The drift velocity of free electrons 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 will be :

Options

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

Answer

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

### Step-by-Step Solution:

1. **Understand the Formula for Drift Velocity**:
The drift velocity \( v_d \) of free electrons in a conductor can be expressed as:
\[ v_d = \frac{I}{n \cdot A \cdot e} \]
where:
- \( I \) is the current,
- \( n \) is the number of charge carriers per unit volume,
- \( A \) is the cross-sectional area of the conductor,
- \( e \) is the charge of an electron.

2. **Express the Cross-Sectional Area**:
The cross-sectional area \( A \) of a cylindrical conductor is given by:
\[ A = \pi r^2 \]
where \( r \) is the radius of the conductor.

3. **Initial Conditions**:
Let the initial current be \( I \) and the radius be \( r \). Therefore, the initial area \( A \) can be expressed as:
\[ A = \pi r^2 \]

4. **New Conditions**:
If both the current and the radius are doubled, the new current \( I' \) and new radius \( r' \) are:
\[ I' = 2I \quad \text{and} \quad r' = 2r \]
The new area \( A' \) becomes:
\[ A' = \pi (r')^2 = \pi (2r)^2 = \pi \cdot 4r^2 = 4\pi r^2 \]

5. **Calculate the New Drift Velocity**:
The new drift velocity \( v_d' \) can now be calculated using the new current and area:
\[ v_d' = \frac{I'}{n \cdot A'} = \frac{2I}{n \cdot 4\pi r^2} \]

6. **Relate New Drift Velocity to Initial Drift Velocity**:
We can express the new drift velocity in terms of the original drift velocity:
\[ v_d' = \frac{2I}{4(n \cdot \pi r^2)} = \frac{1}{2} \cdot \frac{I}{n \cdot \pi r^2} = \frac{1}{2} v_d \]

7. **Conclusion**:
Therefore, when both the radius and the current are doubled, the new drift velocity \( v_d' \) is:
\[ v_d' = \frac{v_d}{2} \]
Thus, the drift velocity will be half of the original drift velocity.

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