The following four wires are made of same material. Which of these will have the largest extension when the same tension is applied?

Options

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

Explanation

To determine which of the four wires made of the same material will have the largest extension when the same tension is applied, we can use the formula for extension derived from Young's modulus.

### Step-by-Step Solution:

1. **Understand the Formula for Extension**:
The extension (ΔL) of a wire under tension can be expressed as:
\[ \Delta L = \frac{F L}{A Y} \]
where:
- \(F\) = applied force (tension)
- \(L\) = original length of the wire
- \(A\) = cross-sectional area of the wire
- \(Y\) = Young's modulus of the material

2. **Recognize Constants**:
Since all wires are made of the same material, Young's modulus \(Y\) is constant for all wires. If the same tension \(F\) is applied, it is also constant.

3. **Identify the Relationship**:
The extension ΔL is directly proportional to the length \(L\) of the wire and inversely proportional to the cross-sectional area \(A\):
\[ \Delta L \propto \frac{L}{A} \]

4. **Calculate the Cross-Sectional Area**:
The cross-sectional area \(A\) of a wire with diameter \(d\) is given by:
\[ A = \frac{\pi d^2}{4} \]
Therefore, as the diameter increases, the area increases, leading to a decrease in extension.

5. **Evaluate Each Wire**:
For each wire, calculate the ratio \( \frac{L}{A} \):
- Let’s denote the lengths and diameters of the wires as follows:
- Wire 1: Length = 50 cm, Diameter = 0.5 mm
- Wire 2: Length = 50 cm, Diameter = 1 mm
- Wire 3: Length = 100 cm, Diameter = 1 mm
- Wire 4: Length = 150 cm, Diameter = 1.5 mm

6. **Calculate the Ratios**:
- For Wire 1:
- \(L = 50\) cm, \(A = \frac{\pi (0.05)^2}{4} = 0.0019635\) cm²
- Ratio = \( \frac{50}{0.0019635} \approx 25400\)
- For Wire 2:
- \(L = 50\) cm, \(A = \frac{\pi (0.1)^2}{4} = 0.007854\) cm²
- Ratio = \( \frac{50}{0.007854} \approx 6366.2\)
- For Wire 3:
- \(L = 100\) cm, \(A = \frac{\pi (0.1)^2}{4} = 0.007854\) cm²
- Ratio = \( \frac{100}{0.007854} \approx 12732.4\)
- For Wire 4:
- \(L = 150\) cm, \(A = \frac{\pi (0.15)^2}{4} = 0.017671\) cm²
- Ratio = \( \frac{150}{0.017671} \approx 8485.3\)

7. **Compare Ratios**:
The wire with the largest ratio \( \frac{L}{A} \) will have the largest extension. From our calculations:
- Wire 1: 25400
- Wire 2: 6366.2
- Wire 3: 12732.4
- Wire 4: 8485.3

Therefore, Wire 1 has the largest extension.

8. **Conclusion**:
The wire with the largest extension when the same tension is applied is **Wire 1**.