Question

What is relaxation time? Find the equation of the relation between drift velocity and relaxation time.

Hint:

Relaxation time is the mean time between electron collisions; equate the velocity gained, \( v_d=a\tau \), with \( a=eE/m \).

Answer 1

Step 1: Definition of relaxation time.
In a conductor the free electrons move randomly and frequently collide with the positive ions of the lattice. Relaxation time \( \tau \) is the average time interval between two successive collisions of an electron. After each collision the electron's drift is randomised, so it effectively starts afresh.

Step 2: Force and acceleration on an electron.
When a potential difference is applied, a uniform electric field \( E \) exists in the conductor. The force on an electron of charge \( e \) is \( F = eE \), giving it an acceleration
\[ a = \frac{F}{m} = \frac{eE}{m} \]
where \( m \) is the mass of the electron.

Step 3: Velocity gained between collisions.
Just after a collision an electron's average velocity is zero (random directions cancel). Under the field it accelerates for an average time \( \tau \) before the next collision. The extra velocity it gains is the drift velocity:
\[ v_d = a\,\tau \]

Step 4: Substitute the acceleration.
\[ v_d = \frac{eE}{m}\,\tau \]

Step 5: Result.
\[\boxed{v_d = \frac{eE\tau}{m}}\]
The drift velocity is directly proportional to the applied field \( E \) and to the relaxation time \( \tau \). (For electrons the drift is directed opposite to \( E \).)