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

Options

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

Answer

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

### Step-by-Step Solution:

1. **Understanding Drift Velocity**:
Drift velocity \( v_d \) is the average velocity that charged particles, such as electrons, attain due to an electric field.

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

3. **Relating Potential Difference to Electric Field**:
The potential difference \( V \) can be expressed in terms of the electric field \( E \) and the length \( L \) of the conductor:
\[ V = E \cdot L \]

4. **Substituting into Drift Velocity Formula**:
Substituting \( V \) into the drift velocity formula gives:
\[ v_d = \frac{e(E \cdot L)}{mL \tau} \]

5. **Simplifying the Expression**:
In the equation, the length \( L \) cancels out:
\[ v_d = \frac{eE}{m \tau} \]

6. **Identifying the Relationship**:
From the simplified equation, we can see that:
\[ v_d \propto E \]
This indicates that the drift velocity \( v_d \) is directly proportional to the electric field intensity \( E \) when \( e \), \( m \), and \( \tau \) are constants.

7. **Conclusion**:
Therefore, the relationship between drift velocity and electric field intensity is:
\[ v_d \text{ is directly proportional to } E \]

### Final Answer:
The correct option is **Option 1: \( v_d \) is proportional to the electric field intensity \( E \)**.