Question

An electric field \(E\) is established across the ends of a cylindrical conductor of length \(L\) and area of cross-section \(A\). Discuss how electrons attain an average velocity, independent of time. Hence, obtain a relation between current in the conductor and this ‘average velocity’ of electrons.

Hint:

The most important drift velocity formulas are: \[ v_d=\frac{eE\tau}{m} \] and \[ I=neAv_d \] These are frequently used in numericals related to current electricity.

Answer 1

Concept: In a metallic conductor, a large number of free electrons are continuously moving in random directions due to thermal energy. In the absence of an external electric field, the random motion of electrons does not produce any net current because the average velocity of the electrons is zero. When an electric field is applied across the conductor, each electron experiences an electric force. As a result, the electrons acquire a small average velocity in a direction opposite to the electric field. This average velocity is known as the drift velocity.

Step 1: Force acting on an electron Let an electric field \(E\) be applied across the conductor. The force acting on an electron is \[ F=-eE \] where
• \(e\) is the magnitude of electronic charge,
• \(E\) is the applied electric field. Ignoring the negative sign while considering magnitude, \[ F=eE \] Using Newton's second law, \[ F=ma \] Therefore, \[ ma=eE \] or \[ a=\frac{eE}{m} \] where \(m\) is the mass of the electron.

Step 2: Introduction of relaxation time Electrons do not continue accelerating indefinitely because they frequently collide with the positive ions of the metallic lattice. The average time between two successive collisions is called the relaxation time and is denoted by \(\tau\). Between collisions, an electron accelerates under the influence of the electric field. Hence, the average drift velocity acquired by an electron is \[ v_d=a\tau \] Substituting the value of acceleration, \[ v_d=\frac{eE}{m}\tau \] Thus, \[ \boxed{v_d=\frac{eE\tau}{m}} \] This drift velocity remains constant with time because repeated collisions prevent continuous acceleration and establish a steady average velocity.

Step 3: Number of electrons crossing a cross-section Let
• \(n\) = number of free electrons per unit volume,
• \(A\) = area of cross-section of conductor,
• \(v_d\) = drift velocity. In one second, electrons drift through a distance \[ v_d \] Therefore, the volume swept in one second is \[ Av_d \] Hence, the number of free electrons crossing the cross-section per second is \[ nAv_d \]

Step 4: Charge flowing per second Since each electron carries charge \(e\), the total charge crossing the section per second is \[ I=nAv_de \] Since current is charge flowing per unit time, \[ \boxed{I=neAv_d} \] Final Result: The drift velocity of electrons is \[ \boxed{v_d=\frac{eE\tau}{m}} \] and the relation between electric current and drift velocity is \[ \boxed{I=neAv_d} \] where
• \(n\) = electron number density,
• \(e\) = electronic charge,
• \(A\) = area of cross-section,
• \(v_d\) = drift velocity.