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

Options

A. length 50 cm and diameter 0.5 mm
B. length 100 cm and diameter 1 mm
C. length 200 cm and diameter 2 mm
D. length 300 cm and diameter 3 mm

Explanation

To determine which of the four wires will have the largest extension when the same tension is applied, we can use the relationship derived from Young's modulus. The extension (Δl) of a wire under tension is given by the formula:

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

where:
- \( l \) is the original length of the wire,
- \( d \) is the diameter of the wire.

### Step-by-Step Solution:

1. **Identify the Variables**:
We need to find the ratio \( \frac{l}{d^2} \) for each of the four wires. This will help us determine which wire has the largest extension.

2. **Calculate \( \frac{l}{d^2} \) for Each Wire**:
- **Wire 1**:
- Length \( l_1 = 50 \) units, Diameter \( d_1 = 0.5 \) units.
- \( d_1^2 = (0.5)^2 = 0.25 \)
- \( \frac{l_1}{d_1^2} = \frac{50}{0.25} = 200 \)

- **Wire 2**:
- Length \( l_2 = 100 \) units, Diameter \( d_2 = 1 \) unit.
- \( d_2^2 = (1)^2 = 1 \)
- \( \frac{l_2}{d_2^2} = \frac{100}{1} = 100 \)

- **Wire 3**:
- Length \( l_3 = 200 \) units, Diameter \( d_3 = 2 \) units.
- \( d_3^2 = (2)^2 = 4 \)
- \( \frac{l_3}{d_3^2} = \frac{200}{4} = 50 \)

- **Wire 4**:
- Length \( l_4 = 300 \) units, Diameter \( d_4 = 3 \) units.
- \( d_4^2 = (3)^2 = 9 \)
- \( \frac{l_4}{d_4^2} = \frac{300}{9} \approx 33.33 \)

3. **Compare the Values**:
- Wire 1: \( 200 \)
- Wire 2: \( 100 \)
- Wire 3: \( 50 \)
- Wire 4: \( 33.33 \)

4. **Determine the Largest Extension**:
The wire with the largest value of \( \frac{l}{d^2} \) will have the largest extension. From our calculations, Wire 1 has the largest value of \( 200 \).

### Conclusion:
Thus, **Wire 1** will have the largest extension when the same tension is applied.