The centres 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, the ratio a/d is equal to:
Here are the various interference conditions for interference and diffraction:
Interference conditions for double slit:
MAX: dsinθ = mλ
MIN: dsinθ = (m+½)λ
Diffraction conditions for a single slit:
MAX: asinθ = (m+½)λ
MIN: asinθ = mλ
Diffraction conditions for diffraction grating:
MAX: asinθ = mλ
MIN: asinθ = (m+½)λ
The Attempt at a Solution
I will walk through my reasoning...
I've classified the diffraction component of this problem as diffraction grating rather than diffraction through a single slit, because based on this setup, there are two slits for diffraction to occur through. While we normally see grating in the order of 2500grates/cm, 2grates/cm would still be considered grating.
So based on that logic, the conditions for minimum for both are:
Diffraction: dsinθ = (m+½)λ, where m=1
Interference: asinθ = (m+½)λ, where m=4
Now, when I plug in all the values for m, and cancel out all the similarities (sinθ, λ):
asinθ=(1+½)λ --> a=1.5
dsinθ=(4+½)λ --> d=4.5
The ratio I get for a/d = 1.5/4.5 = ⅓
HOWEVER, I noticed that if I cancel out ½ rather than adding it to m like I did above, then the ratio for a/d=¼.
What am I doing wrong?