How Do You Derive the Diffraction Formula nλ = d sinθ?

[pianogirl]

Homework Statement

Basically there's a question that I'm doing which asks me to show how to derive:
n lambda=d sin theta
We haven't learned it yet and so how do you do it?

Homework Equations

n lambda=d sin theta

The Attempt at a Solution

Not a clue!

Thanks!

 

[Delphi51]

Take a look at this explanation
http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/slits.html#c1
Basically the little triangle with the δ in it is similar to the big one with the D and y, as long as D is very large. You write a sinθ formula for both of them and combine them to get the formula you want.

 

[kerimek]

Hello,

Deriving the diffraction formula involves using principles from wave optics and trigonometry. First, we need to understand that when light passes through a narrow slit, it diffracts or bends around the edges of the slit. This creates a pattern of bright and dark spots, known as the diffraction pattern.

To derive the formula, we start by considering a single slit of width d. When light passes through this slit, it creates a wavefront that spreads out in all directions. However, due to the narrow width of the slit, the wavefronts become curved. This can be represented by a series of circular arcs, with the center of the slit as the center of each arc.

Next, we consider a point P on the screen where the diffraction pattern is formed. This point is a distance L away from the slit. We can draw a line from the top edge of the slit to point P, creating a right triangle. The angle made by this line with the horizontal is theta.

Using trigonometry, we can see that the length of the arc from the center of the slit to point P is equal to d sin theta. This is because the sine of an angle is equal to the opposite side (in this case, the distance from the center of the slit to point P) divided by the hypotenuse (the distance from the slit to point P).

Now, we know that the distance between bright spots in the diffraction pattern is equal to the wavelength of light, denoted by lambda. This means that the distance between the center of the slit and the first bright spot is also equal to lambda. Using this information, we can set up the following equation:

n lambda = d sin theta

Where n is the number of bright spots away from the center of the slit. This is known as the diffraction formula, and it tells us the relationship between the distance between bright spots, the wavelength of light, and the angle of diffraction.

I hope this explanation helps you understand how to derive the diffraction formula. It is important to have a good understanding of wave optics and trigonometry in order to fully understand the derivation. Let me know if you have any further questions.

Best,