To solve this problem, we need to delve into the principles of interference and the relationship between intensity and path difference in coherent light sources. When two coherent sources emit light, they can interfere constructively or destructively depending on their phase relationship and the path difference between the light waves reaching a point.
Understanding the Setup
We have two coherent sources, A and B, emitting light of wavelength λ. Since they are initially in phase, the light waves from both sources will reinforce each other at certain points, leading to constructive interference, while at other points, they will cancel each other out, resulting in destructive interference.
Intensity and Path Difference
The intensity at a point in an interference pattern is related to the path difference between the waves arriving from the two sources. The maximum intensity occurs when the path difference is an integer multiple of the wavelength (nλ, where n is an integer). Conversely, the minimum intensity occurs when the path difference is an odd multiple of half the wavelength ((n + 0.5)λ).
Given Conditions
In this scenario, we know that the intensity at point P is half of the maximum intensity. This situation indicates that the path difference at point P is such that it leads to a specific fraction of the maximum intensity. The relationship between intensity and path difference can be expressed mathematically:
- Maximum intensity (I_max) occurs at path difference = nλ.
- Minimum intensity (I_min) occurs at path difference = (n + 0.5)λ.
For the intensity at point P to be half of the maximum intensity, we can use the formula:
I = I_max * cos²(Δφ/2), where Δφ is the phase difference.
Since I = 0.5 * I_max, we can set up the equation:
0.5 = cos²(Δφ/2).
Calculating the Phase Difference
From the equation above, we find that:
cos²(Δφ/2) = 0.5 implies that Δφ/2 = π/4 or 3π/4.
This means that the phase difference Δφ can be either π/2 or 3π/2, corresponding to a path difference of:
Δx = (Δφ * λ) / (2π) = (π/2 * λ) / (2π) = λ/4.
Relating Path Difference to Source Separation
The path difference Δx is also related to the distance D from source A to point P and the separation d between the two sources A and B. The path difference can be expressed as:
Δx = D * sin(θ), where θ is the angle between the line connecting the sources and the line to point P.
For small angles, sin(θ) can be approximated as θ (in radians), leading to:
Δx = D * (d / (D + d)) for small angles, which simplifies to Δx ≈ D * (d / D) = d.
Final Calculation
Given that the path difference we calculated is λ/4, we can equate this to the minimum separation d:
d = λ/4.
Thus, the minimum separation between the sources A and B must be equal to λ/4 for the intensity at point P to be half of the maximum intensity.
Summary
In summary, when dealing with coherent sources and interference patterns, the relationship between intensity and path difference is crucial. For the given problem, the minimum separation between the two sources A and B, when the intensity at point P is half of the maximum intensity, is determined to be:
d = λ/4.
