Interference vs. diffraction patterns

In summary: So, in summary, the ratio a/d for the centres of two slits of width a being a distance d apart, where the fourth minimum of the interference pattern occurs at the location of the first minimum of the diffraction pattern for light, is equal to 1/4.
  • #1
vetgirl1990
85
3

Homework Statement


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:

ANS: 1/4

Homework Equations


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?
 
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  • #2
vetgirl1990 said:
If the fourth minimum of the interference
Could you please check again in the original problem, is it minimum or maximum?
 
  • #3
blue_leaf77 said:
Could you please check again in the original problem, is it minimum or maximum?
The original question does indeed say minimum.
 

1. What is the difference between interference and diffraction patterns?

Interference patterns occur when two or more waves overlap and interact with each other, resulting in a distinct pattern of alternating bright and dark regions. Diffraction patterns, on the other hand, occur when a wave passes through a small opening or around an obstacle, resulting in a spreading out or bending of the wave. In short, interference involves the interaction of multiple waves, while diffraction involves the bending or spreading of a single wave.

2. What causes interference patterns?

Interference patterns are caused by the superposition of two or more waves. This means that the waves overlap and their amplitudes either add together (constructive interference) or cancel each other out (destructive interference), creating a distinct pattern of bright and dark regions.

3. How are diffraction patterns formed?

Diffraction patterns are formed when a wave encounters an obstacle or passes through a narrow opening. The wave spreads out as it passes through the opening or around the obstacle, resulting in a pattern of alternating bright and dark regions. The size of the opening or obstacle and the wavelength of the wave determine the shape and size of the diffraction pattern.

4. What is the difference between single-slit and double-slit diffraction?

In single-slit diffraction, a wave passes through a single narrow opening or slit, resulting in a diffraction pattern with a single bright spot in the center and alternating dark and bright regions on either side. In double-slit diffraction, a wave passes through two narrow slits close together, resulting in a more complex pattern of alternating bright and dark regions. This is because the waves from each slit overlap and interfere with each other.

5. What are some real-life applications of interference and diffraction patterns?

Interference and diffraction patterns have many practical applications in fields such as optics, acoustics, and signal processing. Examples include diffraction gratings used in spectroscopy and optical communication systems, and the use of interference patterns in technologies such as holography and interferometry. These patterns also play a crucial role in understanding and analyzing natural phenomena, such as the behavior of light and sound waves in various environments.

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