The drift velocity of free elecrons in a conductor is `v`, when a current i is flowing in it. If both the radius and current are doubled, then the drift velocity wil be

Options

A. v
B. `v//2`
C. `v//4`
D. `v//8`

Explanation

To solve the problem, we need to analyze how the drift velocity of free electrons in a conductor changes when both the radius of the conductor and the current flowing through it are doubled.

### Step-by-Step Solution:

1. **Understanding Drift Velocity**:
The drift velocity \( v_d \) of electrons in a conductor is given by the formula:
\[ I = n e v_d A \]
where:
- \( I \) is the current,
- \( n \) is the number density of charge carriers (electrons),
- \( e \) is the charge of an electron,
- \( A \) is the cross-sectional area of the conductor.

2. **Initial Conditions**:
Let the initial current be \( I \) and the initial drift velocity be \( v_d \). The initial cross-sectional area \( A \) can be expressed in terms of the radius \( r \):
\[ A = \pi r^2 \]

3. **New Conditions**:
When both the radius and the current are doubled:
- New current \( I' = 2I \)
- New radius \( r' = 2r \)
- New cross-sectional area \( A' \):
\[ A' = \pi (r')^2 = \pi (2r)^2 = 4\pi r^2 = 4A \]

4. **Applying the New Conditions**:
Substitute the new values into the drift velocity equation:
\[ I' = n e v_d' A' \]
Replacing \( I' \) and \( A' \):
\[ 2I = n e v_d' (4A) \]

5. **Setting Up the Equation**:
We can now set up the equation:
\[ 2I = n e v_d' (4A) \]
We can also express the original current \( I \) in terms of the original drift velocity:
\[ I = n e v_d A \]
Substituting this into the equation gives:
\[ 2(n e v_d A) = n e v_d' (4A) \]

6. **Canceling Common Terms**:
Since \( n \), \( e \), and \( A \) are common in both sides, we can cancel them out:
\[ 2 = 4 \frac{v_d'}{v_d} \]

7. **Solving for the New Drift Velocity**:
Rearranging gives:
\[ \frac{v_d'}{v_d} = \frac{2}{4} = \frac{1}{2} \]
Therefore, we find:
\[ v_d' = \frac{v_d}{2} \]

8. **Conclusion**:
If the initial drift velocity was \( v \), then the new drift velocity when both the radius and the current are doubled is:
\[ v_d' = \frac{v}{2} \]

### Final Answer:
The drift velocity will be \( \frac{v}{2} \).