The diagram shows a single slit with the direction to a point P on a distant screen shown. At P, the pattern has its second minimum (from its central maximum). If X and Y are the edges of the slit, what is the path length difference (PX) - (PY)?
A) $\lambda$/2
B) $\lambda$
C) 3$\lambda$/2
D) 2$\lambda$
E) 5$\lambda$/2

Question

The diagram shows a single slit with the direction to a point P on a distant screen shown. At P, the pattern has its second minimum (from its central maximum). If X and Y are the edges of the slit, what is the path length difference (PX) - (PY)? 
A) $\lambda$/2 
B) $\lambda$ 
C) 3$\lambda$/2 
D) 2$\lambda$ 
E) 5$\lambda$/2

Quizplus · Accepted Answer

For the second minimum, we know that the path length difference between waves from the top and bottom edges of the slit must be half a wavelength, as they will interfere destructively. Therefore, we have:
(PX) - (PY) = (XY)sinθ = (XY) / 2
Using the geometry of the triangle, we can see that:
sinθ = h / L, where h is the distance between the center of the pattern and the second minimum, and L is the distance from the slit to the screen.
Therefore, (XY) / 2 = hL / (L^2 + d^2) where d is the width of the slit.
Using the small angle approximation (sinθ ≈ θ), we can simplify this to:
(XY) / 2 = hL / L^2 = h / L
Therefore, (PX) - (PY) = (XY) / 2 = h / L
Since we are not given any specific values for h or L, we can express our answer purely in terms of the wavelength and the slit width, which gives us choice D.

The Diagram Shows a Single Slit with the Direction to a Point