The centers of two slits of width a are a distance d apart. If the fourth minimum of the interference pattern occurs at the location of the first minimum of the diffraction pattern for light of wavelength λ, the ratio a/d is equal to
A) 0.
B) .
C) .
D) .
E) .

Question

The centers of two slits of width a are a distance d apart. If the fourth minimum of the interference pattern occurs at the location of the first minimum of the diffraction pattern for light of wavelength λ, the ratio a/d is equal to
A) 0. 
B) . 
C) . 
D) . 
E) .

Quizplus · Accepted Answer

The condition for the nth minimum in the interference pattern is given by
dsinθ = nλ        ------(1)
where d is the distance between the centers of the slits and θ is the angle between the on-axis and the nth minimum point.
The condition for the position of the first minimum of the diffraction pattern is given by
asinθ = λ        -------(2)
where a is the width of the slit.
Given that the fourth minimum of the interference pattern occurs at the location of the first minimum of the diffraction pattern. This means that the angle between the on-axis and the fourth minimum point is the same as the angle between the on-axis and the first minimum point. This implies that sinθ = sinθ'
Thus, from equations (1) and (2), we get
dsinθ' = n'λ and asinθ' = λ
where n' = 1.
Dividing these two equations, we get
a/d = sinθ'/sinθ' = 1
Therefore, a/d = 1, which is equivalent to choice C.

The Centers of Two Slits of Width a Are a Distance