Question

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

Answer 1

Hint: A drift velocity is an average velocity attained by charged particles, such as electrons, in a material due to an electric field. In general, an electron in a conductor will propagate randomly at the Fermi velocity, resulting in an average velocity of zero. Applying an electric field adds to this random motion a small net flow in one direction which is the drift.

Complete step by step answer:
Step 1: When we apply a potential difference between two points of a conductor the free electrons experience an electric force due to the generated electric field in the conductor. The direction of the force is towards the positive end of the conductor. As a result, free electrons start to move towards the positive end of the conductor. During its accelerated motion it collides with the other electrons and positive ions of the conductor. This motion of the electron is known as 'Drift motion' and the average velocity between two successive collisions is known as Drift velocity.
Step 2: drift velocity is denoted by
${v_d} = \dfrac{{e\tau }}{m}\dfrac{V}{l}$ , where $\tau $ is the relaxation time, $V$ is the potential difference, $m$ is the mass of the electron, $l$ is the length of the conductor.
The drift velocity in terms of current $i$ is given by
${v_d} = \dfrac{i}{{Ane}}$ , where $A$ is the cross-section area of the conductor, $n$ is the number of electrons.
Step 3: compare both the equations
$\therefore \dfrac{{e\tau }}{m}\dfrac{V}{l} = \dfrac{i}{{Ane}}$
$ \Rightarrow V = \dfrac{{mli}}{{e\tau Ane}}$
$ \Rightarrow V = \dfrac{m}{{n{e^2}\tau }}\dfrac{l}{A}i$ ……equation (1).
Step 4: from equation 1.
$\therefore \dfrac{V}{l} = \dfrac{m}{{n{e^2}\tau }}\dfrac{1}{A}i$
$ \Rightarrow \overrightarrow E = \dfrac{m}{{n{e^2}\tau }}\overrightarrow J $ , where $\overrightarrow E = \dfrac{V}{l}$ is the electric field due to potential $V$ , and $\overrightarrow J = \dfrac{1}{A}i$ is the current density.
Step 5: The term $\dfrac{m}{{n{e^2}\tau }}$ is constant for a specific conductor. This is called the resistivity of the conductor. Let us denote it by $\rho $ . then
$\therefore \overrightarrow E = \rho \overrightarrow J $
We know that \[\rho = \dfrac{1}{\sigma }\] where $\sigma $ is specific conductivity of the conductor.
$\therefore \overrightarrow E = \dfrac{1}{\sigma }\overrightarrow J $
$ \Rightarrow \overrightarrow E \sigma = \overrightarrow J $

Note:
An above expression is a vector form of the ohm's law. With the help of the above expression, we can define the conductivity of the conductor as the ratio of the electric field and the current density. This is the general form of Ohm’s law. Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points. The constant is the resistance of the conductor.