The dimensions of four wires of the same material an given below. In which wire the increase in the length will be maximum?

Options

A. Length `100 cm`, diameter `1 mm`
B. Length `200 cm`, diameter `2 mm`
C. Length `300 cm`, diameter `3 mm`
D. Length `50 cm`, diameter `0.5 mm`

Explanation

To determine which wire experiences the maximum increase in length when subjected to the same force, we can use the relationship derived from Young's modulus. The increase in length (ΔL) of a wire is given by the formula:

\[ \Delta L = \frac{F \cdot L}{A \cdot Y} \]

Where:
- \(F\) is the force applied,
- \(L\) is the original length of the wire,
- \(A\) is the cross-sectional area of the wire,
- \(Y\) is Young's modulus (which is constant for wires made of the same material).

The cross-sectional area \(A\) of a wire with diameter \(d\) is given by:

\[ A = \frac{\pi}{4} d^2 \]

Substituting this into the equation for ΔL, we get:

\[ \Delta L = \frac{F \cdot L}{\frac{\pi}{4} d^2 \cdot Y} \]

This simplifies to:

\[ \Delta L \propto \frac{L}{d^2} \]

Thus, the increase in length is directly proportional to the ratio of the original length \(L\) to the square of the diameter \(d^2\). To find which wire has the maximum increase in length, we need to calculate \( \frac{L}{d^2} \) for each wire.

### Step-by-Step Calculation:

1. **Wire 1:**
- Length = 100 cm = 1000 mm
- Diameter = 1 mm
- \( \frac{L}{d^2} = \frac{1000}{1^2} = 1000 \)

2. **Wire 2:**
- Length = 200 cm = 2000 mm
- Diameter = 2 mm
- \( \frac{L}{d^2} = \frac{2000}{2^2} = \frac{2000}{4} = 500 \)

3. **Wire 3:**
- Length = 300 cm = 3000 mm
- Diameter = 3 mm
- \( \frac{L}{d^2} = \frac{3000}{3^2} = \frac{3000}{9} \approx 333.33 \)

4. **Wire 4:**
- Length = 50 cm = 500 mm
- Diameter = 0.5 mm
- \( \frac{L}{d^2} = \frac{500}{(0.5)^2} = \frac{500}{0.25} = 2000 \)

### Conclusion:
Comparing the values of \( \frac{L}{d^2} \):
- Wire 1: 1000
- Wire 2: 500
- Wire 3: 333.33
- Wire 4: 2000

The maximum value is for Wire 4, which has a value of 2000. Therefore, the wire with the maximum increase in length is **Wire 4**.