In Young double slit interference experiment, the distance between two sources is 0.1 mm. The distance of the screen fr0m the sources is 20cm. Wavelength of light used is `5460Å`. Then the angular position of the first dark fringe is

Options

A. `0.08^(@)`
B. `0.16^(@)`
C. `0.20^(@)`
D. `0.32^(@)`

Answer

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

### Step 1: Define the given values
- Distance between the two slits (sources), \( D = 0.1 \, \text{mm} = 0.1 \times 10^{-3} \, \text{m} = 1 \times 10^{-4} \, \text{m} \)
- Distance from the slits to the screen, \( L = 20 \, \text{cm} = 0.2 \, \text{m} \)
- Wavelength of light, \( \lambda = 5460 \, \text{Å} = 5460 \times 10^{-10} \, \text{m} = 5.46 \times 10^{-7} \, \text{m} \)

### Step 2: Use the formula for the angular position of dark fringes
The angular position \( \theta \) of the \( n \)-th dark fringe is given by the formula:
\[ \theta = \frac{(2n - 1) \lambda}{2D} \]
For the first dark fringe, \( n = 1 \):
\[ \theta = \frac{(2 \times 1 - 1) \lambda}{2D} = \frac{\lambda}{2D} \]

### Step 3: Substitute the values into the formula
Now, substituting the values of \( \lambda \) and \( D \):
\[ \theta = \frac{5.46 \times 10^{-7} \, \text{m}}{2 \times (1 \times 10^{-4} \, \text{m})} \]
\[ \theta = \frac{5.46 \times 10^{-7}}{2 \times 10^{-4}} = \frac{5.46 \times 10^{-7}}{2 \times 10^{-4}} = \frac{5.46}{2} \times 10^{-3} = 2.73 \times 10^{-3} \, \text{radians} \]

### Step 4: Convert radians to degrees
To convert radians to degrees, we use the conversion factor \( \frac{180}{\pi} \):
\[ \theta_{\text{degrees}} = \theta \times \frac{180}{\pi} \]
Substituting the value of \( \theta \):
\[ \theta_{\text{degrees}} = 2.73 \times 10^{-3} \times \frac{180}{3.14} \]
Calculating this gives:
\[ \theta_{\text{degrees}} \approx 0.156 \, \text{degrees} \]

### Step 5: Final result
Thus, the angular position of the first dark fringe is approximately:
\[ \theta \approx 0.156 \, \text{degrees} \]