If a single slit fraunhoffer diffraction set up is used with light of wavelength `4000Å`. Distance D between central maximum and first minimum found to be `0.3 cm` in set up if the wavelength changed to `6000Å` the corresponding value of D will be

Options

A. `0.20 cm`
B. `0.24 cm`
C. `0.30 cm`
D. `0.45 cm`

Explanation

To solve the problem, we will use the formula for the distance \( y \) between the central maximum and the first minimum in a single slit Fraunhofer diffraction setup. The formula is given by:

\[ y = \frac{D \cdot \lambda}{d} \]

where:
- \( y \) is the distance from the central maximum to the first minimum,
- \( D \) is the distance from the slit to the screen,
- \( \lambda \) is the wavelength of the light,
- \( d \) is the width of the slit.

### Step 1: Understand the relationship between distances and wavelengths

From the problem, we know that when the wavelength changes, the distance \( D \) will also change according to the relationship:

\[ \frac{y_1}{y_2} = \frac{\lambda_1}{\lambda_2} \]

where:
- \( y_1 \) is the distance for the first wavelength,
- \( y_2 \) is the distance for the second wavelength,
- \( \lambda_1 \) is the first wavelength (4000 Å),
- \( \lambda_2 \) is the second wavelength (6000 Å).

### Step 2: Substitute the known values

Given:
- \( y_2 = 0.3 \) cm (for \( \lambda_2 = 6000 \) Å),
- \( \lambda_1 = 4000 \) Å,
- \( \lambda_2 = 6000 \) Å.

We can rearrange the equation to find \( y_1 \):

\[ y_1 = y_2 \cdot \frac{\lambda_1}{\lambda_2} \]

### Step 3: Calculate \( y_1 \)

Substituting the values:

\[ y_1 = 0.3 \, \text{cm} \cdot \frac{4000 \, \text{Å}}{6000 \, \text{Å}} \]

### Step 4: Simplify the expression

Calculating the fraction:

\[ \frac{4000}{6000} = \frac{2}{3} \]

Thus,

\[ y_1 = 0.3 \, \text{cm} \cdot \frac{2}{3} = 0.3 \cdot 0.6667 = 0.2 \, \text{cm} \]

### Step 5: Final calculation for \( y_1 \)

Now, we can compute:

\[ y_1 = 0.3 \cdot \frac{2}{3} = 0.2 \, \text{cm} \]

### Conclusion

So, the corresponding value of \( D \) when the wavelength is changed to \( 6000 \) Å is:

\[ \boxed{0.2 \, \text{cm}} \]