Drift velocity `v_(d)` varies with the intensity of electric field as per the relation

Options

A. `v_(d) prop E`
B. `v_(d) prop (1)/(E)`
C. `v_(d) =` constant
D. `v_(d) prop E^(2)`

Explanation

To find the relationship between drift velocity \( v_d \) and the intensity of the electric field \( E \), we can follow these steps:

### Step 1: Understand the Definition of Drift Velocity
Drift velocity \( v_d \) is defined as the average velocity that a charged particle, such as an electron, attains due to an electric field.

### Step 2: Use the Formula for Drift Velocity
The formula for drift velocity can be expressed as:
\[ v_d = \frac{e}{m} \cdot \frac{V}{L} \cdot \tau \]
where:
- \( e \) is the charge of the electron,
- \( m \) is the mass of the electron,
- \( V \) is the potential difference,
- \( L \) is the length of the conductor,
- \( \tau \) is the relaxation time.

### Step 3: Relate Potential Difference to Electric Field
The potential difference \( V \) can be related to the electric field \( E \) and the length \( L \) of the conductor by the equation:
\[ V = E \cdot L \]

### Step 4: Substitute the Potential in the Drift Velocity Formula
Substituting \( V \) in the drift velocity formula gives:
\[ v_d = \frac{e}{m} \cdot \frac{E \cdot L}{L} \cdot \tau \]

### Step 5: Simplify the Equation
The length \( L \) cancels out:
\[ v_d = \frac{e}{m} \cdot E \cdot \tau \]

### Step 6: Analyze the Result
In this equation, \( \frac{e}{m} \) and \( \tau \) are constants for a given material and condition. Therefore, we can conclude that:
\[ v_d \propto E \]
This indicates that the drift velocity \( v_d \) is directly proportional to the electric field \( E \).

### Conclusion
Thus, the relationship between drift velocity and the intensity of the electric field is that drift velocity increases linearly with an increase in the electric field.