# Finding highest order number for a diffraction grating

• Oijl
In summary, the diffraction grating orders for light of wavelength 680 nm show that the 4th and 8th order maxima are missing due to coincidence with the minima of the single-slit diffraction pattern. The highest visible orders are 9, 7, and 6, with the 10th order being a limiting case.
Oijl

## Homework Statement

Light of wavelength 680 nm is incident normally on a diffraction grating. Two adjacent maxima occur at angles given by sin θ = 0.2 and sin θ = 0.3, respectively. The fourth-order maxima are missing.

Slit separation d = 6.8 µm
Slit width a = 1.7 µm

What are the largest, second largest, and third largest values of the order number m of the maxima produced by the grating?

## Homework Equations

((y is wavelength))
[diffraction grating, maxima]: dsinθ=my
[single slit, minima]: asinθ=my

## The Attempt at a Solution

I know, from the previous part of this problem, that d = 4a. So, the equation for maxima for the grating can be rewritten

4asinθ=my

I know that sinθ can at most be 1. Therefore, the equation for greatest m is 4a=my, and this solves where m = 10.

So I know the greatest value of m is 10. But how can I figure the penultimate and antepenultimate values of m? If I decrease m by one, so that m = 9, I have the equation

4asinθ=9y

for which a θ exists that will solve it. And for any m less than 10, there is a θ that will make the statement (4asinθ=my) true.

But the answer is not m = 10, 9, 8. How do I find the other order numbers?

This is a very old (12+ years at the time of answering) question. Since there is no possibility of guiding the OP through the solution, here’s a fairly complete explanation.

Using ‘nλ = dsinθ’ the diffraction grating orders are at:
sinθ₁ = 1*680e-9 / 6.8e-6 = 0.1
sinθ₂ = 2*680e-9 / 6.8e-6 = 0.2
.
.
sinθ₁₀ = 10*680e-9 / 6.8e-6 = 1

Some of the diffraction grating orders will be missing. These are the ones which coincide with the minima of the single-slit diffraction pattern. For slit width = a, the single-slit minima are given by mλ = asinφ and there will be two of them:
sinφ₁ = 1*680e-9 / 1.7e-6 = 0.4
sinφ₂ = 2*680e-9 / 1.7e-6 = 0.8

As a result, the 4th and 8th order diffraction grating maxima (at sinθ₄ = 0.4 and sinθ₈ = 0.8) will both be missing.

n=10 is a limiting case. The 10th order cannot actually be seen in practice. The highest order visible is the 9th . So the three highest orders of grating maximum are 9, 7 and 6.

(If you wanted to argue that the 10th order of the grating should count, the answer would be 10, 9, and 7.)

## 1. How do I calculate the highest order number for a diffraction grating?

The highest order number for a diffraction grating can be calculated using the formula n = d/λ, where n is the highest order number, d is the distance between the slits on the grating, and λ is the wavelength of the incident light.

## 2. What is the significance of the highest order number in a diffraction grating?

The highest order number determines the maximum number of diffraction patterns that can be observed on the screen. It also affects the intensity and spacing of the diffraction peaks.

## 3. Can the highest order number be greater than the number of slits on the grating?

No, the highest order number cannot be greater than the number of slits on the grating. This is because the diffraction pattern is formed by light passing through the slits, and if there are more slits than the highest order number, some of the light will not contribute to the pattern.

## 4. How does the highest order number change with different wavelengths of light?

The highest order number is directly proportional to the wavelength of light. This means that as the wavelength increases, the highest order number also increases. This relationship can be seen in the equation n = d/λ, where n and λ are inversely proportional.

## 5. What factors can affect the accuracy of calculating the highest order number for a diffraction grating?

The accuracy of calculating the highest order number can be affected by factors such as the accuracy of the distance between the slits, the accuracy of the wavelength of light, and any errors in the measurement or calculation process. Additionally, environmental factors such as temperature and air currents can also impact the accuracy of the results.

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