Why is there Proportionality instead of Equality in Young's Experiment?

[Syrus]

Homework Statement

The picture below shows line 2.14, which is derived based on some of the discussion from the text above (I assume it's fairly obvious the experiment is Young's). My question is, why is there a proportionality instead of an equality? No amplitudes are mentioned for the constituent waves eminating from each of the slits, so I figure this should be an equality?

Also, some lines below, the sentence "Note that we are adding the wave amplitudes here." appears. I don't, however, see any amplitude additions. Is this a mistake for "..adding the wave expressions here"?

Homework Equations

The Attempt at a Solution

 

[jedishrfu]

It looks right. They are adding the wave intensities from each slit to determine the intensity on the screen and they show it from a top down perspective.
the sqrt expressions are the pythagorean distance from a given slit to a point on the screen (notice they take the center between the slits as x=0 and so each slit is +-s/2 from the center.

 

[Syrus]

Yes, I unserstand that much, the only thing I am unsure of is why a proportionality is used as opposed to an equality. Any ideas?

 

[haruspex]

They don't know and don't care what the actual intensities are. because they're assuming only small deviations from the midpoint, the two intensities will be almost the same.

 

[paranormal]

I can explain the use of proportionality in Young's Experiment by considering the fundamental principles of wave interference. In this experiment, light waves from two slits interfere with each other to produce a pattern of light and dark fringes on a screen placed behind the slits. These fringes are a result of the superposition of the two waves from the slits, and the intensity of the light at each point on the screen is proportional to the square of the amplitude of the resulting wave.

This proportionality can be seen in the equation for the intensity of the light at a point on the screen, which is given by I ∝ A^2, where A is the amplitude of the resulting wave. This means that the intensity of the light at a point is directly proportional to the square of the amplitude of the resulting wave.

In Young's Experiment, the amplitudes of the waves from each slit are not equal, as the waves travel different distances to reach the screen. This difference in path length results in a phase difference between the two waves, which affects the amplitude of the resulting wave at each point on the screen. Therefore, the intensity of the light at each point is not equal, but rather proportional to the square of the amplitude of the resulting wave.

The sentence "Note that we are adding the wave amplitudes here." refers to the mathematical process of superposition, where the amplitudes of the waves are added together to determine the resulting wave. This is not a physical addition of amplitudes, but rather a mathematical representation of the interference of waves.

In conclusion, the use of proportionality in Young's Experiment is a result of the principles of wave interference, where the intensity of the light at each point on the screen is directly proportional to the square of the amplitude of the resulting wave. This is due to the unequal amplitudes of the waves from each slit and the process of superposition.