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MJ 11 P11 Question number 27. It's quite difficult I think so anybody wanna try?
It's a tricky question in that you need to understand how orders behave around the central maxima.
A diffraction grating with 500 lines per mm, with light of wavelength 600 nm passing through it.
For diffraction gratings, there are two formulas you always need to known. The first being: d sin θ = n λ.
d being the width of each slit.
theta being the angle between the central maxima and each order.
n being the number of orders.
and λ being the wavelength.
The other one deals with finding out the spacing of each slit, from the number of lines there are.
If there are 100 lines, the spacing for 1 slit will be 1/100.
This gives us the formula, d = 1/N, with N being the number of lines.
With all the info they have already given us, we can first find out how many orders there are, in order to determine the number of images produced.
They tell us, 500 lines " per " mm. The number of lines present will be per metre, so we convert the mm to metres. This gives us 500 / 1*10^-3 m
Which is 500,000 lines.
Plug this into d = 1/N to find the split spacing, which is 1/500,000 = 2 x 10^-6 m
Now before we go any further, we need to know, that the image produced at Y, is along the central maxima. Which follows an order of n = 0.
They mention that X and Z are "parallel" to the grating, this means that the angle at which they are being viewed at is 90 degrees.
That gives us,
2 x 10^-6 ( Sin 90 ) = n ( 600 x10^-9)
Since Sin 90 = 1,
n = 2 x 10^-6 / 600 x 10^-9
n = 3.333
3.333 might seem like an odd number to work with, but it is completely fine, since 3.33 still belongs to the 3rd order.
Hence we consider there to be 3 orders from the central maxima.
Now as for the number of images produced, there are 3 orders, on BOTH sides of the central maxima. That is 3 orders below n = 0, and 3 orders above n = 0.
That is a total of 7 images produced as he moves his head from X to Z, since the central maxima AKA "Y", is where he would also see an image.
They did mention " different " angles, along X to Z, that is how you are able to know that his head need not be at X and Z only.
With 7 images being produced, your only answer will be D.
Hope that helped