In Young.s double slit experiment the distance between the sources is `7.7 mu""m`. If the wavelength of light used is 500 nanometre, the angular position of the third dark ringe from the centre fringe is

Options

A. `10.9^(0)`
B. `0.15^(0)`
C. `11.3^(0)`
D. `9.4^(0)`

Answer

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

### Step 1: Understand the parameters given
- Distance between the slits (d) = 7.7 µm = \(7.7 \times 10^{-6}\) m
- Wavelength of light (λ) = 500 nm = \(500 \times 10^{-9}\) m
- We need to find the angular position of the third dark fringe, so n = 3.

### Step 2: Use the formula for dark fringes
The angular position (θ) of the dark fringes in Young's double slit experiment can be calculated using the formula:
\[ \theta = \sin^{-1}\left(\frac{(n + 0.5) \lambda}{d}\right) \]
For the third dark fringe, we substitute n = 3.

### Step 3: Substitute the values into the formula
Substituting the values into the formula gives:
\[ \theta = \sin^{-1}\left(\frac{(3 + 0.5) \times 500 \times 10^{-9}}{7.7 \times 10^{-6}}\right) \]
This simplifies to:
\[ \theta = \sin^{-1}\left(\frac{3.5 \times 500 \times 10^{-9}}{7.7 \times 10^{-6}}\right) \]

### Step 4: Calculate the argument of the sine function
Calculating the argument:
\[ \theta = \sin^{-1}\left(\frac{1750 \times 10^{-9}}{7.7 \times 10^{-6}}\right) \]
\[ \theta = \sin^{-1}\left(\frac{1750}{7.7} \times 10^{-3}\right) \]
\[ \theta = \sin^{-1}(0.22727) \]

### Step 5: Calculate the angle in radians
Using a calculator, we find:
\[ \theta \approx 0.2309 \text{ radians} \]

### Step 6: Convert radians to degrees
To convert radians to degrees, we use the conversion factor \( \frac{180}{\pi} \):
\[ \theta_{degrees} = 0.2309 \times \frac{180}{\pi} \approx 13.23 \text{ degrees} \]

### Step 7: Conclusion
The angular position of the third dark fringe from the center fringe is approximately **13.23 degrees**.