One end of a horizontal thick copper wire of length `2L` and radius `2R` is weded to an end of another horizontal thin copper wire of length `L` and radius `R` .When the arrangement is stretched by applying forces at two ends , the ratio of the elongation in the thin wire to that in the thick wire is

Options

A. `0.25`
B. `0.50`
C. `2.00`
D. `4.00`

Explanation

To solve the problem of finding the ratio of the elongation in the thin wire to that in the thick wire, we can follow these steps:

### Step 1: Identify the parameters of the wires
- **Thick Wire (Wire 1)**:
- Length, \( L_1 = 2L \)
- Radius, \( R_1 = 2R \)
- Area of cross-section, \( A_1 = \pi (R_1)^2 = \pi (2R)^2 = 4\pi R^2 \)

- **Thin Wire (Wire 2)**:
- Length, \( L_2 = L \)
- Radius, \( R_2 = R \)
- Area of cross-section, \( A_2 = \pi (R_2)^2 = \pi R^2 \)

### Step 2: Write the formula for elongation
The elongation (\( \Delta l \)) of a wire when a force \( F \) is applied is given by the formula:
\[ \Delta l = \frac{F \cdot L}{A \cdot Y} \]
where:
- \( F \) is the force applied,
- \( L \) is the length of the wire,
- \( A \) is the area of cross-section,
- \( Y \) is the Young's modulus of the material.

### Step 3: Calculate elongation for both wires
- **Elongation of the thick wire (Wire 1)**:
\[ \Delta l_1 = \frac{F \cdot L_1}{A_1 \cdot Y} = \frac{F \cdot (2L)}{(4\pi R^2) \cdot Y} = \frac{2F L}{4\pi R^2 Y} = \frac{F L}{2\pi R^2 Y} \]

- **Elongation of the thin wire (Wire 2)**:
\[ \Delta l_2 = \frac{F \cdot L_2}{A_2 \cdot Y} = \frac{F \cdot L}{(\pi R^2) \cdot Y} = \frac{F L}{\pi R^2 Y} \]

### Step 4: Find the ratio of elongations
Now, we need to find the ratio of elongation in the thin wire to that in the thick wire:
\[ \frac{\Delta l_2}{\Delta l_1} = \frac{\frac{F L}{\pi R^2 Y}}{\frac{F L}{2\pi R^2 Y}} = \frac{F L}{\pi R^2 Y} \cdot \frac{2\pi R^2 Y}{F L} = 2 \]

### Conclusion
The ratio of the elongation in the thin wire to that in the thick wire is:
\[ \frac{\Delta l_2}{\Delta l_1} = 2 \]

### Final Answer
Thus, the answer is **2** (Option 3).

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