In Young's double slit experiment, the distance between two sources is `0.1mm`. The distance of screen from the sources is `20cm`. Wavelength of light used is `5460Å`. Then angular position of the first dark fringe is

Options

A. `0.08^(@)`
B. `0.16^(@)`
C. `0.20^(@)`
D. `0.31^(@)`

Explanation

To find the angular position of the first dark fringe in Young's double slit experiment, we can follow these steps:

### Step 1: Understand the formula for dark fringes
In Young's double slit experiment, the position of dark fringes on the screen can be calculated using the formula:
\[ x_n = \frac{(2n - 1) \lambda D}{2d} \]
where:
- \( x_n \) is the position of the nth dark fringe,
- \( n \) is the order of the dark fringe (for the first dark fringe, \( n = 1 \)),
- \( \lambda \) is the wavelength of light,
- \( D \) is the distance from the slits to the screen,
- \( d \) is the distance between the two slits.

### Step 2: Identify the given values
From the problem statement, we have:
- Distance between the two sources (slits), \( d = 0.1 \, \text{mm} = 0.1 \times 10^{-3} \, \text{m} \)
- Distance of the screen from the sources, \( D = 20 \, \text{cm} = 20 \times 10^{-2} \, \text{m} \)
- Wavelength of light, \( \lambda = 5460 \, \text{Å} = 5460 \times 10^{-10} \, \text{m} \)

### Step 3: Substitute values into the formula for the first dark fringe
For the first dark fringe, we set \( n = 1 \):
\[ x_1 = \frac{(2 \cdot 1 - 1) \lambda D}{2d} = \frac{\lambda D}{2d} \]

Substituting the values:
\[ x_1 = \frac{(5460 \times 10^{-10} \, \text{m}) \cdot (20 \times 10^{-2} \, \text{m})}{2 \cdot (0.1 \times 10^{-3} \, \text{m})} \]

### Step 4: Calculate the value
Calculating the numerator:
\[ \text{Numerator} = 5460 \times 10^{-10} \times 20 \times 10^{-2} = 1.092 \times 10^{-7} \, \text{m}^2 \]

Calculating the denominator:
\[ \text{Denominator} = 2 \cdot 0.1 \times 10^{-3} = 0.2 \times 10^{-3} \, \text{m} \]

Now, substituting these values into the equation:
\[ x_1 = \frac{1.092 \times 10^{-7}}{0.2 \times 10^{-3}} = 5.46 \times 10^{-4} \, \text{m} = 0.546 \, \text{mm} \]

### Step 5: Find the angular position
The angular position \( \theta \) of the first dark fringe can be found using the small angle approximation:
\[ \theta \approx \tan(\theta) = \frac{x_1}{D} \]
Thus,
\[ \theta \approx \frac{0.546 \times 10^{-3}}{20 \times 10^{-2}} = 0.0273 \, \text{radians} \]

Converting radians to degrees:
\[ \theta \approx 0.0273 \times \frac{180}{\pi} \approx 1.56 \, \text{degrees} \]

### Final Answer
The angular position of the first dark fringe is approximately \( 1.56 \, \text{degrees} \).