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 find the ratio of the resistances of two wires made of the same material and having equal cross-sectional areas but different lengths, we can follow these steps:

### Step 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.

### Step 2: Define the lengths and resistances of the wires
Let:
- The length of the first wire be \( L_1 = L \)
- The length of the second wire be \( L_2 = 2L \)

### Step 3: Calculate the resistance of the first wire
Using the resistance formula for the first wire:
\[ R_1 = \frac{\rho L_1}{A} = \frac{\rho L}{A} \]

### Step 4: Calculate the resistance of the second wire
Using the resistance formula for the second wire:
\[ R_2 = \frac{\rho L_2}{A} = \frac{\rho (2L)}{A} = \frac{2\rho L}{A} \]

### Step 5: Find the ratio of the resistances
Now, we can find the ratio of the resistances \( R_1 \) and \( R_2 \):
\[ \frac{R_1}{R_2} = \frac{\frac{\rho L}{A}}{\frac{2\rho L}{A}} = \frac{1}{2} \]

### Conclusion
Thus, the ratio of the resistances \( R_1 : R_2 \) is:
\[ 1 : 2 \]