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 can follow these steps:

### Step 1: Understand the Resonance Tube
A resonance tube is a tube that is open at one end and closed at the other. When a tuning fork is sounded, it creates sound waves that resonate within the tube. The closed end of the tube is a node (minimum displacement), and the open end is an antinode (maximum displacement).

### Step 2: Identify the Lengths of the Air Column
Let \( l_1 \) be the length of the air column for the first resonance, and \( l_2 \) be the length of the air column for the second resonance. The first resonance corresponds to the first harmonic (fundamental frequency), and the second resonance corresponds to the third harmonic.

### Step 3: Relationship Between Resonances
For a tube closed at one end, the lengths of the air column for the first and second resonance can be expressed as:
- First resonance (1st harmonic): \( l_1 = \frac{1}{4} \lambda \)
- Second resonance (3rd harmonic): \( l_2 = \frac{3}{4} \lambda \)

Where \( \lambda \) is the wavelength of the sound wave produced by the tuning fork.

### Step 4: Calculate the Wavelength
From the relationship between the lengths of the air column, we can find the wavelength:
\[ l_2 - l_1 = \frac{3}{4} \lambda - \frac{1}{4} \lambda = \frac{1}{2} \lambda \]
Thus, we can express the wavelength as:
\[ \lambda = 2(l_2 - l_1) \]

### Step 5: Determine the Distance of Antinode from the Top
The distance of the antinode from the top end of the resonance tube can be calculated using the length of the air column at the first resonance:
\[ \text{Distance of antinode from top} = l_1 \]

### Final Result
The distance of the antinode from the top end of the resonance tube is \( l_1 \).

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