When the current i is flowing through a conductor, the drift velocity is v . If 2i current is flowed through the same metal but having double the area of cross-section, then the drift velocity will be

Options

A. `v/4`
B. `(v)/(2)`
C. `v`
D. `4v`

Answer

To solve the problem, we need to understand the relationship between current, drift velocity, area of cross-section, and the number density of charge carriers.

### Step-by-Step Solution:

1. **Understand the Formula for Current**:
The current \( I \) flowing through a conductor can be expressed as:
\[ I = n \cdot e \cdot A \cdot v_d \]
where:
- \( I \) is the current,
- \( n \) is the number density of charge carriers (electrons),
- \( e \) is the charge of an electron,
- \( A \) is the area of cross-section,
- \( v_d \) is the drift velocity.

2. **Initial Conditions**:
When the current \( i \) is flowing through the conductor, we have:
\[ i = n \cdot e \cdot A \cdot v \]
where \( v \) is the initial drift velocity.

3. **New Conditions**:
Now, we are given that the current is increased to \( 2i \) and the area of cross-section is doubled (i.e., \( A' = 2A \)). We need to find the new drift velocity \( v_d' \).

4. **Applying the Formula for New Conditions**:
The new current can be expressed as:
\[ 2i = n \cdot e \cdot A' \cdot v_d' \]
Substituting \( A' = 2A \):
\[ 2i = n \cdot e \cdot (2A) \cdot v_d' \]
This simplifies to:
\[ 2i = 2n \cdot e \cdot A \cdot v_d' \]

5. **Relating the Two Currents**:
Since we know from the initial condition that:
\[ i = n \cdot e \cdot A \cdot v \]
We can substitute \( i \) in the equation:
\[ 2(n \cdot e \cdot A \cdot v) = 2n \cdot e \cdot A \cdot v_d' \]
Dividing both sides by \( 2n \cdot e \cdot A \):
\[ v = v_d' \]

6. **Conclusion**:
Therefore, the new drift velocity \( v_d' \) when the current is \( 2i \) and the area is doubled remains the same as the initial drift velocity \( v \):
\[ v_d' = v \]

### Final Answer:
The drift velocity will be \( v \).

---