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

An applied electric field \( E \) causes free electrons to accelerate, resulting in an average drift velocity \( v_d \). The equation of motion is:

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

Step 2: Current Density \( J \)

Current density \( J \) is the product of electron number density, electron charge, and drift velocity:

\[ J = n e v_d \]

Substituting \( v_d \):

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

Step 3: Relation Between Current Density and Electric Field

Ohm's Law states the relationship between current density \( J \) and electric field \( E \):

\[ J = \sigma E \]

where \( \sigma \) is conductivity. Equating the coefficients of \( E \) from both expressions for \( J \) yields conductivity \( \sigma \):

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

Step 4: Resistivity \( \rho \)

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

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

Final Answer:

The resistivity \( \rho \) is calculated as:

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