Question

Derive an expression for the resistivity of a conductor in terms of number density of free electrons and relaxation time.

Hint:

To compare charge using a current-time graph, always calculate the area under the curve. For constant current, it is a rectangle; for linearly increasing current, it's a triangle.

Answer 1

Derivation of Resistivity from Drift Velocity

Given:

• n = number density of free electrons (number of free electrons per unit volume)
• e = electronic charge
• m = mass of an electron
• \( \tau \) = relaxation time (time between collisions of electrons)

Step 1: Drift Velocity of Electrons

When an electric field \( E \) is applied to the conductor, the free electrons experience a force and accelerate, attaining an average drift velocity \( v_d \). The equation of motion for the drift velocity is given by:

\[ v_d = \frac{e E \tau}{m} \]

Step 2: Current Density \( J \)

The current density \( J \) is given by the product of the number density of free electrons, the electron charge, and the drift velocity:

\[ J = n e v_d \]

Substituting the expression for \( v_d \):

\[ J = n e \left( \frac{e E \tau}{m} \right) = \frac{n e^2 \tau}{m} E \]

Step 3: Relation Between Current Density and Electric Field

The relationship between current density \( J \) and the applied electric field \( E \) is given by Ohm's Law:

\[ J = \sigma E \]

where \( \sigma \) is the conductivity of the material. Comparing the two expressions for \( J \), we can equate the coefficients of \( E \) to find the conductivity \( \sigma \):

\[ \sigma = \frac{n e^2 \tau}{m} \]

Step 4: Resistivity \( \rho \)

Resistivity \( \rho \) is the reciprocal of conductivity. Thus:

\[ \rho = \frac{1}{\sigma} = \frac{m}{n e^2 \tau} \]

Final Answer:

The resistivity \( \rho \) is given by:

\[ \rho = \frac{m}{n e^2 \tau} \]