If ` l_(1) and l_(2)` are the lengths of air column for the first and second resonance when a tuning fork frequency n is sounded on a resonance tube, there the distance of the antinode from the top end of resonance tube is

Options

A. `2(l_(2)-l_(1))`
B. `1/(2)2(l_(1)-l_(2))`
C. `(l_(2) - 3l_(1))/2`
D. `(l_(2) - l_(1))/2`

Answer

To find the distance of the antinode from the top end of the resonance tube when a tuning fork of frequency \( n \) is sounded, we will analyze the lengths of the air column at the first and second resonance.

### Step-by-Step Solution:

1. **Understanding Resonance in a Tube**:
- In a resonance tube, the first resonance occurs when the length of the air column \( l_1 \) is equal to \( \frac{\lambda}{4} \), where \( \lambda \) is the wavelength of the sound.
- The second resonance occurs when the length of the air column \( l_2 \) is equal to \( \frac{3\lambda}{4} \).

2. **Setting Up the Equations**:
- For the first resonance:
\[ l_1 = \frac{\lambda}{4} \]
- For the second resonance:
\[ l_2 = \frac{3\lambda}{4} \]

3. **Relating the Wavelength to the Lengths**:
- From the equations above, we can express the wavelength \( \lambda \) in terms of \( l_1 \) and \( l_2 \):
\[ \lambda = 4l_1 \quad \text{(from the first resonance)} \]
\[ \lambda = \frac{4l_2}{3} \quad \text{(from the second resonance)} \]

4. **Equating the Two Expressions for Wavelength**:
- Since both expressions represent the same wavelength \( \lambda \), we can set them equal to each other:
\[ 4l_1 = \frac{4l_2}{3} \]

5. **Solving for the Antinode Distance**:
- Rearranging the equation gives us:
\[ 3l_1 = l_2 \]
- This indicates that the second resonance length \( l_2 \) is three times the first resonance length \( l_1 \).

6. **Finding the Distance of the Antinode**:
- The distance of the antinode from the top of the resonance tube is at the open end, which is considered to be at a distance of 0. Therefore, the distance of the antinode from the top end of the resonance tube is:
\[ \text{Distance} = 0 \]

### Final Answer:
The distance of the antinode from the top end of the resonance tube is **0**.

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