Diffraction Grating problem (waves and optics)

In summary, using the equation d*sinΘ=mλ and the given values of the grating's slit density and the separation between maxima, we can find the difference between the two wavelengths in the first-order spectrum. By calculating the difference in angle (ΔΘ) and then using the small angle approximation, we can determine the difference in wavelength (Δλ). The difference in angle can be found by subtracting the sine values of the two angles, and using the total distance on the screen as the opposite side in the equation. The result for ΔΘ is 0.001225 radians, which leads to a difference in wavelength of 1.36*10^-8 meters.
  • #1
pondzo
169
0

Homework Statement



Visible light passes through a diffraction grating that has 900 slits per centimeter, and the interference pattern is observed on a screen that is 2.58m from the grating. In the first-order spectrum, maxima for two different wavelengths are separated on the screen by 3.16 mm . What is the difference between these wavelengths?

Homework Equations



d*sinΘ=mλ

The Attempt at a Solution



d = 1/90000 = 1.111*10-5

d*sin(Θ1)=λ1 and d*sin(Θ2)=λ2

using subtraction d*ΔsinΘ=Δλ (and using small angle approx d*ΔΘ=Δλ)

But I am not sure how to find ΔΘ (or how even to visualize ΔΘ)

thanks in advance.
 
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  • #2
This help ?
Azd61G7.jpg

On the left is the grating, on the right is the screen.
(grating details not shown on this scale)
 
  • #3
Thank you for making it clearer! i was originally thinking this but i didn't think i could calculate it since i didn't have all of the distance of the opposite side (the screen).

Is this the correct way to go about it?

Sin(ΔΘ) = sin(Θ2) - sin(Θ1)
= x/2.58 + (3.16*10-3-x)/2.58 (where x is the total distance on the screen)
= (3.16*10-3)/2.58
ΔΘ=0.001225

and λ=1.225*10-3*1.11*10-5
=1.36*10-8
 

FAQ: Diffraction Grating problem (waves and optics)

1. What is a diffraction grating?

A diffraction grating is a device that consists of a large number of closely spaced parallel slits or grooves. It is used to separate light into its component wavelengths, similar to a prism.

2. How does a diffraction grating work?

When light passes through a diffraction grating, it is diffracted or bent by each of the slits or grooves in the grating. This causes the light to interfere with itself, resulting in a pattern of bright and dark fringes. Each fringe corresponds to a different wavelength of light, allowing the grating to separate the light into its individual wavelengths.

3. What is the equation for calculating the angle of diffraction for a diffraction grating?

The equation is given by: dsinθ = mλ, where d is the distance between adjacent slits in the grating, θ is the angle of diffraction, m is the order of the fringe, and λ is the wavelength of the light.

4. How does the number of slits in a diffraction grating affect the resulting pattern?

The number of slits in a diffraction grating directly affects the number of fringes in the resulting pattern. As the number of slits increases, the fringes become more closely spaced and the resolution of the grating improves. However, too many slits can also cause the fringes to overlap, making it difficult to distinguish them.

5. How is a diffraction grating different from a prism?

A diffraction grating and a prism both separate light into its component wavelengths, but they do so in different ways. A prism uses refraction to bend light, while a diffraction grating uses diffraction. Additionally, a prism produces a continuous spectrum of colors, whereas a diffraction grating produces discrete lines or fringes corresponding to specific wavelengths of light.

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