To understand why the farthest minimum occurs at a point P where the path difference is λ/2, we first need to delve into the principles of interference. When light waves from two sources, S1 and S2, meet at a point, they can either constructively interfere (resulting in a bright spot) or destructively interfere (resulting in a dark spot). The condition for destructive interference is that the path difference between the two waves arriving at the detector must be an odd multiple of half the wavelength, which can be expressed mathematically as:

Understanding Path Difference

The path difference (Δ) is the difference in the distance traveled by the two waves from the sources S1 and S2 to the point P. For destructive interference, we want:

• Δ = (m + 0.5)λ, where m is an integer (0, 1, 2, ...)

For the first minimum, we set m = 0, which gives us:

• Δ = 0.5λ

For the second minimum, we set m = 1, leading to:

• Δ = 1.5λ

Continuing this pattern, we see that the farthest minimum corresponds to the largest value of m that still results in a detectable minimum. However, in practical terms, we often refer to the first minimum as the farthest minimum because it is the first point where we observe a dark fringe.

Setting Up the Problem

Now, let’s solve the problem with the given parameters:

• Wavelength (λ) = 600 nm = 600 x 10^-9 m
• Distance between sources (d) = 1.0 x 10^-2 cm = 1.0 x 10^-4 m

Calculating the Position of the Farthest Minimum

To find the position of the farthest minimum, we can use the geometry of the situation. The path difference Δ at a distance y from the midpoint between S1 and S2 can be approximated using the small angle approximation:

• Δ = d * sin(θ)

For small angles, sin(θ) can be approximated by θ (in radians), which gives us:

• Δ ≈ d * (y / L)

Here, L is the distance from the sources to the detector. Setting Δ = 0.5λ for the first minimum, we have:

• d * (y / L) = 0.5λ

Rearranging this gives:

• y = (0.5λ * L) / d

Finding y for the Farthest Minimum

To find the farthest minimum, we need to determine the maximum value of y that still satisfies the condition for destructive interference. The maximum path difference occurs when the angle θ approaches 90 degrees, which means the detector is directly in line with one of the sources. However, for practical purposes, we will consider the first minimum:

• y = (0.5 * 600 x 10^-9 m * L) / (1.0 x 10^-4 m)

Assuming L is sufficiently large, we can simplify our calculations. If we take L to be a large distance, the value of y will increase proportionally. However, the exact position of the farthest minimum will depend on the specific setup and distance L.

Conclusion

In summary, the farthest minimum occurs at a point where the path difference is λ/2 because this is the condition for the first instance of destructive interference. By applying the formula for path difference and understanding the geometry of the setup, we can calculate the position of the minimum. The actual distance y will depend on the distance L from the sources to the detector, but the relationship remains consistent across setups.