Question

The two wires A and B of the same material have their lengths in the ratio 1 : 2 and their diameters in the ratio 2 : 1. If they are stretched with the same force, the ratio of the increase in the length of A to that of B will be:

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

Hint:

The extension in the wire is inversely proportional to the cross-sectional area and directly proportional to the length of the wire.

Answer 1

The Correct Option is B

Step 1: Elongation scales as \( \Delta L \propto \dfrac{L}{d^2} \) for the same force and material, since \( A \propto d^2 \).

Step 2: Write the ratio symbolically before plugging in numbers: \( \dfrac{\Delta L_A}{\Delta L_B} = \dfrac{L_A}{L_B}\cdot\dfrac{d_B^2}{d_A^2} \).

Step 3: Substitute the given ratios \( \dfrac{L_A}{L_B} = \dfrac{1}{2} \) and \( \dfrac{d_A}{d_B} = \dfrac{2}{1} \), so \( \dfrac{d_B^2}{d_A^2} = \dfrac{1}{4} \): \( \dfrac{\Delta L_A}{\Delta L_B} = \dfrac{1}{2}\times\dfrac{1}{4} = \dfrac{1}{8} \).

\[ \boxed{\Delta L_A : \Delta L_B = 1:8} \]

Answer 2

Rather than working with symbols throughout, we can plug in concrete numbers and see the ratio emerge directly. Let \( F = 100 \, \text{N} \), and suppose wire A has diameter \( d_A = 2 \, \text{mm} = 0.002 \, \text{m} \) and length \( L_A = 1 \, \text{m} \). Then wire B, with diameter half and length twice that of A, has \( d_B = 1 \, \text{mm} = 0.001 \, \text{m} \) and \( L_B = 2 \, \text{m} \). Take \( E = 2\times10^{11} \, \text{Pa} \) for both, since the material is the same.

Area of A: \( A_A = \dfrac{\pi (0.002)^2}{4} \approx 3.1416\times10^{-6} \, \text{m}^2 \).

Area of B: \( A_B = \dfrac{\pi (0.001)^2}{4} \approx 0.7854\times10^{-6} \, \text{m}^2 \).

Elongation of A: \( \Delta L_A = \dfrac{100 \times 1}{3.1416\times10^{-6}\times 2\times10^{11}} \approx 1.59\times10^{-4} \, \text{m} \).

Elongation of B: \( \Delta L_B = \dfrac{100 \times 2}{0.7854\times10^{-6}\times 2\times10^{11}} \approx 1.273\times10^{-3} \, \text{m} \).

Ratio: \( \dfrac{\Delta L_A}{\Delta L_B} = \dfrac{1.59\times10^{-4}}{1.273\times10^{-3}} \approx \dfrac{1}{8} \).

Comparing with the options:

1. Option A, \( 1:2 \): the numeric substitution gives a much smaller ratio than this.
2. Option B, \( 4:1 \): this is not just the wrong magnitude but the wrong direction, since the computed \( \Delta L_A \) is smaller than \( \Delta L_B \), not larger.
3. Option C, \( 1:8 \): matches the numeric result closely.
4. Option D, \( 1:4 \): is twice as large as what the actual numbers give.

The concrete numeric check confirms option C.

Therefore, the correct answer is 1 : 8.