The resistance of a wire of length `l` and diameter `d` is `R`. The wire is stretched to double its length. The resistance of the wire will now be

Options

A. `(R )/(2)`
B. `2R`
C. `4R`
D. `16R`

Explanation

To solve the problem of how the resistance of a wire changes when it is stretched to double its length, we can follow these steps:

### Step 1: Understand the relationship between resistance, length, and cross-sectional area.
The resistance \( R \) of a wire is given by the formula:
\[ R = \rho \frac{l}{A} \]
where:
- \( R \) is the resistance,
- \( \rho \) is the resistivity of the material,
- \( l \) is the length of the wire,
- \( A \) is the cross-sectional area of the wire.

### Step 2: Define the initial conditions.
Let the initial length of the wire be \( l \) and the diameter be \( d \). The initial cross-sectional area \( A_1 \) can be calculated using the formula for the area of a circle:
\[ A_1 = \pi \left(\frac{d}{2}\right)^2 = \frac{\pi d^2}{4} \]
The initial resistance is given as \( R = \rho \frac{l}{A_1} \).

### Step 3: Determine the new length and area after stretching.
When the wire is stretched to double its length, the new length \( l_2 \) becomes:
\[ l_2 = 2l \]
Since the volume of the wire remains constant during stretching, we can express the volume before and after stretching:
\[ V = A_1 l = A_2 l_2 \]
Substituting \( l_2 \):
\[ A_1 l = A_2 (2l) \]
Cancelling \( l \) from both sides (assuming \( l \neq 0 \)):
\[ A_1 = 2 A_2 \]
Thus, the new cross-sectional area \( A_2 \) is:
\[ A_2 = \frac{A_1}{2} \]

### Step 4: Calculate the new resistance.
Now we can find the new resistance \( R' \) using the new length and area:
\[ R' = \rho \frac{l_2}{A_2} = \rho \frac{2l}{A_2} \]
Substituting \( A_2 \):
\[ R' = \rho \frac{2l}{\frac{A_1}{2}} = \rho \frac{2l \cdot 2}{A_1} = \rho \frac{4l}{A_1} \]
Since \( R = \rho \frac{l}{A_1} \), we can express \( R' \) in terms of \( R \):
\[ R' = 4R \]

### Conclusion:
The resistance of the wire after being stretched to double its length will be:
\[ R' = 4R \]