Two wires made of same material have their electrical resistances in the ratio `1:4` if their lengths are in the ratio `1:2`, the ratio of their masses is

Options

A. `1:1`
B. `1:8`
C. `8:1`
D. `2:1`

Explanation

To solve the problem, we need to find the ratio of the masses of two wires made of the same material, given their electrical resistances and lengths.

### Given:
- The ratio of resistances \( R_1 : R_2 = 1 : 4 \)
- The ratio of lengths \( L_1 : L_2 = 1 : 2 \)

### Step 1: Understand the relationship between resistance, length, and area
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 (constant for both wires since they are made of the same material),
- \( L \) is the length of the wire,
- \( A \) is the cross-sectional area of the wire.

### Step 2: Set up the equations for the two wires
From the resistance formula, we can express the resistances of the two wires as:
\[ R_1 = \frac{\rho L_1}{A_1} \quad \text{and} \quad R_2 = \frac{\rho L_2}{A_2} \]

### Step 3: Use the given resistance ratio
Using the given ratio of resistances:
\[ \frac{R_1}{R_2} = \frac{1}{4} \]
Substituting the expressions for \( R_1 \) and \( R_2 \):
\[ \frac{\frac{\rho L_1}{A_1}}{\frac{\rho L_2}{A_2}} = \frac{1}{4} \]
The resistivity \( \rho \) cancels out:
\[ \frac{L_1 A_2}{L_2 A_1} = \frac{1}{4} \]

### Step 4: Substitute the length ratio
Using the length ratio \( \frac{L_1}{L_2} = \frac{1}{2} \):
Let \( L_1 = x \) and \( L_2 = 2x \). Then:
\[ \frac{x A_2}{2x A_1} = \frac{1}{4} \]
This simplifies to:
\[ \frac{A_2}{2 A_1} = \frac{1}{4} \]
Multiplying both sides by 2:
\[ \frac{A_2}{A_1} = \frac{1}{2} \]

### Step 5: Find the mass ratio
The mass \( m \) of a wire can be expressed as:
\[ m = \rho \cdot V = \rho \cdot (A \cdot L) \]
Thus, for the two wires:
\[ m_1 = \rho A_1 L_1 \quad \text{and} \quad m_2 = \rho A_2 L_2 \]
Now, the ratio of masses is:
\[ \frac{m_1}{m_2} = \frac{\rho A_1 L_1}{\rho A_2 L_2} = \frac{A_1 L_1}{A_2 L_2} \]
Substituting the ratios we found:
\[ \frac{m_1}{m_2} = \frac{A_1 \cdot L_1}{A_2 \cdot L_2} = \frac{A_1 \cdot x}{\frac{1}{2} A_1 \cdot 2x} \]
This simplifies to:
\[ \frac{m_1}{m_2} = \frac{A_1 \cdot x}{A_1 \cdot x} = 1 \]

### Final Answer:
The ratio of their masses is:
\[ \frac{m_1}{m_2} = 1 : 1 \]