A metal wire of circular cross-section has a resistance `R_(1)`. The wire is now stretched without breaking, so that its length is doubled and the density is assumed to remain the same. If the resistance of the wire now becomes `R_(2)`, then `R_(2) : R_(1)` is

Options

A. `1 : 1`
B. `1 : 2`
C. `4 : 1`
D. `1 : 4`

Answer

To solve the problem, we need to analyze the relationship between the resistance of a wire before and after it is stretched.

### Step-by-Step Solution:

1. **Understand the Initial Conditions**:
- Let the initial length of the wire be \( L \).
- Let the initial cross-sectional area be \( A \).
- The initial resistance of the wire is given by:
\[ R_1 = \frac{\rho L}{A} \]
where \( \rho \) is the resistivity of the material.

2. **Determine the New Length and Area After Stretching**:
- The wire is stretched to double its length, so the new length \( L' \) is:
\[ L' = 2L \]
- The density of the wire remains constant. Therefore, the mass before stretching is equal to the mass after stretching:
\[ m_1 = m_2 \]
- The mass can be expressed as:
\[ m = \text{density} \times \text{volume} \]
- Initially, the volume \( V_1 \) is:
\[ V_1 = A \times L \]
- After stretching, the volume \( V_2 \) becomes:
\[ V_2 = A' \times L' = A' \times 2L \]
- Setting the two volumes equal (since mass is constant):
\[ A \times L = A' \times 2L \]
- Canceling \( L \) from both sides:
\[ A = 2A' \]
- Therefore, the new cross-sectional area \( A' \) is:
\[ A' = \frac{A}{2} \]

3. **Calculate the New Resistance**:
- The new resistance \( R_2 \) after stretching is given by:
\[ R_2 = \frac{\rho L'}{A'} \]
- Substituting \( L' \) and \( A' \):
\[ R_2 = \frac{\rho (2L)}{A/2} \]
- Simplifying this:
\[ R_2 = \frac{2\rho L}{A} \times 2 = \frac{4\rho L}{A} \]
- We know from the initial resistance:
\[ R_1 = \frac{\rho L}{A} \]
- Therefore, substituting \( R_1 \) into the equation for \( R_2 \):
\[ R_2 = 4R_1 \]

4. **Find the Ratio of Resistances**:
- The ratio of the new resistance to the old resistance is:
\[ \frac{R_2}{R_1} = 4 \]
- Thus, we can express this as:
\[ R_2 : R_1 = 4 : 1 \]

### Final Answer:
The ratio \( R_2 : R_1 \) is \( 4 : 1 \).