# Small Angle Approximation in Single Slit Interference

• lulzury
In summary: And it's not just "very small", it's "small enough to make the math easy and accurate enough for the purpose at hand".
lulzury

## Homework Statement

A monochromatic light source is used with a double slit to create an interference pattern on a screen that is 2.00 meters away. If the 2nd bright spot is observed 8.73 mm above the central maximum, can the small angle approximation be used? Show and/or explain your reasoning

## d\sinθ=mλ ##

## The Attempt at a Solution

I might be over-thinking this. I know you can have a small angle approximation if D >> d, but in this case I don't know d yet, so I first wanted to relate d to y without first assuming a small angle approximation (sinθ ≈ tanθ ≈ θ), but I get stuck:

I get the following relation:
## D=\frac{d}{2\tan(γ)} ##
## D= (y+\frac{d}{2}) * \frac{1}{ \tan(θ)} ##

## \frac{d}{2}\cot(γ) = y\cot(θ) + \frac{d}{2}\cot(θ) ##

I'm not sure how to solve for d here so that I can show that D >> d.

D>>d/2 is necessary for other approximations (all your formulas wouldn't work without that condition).

The small angle approximation that is relevant here is a small θ. You can calculate θ with D and y alone in a single step.

lulzury
mfb, that makes sense thank you!

So in this case tan(θ) = y/D

## θ = \arctan(\frac{y}{D}) ##
## θ = \arctan(\frac{8.73*10e-3}{2}) ##

θ = 0.00436 rad ~ 0.25 degrees
That is a very small angle, so I can use a small angle approximation here!

Right.

lulzury

## 2. How is the small angle approximation used in single slit interference?

The small angle approximation is used to simplify the calculation of the phase difference between the waves diffracted from a single slit. It assumes that the angle of diffraction is small, allowing for easy calculation of the phase difference using trigonometric functions.

## 3. What are the limitations of the small angle approximation in single slit interference?

The small angle approximation is only valid for small angles of diffraction, typically less than 10 degrees. It also assumes that the slit width is much smaller than the wavelength of the incident light. If these conditions are not met, the small angle approximation may lead to significant errors in the calculations.

## 4. How does the small angle approximation affect the interference pattern in single slit interference?

The small angle approximation does not significantly affect the overall interference pattern, but it can change the shape and intensity of the fringes. This is because the small angle approximation simplifies the calculation of the phase difference, which affects the positions and intensities of the fringes.

## 5. Can the small angle approximation be used in other types of interference?

Yes, the small angle approximation can also be used in other types of interference, such as double slit interference and diffraction grating interference. However, it is important to note that the conditions for using the small angle approximation may vary depending on the type of interference being studied.

Replies
4
Views
796
Replies
1
Views
2K
Replies
11
Views
3K
Replies
4
Views
2K
Replies
8
Views
2K
Replies
1
Views
7K
Replies
2
Views
2K
Replies
7
Views
4K
Replies
2
Views
2K
Replies
8
Views
2K