\overrightarrow{E}}{m})\tau$ ............(2)
Numerically we can write,
$|\overrightarrow{v_d}|=\frac{eE}{m}\tau$
$\because$ From equation (1), $E=\frac{V}{l}$
$\therefore$ $v_d=\frac{e\tau }{m}\frac{V}{l}$ ..............(3)
Mobility
In the expression of drift velocity (speed), the term $\left(\frac{e\tau }{m}\right)$ is constant because e and m are constant and $\tau$ is the characteristics of specific metal, so for specific metal at constant temperature $\tau$ is constant. Constant term $\left(\frac{e\tau }{m}\right)$ for a given metal is called mobility $\left(\mu \right)$.
$\overrightarrow{v_d}=-\frac{e\tau }{m}\overrightarrow{E}$
$\overrightarrow{v_d}=-\mu \overrightarrow{E}$ ..........[1(a)]
$|\overrightarrow{v_d}|=\mu |\overrightarrow{E}|$
$v_d=\mu E$ ..............[1(b)]
For specific metal, $v_d\propto E$
Magnitude of drift velocity is directly proportional to the applied electric field.
$\mu =\frac{e\tau }{m}=\frac{v_d}{E}$ ...........(2)
(for any charge $q,\mu =\frac{q\tau }{m}$)
So the mobility of ions are defined as the ratio of drift velocity to the electric field.
Mobility is positive and its unit is $\frac{m/s}{volt/m}=m^2{s}^{-1}vol{t}^{-1}$
When electric field is same for given metals
$\frac{{\mu }_1}{{\mu }_2}=\frac{v_{d_1}}{v_{d_2}}$
It means when in any metal drift speed is more than electrons, then its mobility will also high. We will discuss about mobility related to semiconductor in Chapter 16.
Relation between electric current and mobility for semiconductor : Conductivity in semiconductors is due to electrons and holes both.
$\therefore i=i_e+i_h$
$=e{n}_{e}A{v}_e+e{n}_{h}A{v}_h$
$\because \mu =\frac{v_d}{E}\Rightarrow v_d=E\mu$
$\therefore v_e=E{\mu }_{e}A{\mu }_{e}E+e{n}_{h}A{\mu }_{h}E$