Four monochromatic and coherent sources of light, emitting waves in phase of wavelength `lambda`, are placed at the points x = 0, d 2d and 3d on the x-axis. Then

Options

A. point having `|x| gt gt d` appear dark if `d = lambda//4`
B. point having `|x| gt gt d` appear dark if `d = lambda//8`
C. point having `|x| gt gt d` appear maximum bright dark if `d = lambda//4`
D. point having `|x| gt gt d` appear maximum bright dark if `d = lambda//8`

Answer

**Step-by-Step Solution:**

Given four monochromatic and coherent sources of light placed at points \( x = 0, d, 2d, \) and \( 3d \) on the x-axis, we need to analyze the interference pattern produced by these sources.

1. **Identify the Sources:**
The sources are located at:
- Source 1 at \( x = 0 \)
- Source 2 at \( x = d \)
- Source 3 at \( x = 2d \)
- Source 4 at \( x = 3d \)

2. **Determine the Path Difference:**
To find the interference pattern, we need to calculate the path difference between the waves coming from these sources to a point \( P \) on the screen. Let’s denote the distance from the point \( P \) to the sources as follows:
- Distance from Source 1 to \( P \): \( r_1 \)
- Distance from Source 2 to \( P \): \( r_2 \)
- Distance from Source 3 to \( P \): \( r_3 \)
- Distance from Source 4 to \( P \): \( r_4 \)

The path differences between pairs of sources can be calculated as:
- Path difference between Source 1 and Source 2: \( \Delta r_{12} = r_2 - r_1 \)
- Path difference between Source 1 and Source 3: \( \Delta r_{13} = r_3 - r_1 \)
- Path difference between Source 1 and Source 4: \( \Delta r_{14} = r_4 - r_1 \)
- Path difference between Source 2 and Source 3: \( \Delta r_{23} = r_3 - r_2 \)
- Path difference between Source 2 and Source 4: \( \Delta r_{24} = r_4 - r_2 \)
- Path difference between Source 3 and Source 4: \( \Delta r_{34} = r_4 - r_3 \)

3. **Condition for Constructive and Destructive Interference:**
For constructive interference (bright fringes), the path difference must be an integer multiple of the wavelength \( \lambda \):
\[ \Delta r = n\lambda \quad (n = 0, 1, 2, \ldots) \]
For destructive interference (dark fringes), the path difference must be an odd multiple of half the wavelength:
\[ \Delta r = \left(n + \frac{1}{2}\right)\lambda \quad (n = 0, 1, 2, \ldots) \]

4. **Analyze the Interference Pattern:**
Since the sources are coherent and in phase, we can expect an interference pattern with alternating bright and dark fringes. The positions of these fringes will depend on the path differences calculated above.

5. **Final Conclusion:**
The resulting interference pattern will show a series of bright and dark fringes on the screen, with the positions determined by the conditions for constructive and destructive interference.