A single slit of width a is illuminated by violet light of wavelength `400nm` and the width of the diffraction pattern is measured as y. When half of the slit width is covered and illuminated by yellow light of wavelength `600nm`, the width of the diffraction pattern is

Options

A. (a) The pattern vanishes and the width is zero
B. (b) `y//3`
C. (c) `3y`
D. (d) None of these

Explanation

To solve the problem, we need to understand the relationship between the width of the diffraction pattern produced by a single slit and the parameters involved, specifically the wavelength of light and the width of the slit.

### Step-by-Step Solution:

1. **Understanding the Diffraction Pattern Width:**
The width of the central maximum in a single slit diffraction pattern is given by the formula:
\[ Y = \frac{2 \lambda D}{a} \]
where \( Y \) is the width of the central maximum, \( \lambda \) is the wavelength of the light, \( D \) is the distance from the slit to the screen, and \( a \) is the width of the slit.

2. **Calculating the Width for Violet Light:**
For the first case, we have:
- Wavelength of violet light, \( \lambda_1 = 400 \, \text{nm} = 400 \times 10^{-9} \, \text{m} \)
- Width of the slit, \( a \) (let's keep it as \( a \) for now).
- The width of the diffraction pattern is given as \( Y \).

Using the formula:
\[ Y = \frac{2 \lambda_1 D}{a} = \frac{2 \times 400 \times 10^{-9} D}{a} \]

3. **Calculating the Width for Yellow Light:**
In the second case, half of the slit width is covered, so the new width of the slit becomes \( a/2 \). The wavelength of yellow light is:
- \( \lambda_2 = 600 \, \text{nm} = 600 \times 10^{-9} \, \text{m} \)

The width of the diffraction pattern for yellow light is:
\[ Y' = \frac{2 \lambda_2 D}{a/2} = \frac{2 \times 600 \times 10^{-9} D}{a/2} = \frac{4 \times 600 \times 10^{-9} D}{a} \]

4. **Finding the Ratio of Widths:**
Now we can find the ratio of the widths of the diffraction patterns:
\[ \frac{Y'}{Y} = \frac{\frac{4 \times 600 \times 10^{-9} D}{a}}{\frac{2 \times 400 \times 10^{-9} D}{a}} = \frac{4 \times 600}{2 \times 400} = \frac{4 \times 600}{800} = \frac{2400}{800} = 3 \]

5. **Conclusion:**
Therefore, the width of the diffraction pattern when half of the slit width is covered and illuminated by yellow light is:
\[ Y' = 3Y \]

### Final Answer:
The width of the diffraction pattern when half of the slit width is covered and illuminated by yellow light is \( 3Y \).