White light is used to illuminate the two slits in a Young's double slit experiment. The separation between the slits is b and the screen is at a distance `d (gt gtb)` from the slits At a point on the screen directly in front of one of the slits, certain wavelengths are missing some of these missing wavelengths are

Options

A. `lambda=b^(2)=b^(2)//d`
B. `lambda=b^(2)=2b^(2)//d`
C. `lambda=b^(2)=b^(2)//3d`
D. `lambda=2b^(2)=b^(2)//3d`

Answer

To solve the problem of determining the missing wavelengths in a Young's double slit experiment with white light, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have two slits separated by a distance \( b \).
- A screen is placed at a distance \( d \) from the slits.
- We are looking for the wavelengths that are missing at a point on the screen directly in front of one of the slits.

2. **Condition for Dark Fringes**:
- The condition for dark fringes in a double-slit experiment is given by the formula:
\[ y = \frac{(2n - 1) \lambda d}{b} \]
- Here, \( y \) is the distance from the central maximum to the dark fringe, \( n \) is the order of the dark fringe, \( \lambda \) is the wavelength of light, \( d \) is the distance from the slits to the screen, and \( b \) is the separation between the slits.

3. **Finding the Position of Dark Fringes**:
- At the point directly in front of one of the slits, the distance \( y \) can be set to \( \frac{b}{2} \) (half the distance between the slits).
- Substituting \( y = \frac{b}{2} \) into the dark fringe formula gives:
\[ \frac{b}{2} = \frac{(2n - 1) \lambda d}{b} \]

4. **Rearranging the Equation**:
- Cross-multiplying gives:
\[ b^2 = (2n - 1) \lambda d \]
- Rearranging for \( \lambda \) gives:
\[ \lambda = \frac{b^2}{(2n - 1) d} \]

5. **Calculating Missing Wavelengths**:
- For different values of \( n \):
- For \( n = 1 \):
\[ \lambda_1 = \frac{b^2}{1 \cdot d} = \frac{b^2}{d} \]
- For \( n = 2 \):
\[ \lambda_2 = \frac{b^2}{3d} \]
- For \( n = 3 \):
\[ \lambda_3 = \frac{b^2}{5d} \]

6. **Identifying Missing Wavelengths**:
- The wavelengths that are missing at the point directly in front of one of the slits are:
- \( \frac{b^2}{d} \)
- \( \frac{b^2}{3d} \)
- \( \frac{b^2}{5d} \)

7. **Conclusion**:
- The missing wavelengths correspond to the values derived above. Thus, the correct answers for the missing wavelengths are \( \frac{b^2}{d} \) and \( \frac{b^2}{3d} \).