Define drift velocity of free electron . If relaxation time is r then write a vector relation b etween drift velocity and electric field.

Explanation

### Step-by-Step Solution:

**Step 1: Define Drift Velocity**
Drift velocity is defined as the average velocity that free electrons attain due to an electric field applied across a conductor. In the absence of an electric field, electrons move randomly in all directions, but when an electric field is applied, they experience a net movement in the direction of the field. This net movement results in a drift velocity, denoted as \( V_d \).

**Step 2: Understand the Concept of Relaxation Time**
Relaxation time, denoted as \( \tau \) (or small \( r \) in the question), is the average time between successive collisions of the electrons with the lattice ions in the conductor. It is a measure of how quickly the electrons can respond to the applied electric field.

**Step 3: Establish the Relationship Between Drift Velocity and Electric Field**
The drift velocity \( V_d \) can be expressed in terms of the electric field \( E \) and the relaxation time \( \tau \). The relation can be derived from the equation of motion for the electrons under the influence of an electric field. The force acting on an electron due to the electric field is given by \( F = eE \), where \( e \) is the charge of the electron.

Using Newton's second law, we can write:
\[ F = ma \implies eE = m \frac{V_d}{\tau} \]
where \( m \) is the mass of the electron and \( a \) is the acceleration. Rearranging this gives us:
\[ V_d = \frac{eE \tau}{m} \]

**Step 4: Write the Vector Relation**
Since both the drift velocity and the electric field are vector quantities, we can express the relationship in vector form:
\[ \mathbf{V_d} = \frac{e \tau}{m} \mathbf{E} \]
This indicates that the drift velocity \( \mathbf{V_d} \) is directly proportional to the electric field \( \mathbf{E} \) and is in the same direction as the electric field.