In Young's experiment the distance between two slits is `d/3` and the distance between the screen and the slits is 3D. The number of fringes in `1/3` metre on the screen, formed by monochromatic light of wavelength `3lambda`, will be:

Options

A. (a)`d/(9Dlambda)`
B. (b)`d/(27Dlambda)`
C. (c)`d/(81Dlambda)`
D. (d)`d/(Dlambda)`

Answer

To solve the problem, we need to determine the number of fringes formed on the screen in Young's double-slit experiment given the parameters of the setup.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Distance between the slits, \( d = \frac{d}{3} \)
- Distance from the slits to the screen, \( D = 3D \)
- Wavelength of the light, \( \lambda = 3\lambda \)

2. **Calculate the Fringe Width (\( \beta \)):**
The formula for fringe width in Young's experiment is given by:
\[ \beta = \frac{\lambda D}{d} \]
Substituting the values we have:
\[ \beta = \frac{3\lambda \cdot 3D}{\frac{d}{3}} = \frac{9\lambda D}{d} \]

3. **Determine the Number of Fringes in a Given Length:**
The number of fringes (\( n \)) in a length \( L \) on the screen can be calculated using the formula:
\[ n = \frac{L}{\beta} \]
Here, we need to find the number of fringes in \( \frac{1}{3} \) meter:
\[ n = \frac{\frac{1}{3}}{\beta} \]
Substituting for \( \beta \):
\[ n = \frac{\frac{1}{3}}{\frac{9\lambda D}{d}} = \frac{d}{27\lambda D} \]

4. **Final Calculation:**
We can simplify the expression for the number of fringes:
\[ n = \frac{d}{27\lambda D} \]

5. **Conclusion:**
The number of fringes formed in \( \frac{1}{3} \) meter on the screen is:
\[ n = \frac{d}{27\lambda D} \]