In young's double slit experiment, the two slits are `0.2` mm apart. The fringes for light of wavelength `6000Å` are found on the screen `80` cm away. The distance of fifth dark fringe, from the central fringe, will be

Options

A. `6.8` mm
B. `7.8` mm
C. `9.8` mm
D. `10.8` mm

Explanation

To find the distance of the fifth dark fringe from the central 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 the dark fringes is given by the formula:
\[ y_n = \left(n + \frac{1}{2}\right) \frac{\lambda D}{d} \]
where:
- \( y_n \) is the distance of the nth dark fringe from the central maximum,
- \( n \) is the order of the dark fringe (for the fifth dark fringe, \( n = 4 \)),
- \( \lambda \) is the wavelength of the light,
- \( D \) is the distance from the slits to the screen,
- \( d \) is the distance between the slits.

### Step 2: Convert the given values into appropriate units
1. The distance between the slits \( d = 0.2 \, \text{mm} = 0.2 \times 10^{-3} \, \text{m} \).
2. The wavelength \( \lambda = 6000 \, \text{Å} = 6000 \times 10^{-10} \, \text{m} \).
3. The distance to the screen \( D = 80 \, \text{cm} = 0.8 \, \text{m} \).

### Step 3: Substitute the values into the formula
Using \( n = 4 \) for the fifth dark fringe:
\[ y_4 = \left(4 + \frac{1}{2}\right) \frac{\lambda D}{d} \]
\[ y_4 = \left(4 + 0.5\right) \frac{6000 \times 10^{-10} \, \text{m} \times 0.8 \, \text{m}}{0.2 \times 10^{-3} \, \text{m}} \]

### Step 4: Calculate the expression
1. Calculate \( 4 + 0.5 = 4.5 \).
2. Substitute and simplify:
\[ y_4 = 4.5 \times \frac{6000 \times 10^{-10} \times 0.8}{0.2 \times 10^{-3}} \]
\[ y_4 = 4.5 \times \frac{6000 \times 0.8}{0.2} \times 10^{-7} \]
\[ y_4 = 4.5 \times 24000 \times 10^{-7} \]
\[ y_4 = 108000 \times 10^{-7} \]
\[ y_4 = 10.8 \, \text{mm} \]

### Conclusion
The distance of the fifth dark fringe from the central fringe is **10.8 mm**.

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