Define 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.

Relaxation time refers to the time gap between two successive collisions of electrons in a conductor.

The relationship between the relaxation time and drift velocity is as given below.

 \(\begin{array}{l}v_{d} = \left ( e\frac{E}{m} \right )T\end{array}\)

Where

• v_d = drift velocity
• e = charge of electron
• E = field
• m = mass of an electron
• T = Relaxation time

So the expression for relaxation time (T) is

 \(\begin{array}{l}T = \left ( v_{d}\frac{m}{e} \right )E\end{array}\)

Let

• L be the length of the conductor
• A be the area of the conductor
• n is the current density

Then, the current flowing through the conductor is \(\begin{array}{l}I = -neAv_{d}\end{array}\)

\(\begin{array}{l}I = neA\left ( e\frac{E}{m} \right )T\end{array}\)

\(\begin{array}{l}I = \frac{ne^{2}EA}{m}T\end{array}\)

The field E can be therefore expressed as – 

E = V/L

Then the current flowing through the conductor becomes \(\begin{array}{l}I = \frac{ne^{2}VA}{mL}T\end{array}\)

\(\begin{array}{l}\frac{V}{I} = \frac{mL}{ne^{2}TA}\end{array}\)

From Ohm’s law

V = IR

R = V/I

\(\begin{array}{l}R = \left ( \frac{m}{ne^{2}T} \right )\frac{L}{A}\end{array}\)

\(\begin{array}{l}R = \rho \frac{L}{A}\end{array}\)

---

Related Questions

1. What is the relation between drift velocity and electric field?
2. Give two examples of drift velocity.
3. Two conducting wires X and Y of the same diameter but different materials are joined in series across a battery. If the number density of electrons in X is twice that in Y, find the ratio of the drift velocity of electrons in the two wires.
4. Explain the relation between current and drift velocity.
5. It is known that the drift velocity of electrons is only a few mm/s for a current of a few amperes. How is it possible that a current is established almost instantaneously when a circuit is closed? For example, a bulb glows as soon as the connection is switched on. Explain.

---

Read More: