Two wires of the same materical having equal area of cross-section have L and 2L. Their respective resistances are in the ratio

Options

A. `2:1`
B. `1:1`
C. `1:2`
D. `1:3`

Answer

To solve the problem of finding the ratio of the resistances of two wires made of the same material with lengths L and 2L, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula for Resistance**:
The resistance \( R \) of a wire is given by the formula:
\[ R = \frac{\rho L}{A} \]
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.

2. **Identify Parameters for Both Wires**:
- For Wire 1 (length \( L \)):
- \( R_1 = \frac{\rho L}{A} \)
- For Wire 2 (length \( 2L \)):
- \( R_2 = \frac{\rho (2L)}{A} = \frac{2\rho L}{A} \)

3. **Calculate the Ratio of Resistances**:
To find the ratio \( \frac{R_1}{R_2} \):
\[ \frac{R_1}{R_2} = \frac{\frac{\rho L}{A}}{\frac{2\rho L}{A}} \]
- Here, we can simplify the expression:
\[ \frac{R_1}{R_2} = \frac{\rho L}{A} \cdot \frac{A}{2\rho L} \]
- The \( \rho \), \( L \), and \( A \) terms cancel out:
\[ \frac{R_1}{R_2} = \frac{1}{2} \]

4. **Express the Ratio**:
Therefore, the ratio of the resistances \( R_1 : R_2 \) is:
\[ R_1 : R_2 = 1 : 2 \]

### Final Answer:
The respective resistances of the two wires are in the ratio \( 1 : 2 \).

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