For two wires, length ratio is `1:4`, radius ratio is 1:2, specific resistances ratio is 3:1. Compare their resistances.

Options

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

Explanation

To compare the resistances of two wires given their length ratio, radius ratio, and specific resistance ratio, we can follow these steps:

### Step 1: Define the Ratios
Given:
- Length ratio \( \frac{L_1}{L_2} = \frac{1}{4} \)
- Radius ratio \( \frac{R_1}{R_2} = \frac{1}{2} \)
- Specific resistance ratio \( \frac{\rho_1}{\rho_2} = \frac{3}{1} \)

### Step 2: Calculate the Area Ratio
The area \( A \) of a wire (cylindrical shape) is given by the formula:
\[ A = \pi R^2 \]

Thus, the area ratio \( \frac{A_1}{A_2} \) can be calculated as:
\[ \frac{A_1}{A_2} = \frac{\pi R_1^2}{\pi R_2^2} = \frac{R_1^2}{R_2^2} \]
Substituting the radius ratio:
\[ \frac{A_1}{A_2} = \frac{(1)^2}{(2)^2} = \frac{1}{4} \]

### Step 3: Write the Resistance Formula
The resistance \( R \) of a wire is given by:
\[ R = \frac{\rho L}{A} \]
For wires 1 and 2, we can express their resistances as:
\[ R_1 = \frac{\rho_1 L_1}{A_1} \quad \text{and} \quad R_2 = \frac{\rho_2 L_2}{A_2} \]

### Step 4: Find the Ratio of Resistances
Now, we can find the ratio \( \frac{R_1}{R_2} \):
\[ \frac{R_1}{R_2} = \frac{\rho_1 L_1}{A_1} \cdot \frac{A_2}{\rho_2 L_2} \]
Substituting the known ratios:
\[ \frac{R_1}{R_2} = \frac{\rho_1}{\rho_2} \cdot \frac{L_1}{L_2} \cdot \frac{A_2}{A_1} \]

### Step 5: Substitute the Values
Substituting the values we have:
- \( \frac{\rho_1}{\rho_2} = 3 \)
- \( \frac{L_1}{L_2} = \frac{1}{4} \)
- \( \frac{A_1}{A_2} = \frac{1}{4} \) implies \( \frac{A_2}{A_1} = 4 \)

Thus,
\[ \frac{R_1}{R_2} = 3 \cdot \frac{1}{4} \cdot 4 \]
\[ \frac{R_1}{R_2} = 3 \]

### Conclusion
The ratio of the resistances is:
\[ R_1 : R_2 = 3 : 1 \]