Two copper wire of length l and 2l have radii, r and 2r respectively. What si the ratio of their specific resistance.?

Options

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

Explanation

To find the ratio of the specific resistance of two copper wires with different lengths and radii, we can follow these steps:

### Step 1: Understand Specific Resistance
Specific resistance (or resistivity) is a property of the material itself and does not depend on the dimensions of the wire. For copper, the specific resistance is constant regardless of the wire's length or radius.

### Step 2: Identify the Wires
We have two wires:
- Wire 1: Length = \( l \), Radius = \( r \)
- Wire 2: Length = \( 2l \), Radius = \( 2r \)

### Step 3: Recognize the Nature of the Material
Both wires are made of copper, which means they have the same specific resistance (\( \rho \)).

### Step 4: Write the Formula for Specific Resistance
The specific resistance (\( \rho \)) is defined as:
\[ \rho = R \cdot \frac{A}{L} \]
where:
- \( R \) is the resistance,
- \( A \) is the cross-sectional area,
- \( L \) is the length of the wire.

### Step 5: Calculate the Cross-Sectional Areas
The cross-sectional area \( A \) of a wire is given by the formula:
\[ A = \pi r^2 \]
For Wire 1:
\[ A_1 = \pi r^2 \]
For Wire 2:
\[ A_2 = \pi (2r)^2 = \pi (4r^2) = 4\pi r^2 \]

### Step 6: Determine the Specific Resistance for Each Wire
Since the specific resistance is independent of the dimensions:
\[ \rho_1 = \rho_2 = \rho \quad (\text{for copper}) \]

### Step 7: Calculate the Ratio of Specific Resistances
The ratio of specific resistances of the two wires is:
\[ \text{Ratio} = \frac{\rho_1}{\rho_2} = \frac{\rho}{\rho} = 1 \]

### Final Answer
The ratio of their specific resistances is:
\[ \text{Ratio} = 1:1 \]

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