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

The path length difference between two waves that interfere destructively at the second minimum of a single slit diffraction pattern is given by: 
$$(PX) - (PY) = d\sin	heta$$ 
where $d$ is the width of the slit and $	heta$ is the angle between the line connecting the slit with the point of interest and the central axis of the diffraction pattern. 
In this case, since the second minimum is formed symmetrically with respect to the central maximum, we have $	heta = \frac{\pi}{2}$. Therefore, 
$$(PX) - (PY) = d\sin\frac{\pi}{2} = d$$ 
Using the relationship $\sin	heta = \frac{\lambda}{d}$, where $\lambda$ is the wavelength of the wave, we have 
$$(PX) - (PY) = d\sin	heta = d\frac{\lambda}{d} = \lambda$$ 
Therefore, the correct answer is D.

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