Question

Define the relaxation time of the free electrons drifting in a conductor. How is it related to the drift velocity of free electrons? Use this relation to deduce the expression for the electrical resistivity of the material.

Answer 1

1. Relaxation time is the time gap between two successive electron collisions in a conductor.
2. The relationship between the relaxation time (𝛕) and drift velocity (V_d) is given below.

$V_d=-e\left(\frac{E𝛕}{m}\right)$

Where

v_d = drift velocity

e = charge of electron

E = field

m = mass of electron

$𝛕$= Relaxation time

3. So the expression for relaxation time (𝛕) is

$𝛕=\left(\frac{V_{d}m}{e}\right)E$

4. Let

L = Length of the conductor

A = Area of the conductor

n = current density

Then current flowing through the conductor is

$I=neA{V}_d$

⇒ $I=neA\left(\frac{eE}{m}\right)𝛕=\frac{n{e}^{2}AE}{m}𝛕$

5. Also field E can be expressed as

$E=V/L$

Then current flowing through the conductor becomes

$$\begin{gathered} i=\frac{n{e}^{2}VA𝛕}{mL} \\ or\frac{V}{I}=\frac{mL}{n{e}^2𝛕A} \\ orR=\frac{mL}{n{e}^2𝛕A}(fromOhmslawV=IR) \\ orR=\frac{m}{n{e}^2𝛕}\left(\frac{L}{A}\right) \\ orR=\rho \frac{L}{A} \end{gathered}$$