Question

What is called drift velocity? Obtain the equation of Ohm's law ($\overline{J}=\sigma \overline{E}$) on the basis of drift velocity where parameters are in their usual meaning.

Answer 1

When a potential difference (V) is applied across the conductor of length (l), then an electric field ($\overrightarrow{E}$) develops in the conductor ($E=\frac{v}{l}$) Due to this field each free electron of the conductor experiences an electric force ($\overrightarrow{F}=-e\overrightarrow{E}$ towards the positive end of the conductor and hence it starts accelerated motion $(\overrightarrow{a}=\frac{\overrightarrow{F}}{m})$ towards the positive end. During its accelerated motion it collides with the other electrons and positive ions of the conductor. Therefore its velocity always remains changing. This motion of electron is known as 'Drift motion' and the average velocity between two successive collisions is known as 'Drift velocity.' It is denoted by $v_d$.
The relation between drift velocity and potential difference,
$v_d=\frac{e\tau }{m}\frac{V}{l}$ ...............(1)
and relation between drift velocity and electric current,
$v_d=\frac{i}{Ane}$ .........(2)
On comparing equations (1) and (2) we have
$\frac{e\tau }{m}\frac{V}{l}=\frac{i}{Ane}$
Or $V=\frac{ml}{e\tau Ane}i$
or $V=\frac{m}{n{e}^2\tau }\frac{l}{A}i$ ..............(3)
or $V=\rho \frac{l}{A}i$ ..............(4)
Here $\frac{m}{n{e}^2\tau }$ which is characteristic of the substance of the conductor and it is called 'specific resistance or 'resistivity' of the substance.
It is constant for one substance and different for different substances. If physical conditions of the conductor do not change at constant temperature, then I and A will also remain constant. Therefore,
$\rho \frac{l}{A}=constant=R$ (Resistance of the conductor)
From equation (4),
V = Ri .......... (5)
or $V\propto i$
i.e., the potential difference developed across a conductor is directly proportional to current flowing through the conductor provided the physical conditions of the conductor remain unchanged! This is Ohm's law.
Vector form of Ohm's law :
From equation (3),
$V=\frac{m}{n{e}^2\tau }\frac{l}{A}i$
$\frac{V}{l}=\frac{m}{n{e}^2\tau }\frac{i}{A}$
or $\overrightarrow{E}=\frac{m}{n{e}^2\tau }\overrightarrow{J}$
Where $\overrightarrow{E}=\frac{V}{l}$ = intensity of electric field
and $\overrightarrow{J}=\frac{i}{A}$ current density
In vector form
$\overrightarrow{E}=\frac{m}{n{e}^2\tau }\overrightarrow{J}$ ...........(6)
or $\overrightarrow{E}=\rho \overrightarrow{J}$
or $\frac{\overrightarrow{E}}{\rho }=\overrightarrow{J}$
or $\overrightarrow{J}=\sigma \overrightarrow{E}$ ................(7)
Where $\sigma =\frac{1}{\rho }$ = Specific conductivity
Equation (7) is called microscopic form of Ohm's law.
Specific Resistance. "Ratio of intensity of electric field (E) and current density (J) at any point inside the current carrying conductor, is called the specific resistance of the material of the conductor. It is represented by $\rho$.
$\therefore \rho =\frac{E}{J}$
Where A is the area of cross-section of the conductor
$\therefore \rho =\frac{V/l}{i/A}=\frac{V}{i}\times \frac{A}{l}$
or $\rho =R\frac{A}{l}$ ...........(8)
Where $R=\frac{V}{i}$ = Resistance of the conductor,
If A = 1 $m^2$; l = 1 m, then $\rho$ = R
i.e., the specific resistance of a material is equal to the resistance of material when the length and area of cross-section of material take as unity".
Specific resistance can also be obtained by following formula,
$P=\frac{m}{n{e}^2\tau }$ ........(9)
Where m is the mass of electron, n is the free electron density, e is the electronic charge and $\tau$ is the relaxation time.