The resistance of a wire is R . If the length of the wire is doubled by stretching, then the new resistance will be

Options

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

Answer

To solve the problem, we need to understand how the resistance of a wire changes when its length is altered. 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-by-Step Solution:

1. **Initial Resistance**:
- Let the initial length of the wire be \( L \) and the initial cross-sectional area be \( A \).
- The initial resistance is given by:
\[ R = \rho \frac{L}{A} \]

2. **Doubling the Length**:
- When the wire is stretched to double its length, the new length \( L' \) becomes:
\[ L' = 2L \]

3. **Volume Conservation**:
- When the wire is stretched, its volume remains constant. The volume \( V \) of the wire is given by:
\[ V = A \cdot L \]
- After stretching, the new volume \( V' \) can be expressed as:
\[ V' = A' \cdot L' = A' \cdot (2L) \]
- Since the volume remains constant, we have:
\[ A \cdot L = A' \cdot (2L) \]

4. **Finding New Cross-sectional Area**:
- From the volume conservation equation, we can solve for the new cross-sectional area \( A' \):
\[ A' = \frac{A}{2} \]

5. **New Resistance Calculation**:
- Now, we can calculate the new resistance \( R' \) using the new length and the new cross-sectional area:
\[ R' = \rho \frac{L'}{A'} = \rho \frac{2L}{\frac{A}{2}} = \rho \frac{2L \cdot 2}{A} = \rho \frac{4L}{A} \]
- Thus, the new resistance \( R' \) can be expressed in terms of the original resistance \( R \):
\[ R' = 4 \left(\rho \frac{L}{A}\right) = 4R \]

### Conclusion:
The new resistance of the wire after doubling its length will be \( 4R \).