Diffraction Grating ( Waves )

[Nemo's]

Homework Statement

White light is incident normally on a diffraction grating with slit separation d of 2.00x10^-6m.
a)Calculate the angle between the red and violet ends of the first-order spectrum. The visible spectrum has wavelength between 400nm and 700nm.
b)Explain why the second and the third order spectra overlap.

Homework Equations

sinθ=nλ/d

The Attempt at a Solution

I was able to solve part (a) by substituting in the above formula with λ=700nm for red and λ=400nm for violet.
However I don't really know how to explain (b) I was thinking that for the next spectrum the path difference for red would be 1 complete wavelength so 400+400=800, for violet 700+700=1400.
For the third spectrum the path difference would be for red 400+400+400= 1200.
So violet from 2nd spectrum seems to be interfering with red from the 3rd spectrum.
I know this isn't the correct answer so can someone please help ?

 

[technician]

calculate the angle for the red in the second order spectrum (the 'end' of the second order spectrum)
calculate the angle of the blue in the third order spectrum (the 'start' of the third order spectrum)
What do you notice about these angles?...can you explain?

 

[Nemo's]

So how do I calculate the angle for the 2nd spectrum? can I use λ =800 for violet and λ=1400 for red

 

[Nemo's]

O.K I got it the angle for red in the 2nd spectrum would be approx. 44.4° and that for violet in the third spectrum would be approx. 36.9° so the second and third spectra overlap.
Thank you :)

 

[Nickie]

The second and third order spectra overlap because diffraction gratings are designed to produce multiple orders of diffraction. This means that not only the first order, but also the second and third orders will be present in the diffracted light. This is due to the fact that the grating has multiple slits, which causes the light to be diffracted at different angles depending on the wavelength. This results in overlapping spectra because the different orders of diffraction overlap with each other.

To explain this further, we can consider the formula sinθ=nλ/d. As we increase the order (n), the angle (θ) also increases. This means that for higher orders, the diffracted light will be at a larger angle compared to the first order. However, since the slit separation (d) is the same for all orders, the wavelength (λ) also decreases as we increase the order. This results in the different orders of diffraction overlapping with each other, as seen in the second and third order spectra in this problem.

In addition, the overlapping of the spectra also occurs because the visible spectrum has a range of wavelengths, from 400nm to 700nm. This means that even within the first order spectrum, there is already some overlap between the red and violet ends. As we move to higher orders, this overlap becomes more significant and results in the overlapping of the spectra.