To find the angular position for the first dark fringe in Young's double-slit experiment, we can follow these steps:
### Step 1: Understand the parameters given
We have the following values:
- Distance between the slits, \( d = 0.1 \, \text{mm} = 0.1 \times 10^{-3} \, \text{m} = 10^{-4} \, \text{m} \)
- Distance from the slits to the screen, \( D = 20 \, \text{cm} = 0.2 \, \text{m} \)
- 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 position of dark fringes
In Young's double-slit experiment, the position of the dark fringes can be given by the formula:
\[
y_n' = \frac{(2n-1) \lambda D}{2d}
\]
For the first dark fringe, \( n = 1 \):
\[
y_1' = \frac{(2 \cdot 1 - 1) \lambda D}{2d} = \frac{\lambda D}{2d}
\]
### Step 3: Calculate \( y_1' \)
Substituting the values into the equation:
\[
y_1' = \frac{(5.46 \times 10^{-7} \, \text{m})(0.2 \, \text{m})}{2(10^{-4} \, \text{m})}
\]
Calculating this gives:
\[
y_1' = \frac{(5.46 \times 10^{-7})(0.2)}{2 \times 10^{-4}} = \frac{1.092 \times 10^{-7}}{2 \times 10^{-4}} = \frac{1.092 \times 10^{-7}}{2 \times 10^{-4}} = 5.46 \times 10^{-4} \, \text{m}
\]
### Step 4: Find the angular position \( \theta \)
The angular position \( \theta \) can be calculated using the formula:
\[
\theta = \frac{y_1'}{D}
\]
Substituting the values:
\[
\theta = \frac{5.46 \times 10^{-4} \, \text{m}}{0.2 \, \text{m}} = 2.73 \times 10^{-3} \, \text{radians}
\]
### Step 5: Convert radians to degrees
To convert radians to degrees, we use the conversion factor \( \frac{180}{\pi} \):
\[
\theta_{\text{degrees}} = \theta \times \frac{180}{\pi}
\]
Calculating this gives:
\[
\theta_{\text{degrees}} = 2.73 \times 10^{-3} \times \frac{180}{\pi} \approx 0.156 \, \text{degrees}
\]
### Final Answer
Thus, the angular position for the first dark fringe is approximately:
\[
\theta \approx 0.16 \, \text{degrees}
\]
