The ratio of the lengths of two wires A and B of same material is `1:2` and the ratio of their diameter is `2:1`. They are stretched by the same force, then the ratio of increase in length will be

Options

A. `2:1`
B. `1:4`
C. `1:8`
D. `8:1`

Explanation

To solve the problem step by step, we need to find the ratio of the increase in length of two wires A and B, given their length and diameter ratios.

### Step 1: Understand the Given Ratios
- The ratio of lengths of wires A and B is given as \( L_A : L_B = 1 : 2 \).
- The ratio of diameters of wires A and B is given as \( D_A : D_B = 2 : 1 \).

### Step 2: Calculate the Cross-Sectional Areas
The cross-sectional area \( A \) of a wire is given by the formula:
\[ A = \frac{\pi}{4} D^2 \]
Using the diameter ratios, we can find the areas of both wires.

For wire A:
\[ A_A = \frac{\pi}{4} D_A^2 \]

For wire B:
\[ A_B = \frac{\pi}{4} D_B^2 \]

Given \( D_A : D_B = 2 : 1 \), we can express \( D_A \) and \( D_B \) in terms of a variable \( d \):
- Let \( D_A = 2d \)
- Let \( D_B = d \)

Now, substituting these into the area formulas:
\[ A_A = \frac{\pi}{4} (2d)^2 = \frac{\pi}{4} \cdot 4d^2 = \pi d^2 \]
\[ A_B = \frac{\pi}{4} (d)^2 = \frac{\pi}{4} d^2 \]

### Step 3: Find the Ratio of Areas
Now we can find the ratio of the areas:
\[ \frac{A_A}{A_B} = \frac{\pi d^2}{\frac{\pi}{4} d^2} = \frac{4}{1} = 4 \]

### Step 4: Use the Formula for Increase in Length
The increase in length \( \Delta L \) for a wire under tension is given by:
\[ \Delta L = \frac{F L}{A Y} \]
Where:
- \( F \) is the force applied,
- \( L \) is the original length of the wire,
- \( A \) is the cross-sectional area,
- \( Y \) is Young's modulus (which is the same for both wires since they are of the same material).

### Step 5: Set Up the Ratio of Increase in Lengths
Since both wires are stretched by the same force \( F \), we can set up the ratio of the increases in length for wires A and B:
\[ \frac{\Delta L_A}{\Delta L_B} = \frac{F L_A / A_A Y}{F L_B / A_B Y} = \frac{L_A A_B}{L_B A_A} \]

### Step 6: Substitute the Ratios
Now substituting the known ratios:
- \( L_A : L_B = 1 : 2 \) implies \( L_A = 1 \) and \( L_B = 2 \).
- \( A_A : A_B = 4 : 1 \) implies \( A_A = 4 \) and \( A_B = 1 \).

Thus:
\[ \frac{\Delta L_A}{\Delta L_B} = \frac{1 \cdot 1}{2 \cdot 4} = \frac{1}{8} \]

### Conclusion
The ratio of the increase in length of wires A and B is:
\[ \Delta L_A : \Delta L_B = 1 : 8 \]