Light is incident normally on a diffraction grating, the ruling separation of the lines:

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  • #1
hidemi
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Homework Statement:
If 550-nm light is incident normally on a diffraction grating and exactly 6 lines are produced, the ruling separation must be:

The answer is between 2.20 ´ 10-6 m and 3.30 ´ 10-6 m
Relevant Equations:
d·sin(θ) = n·λ
I get one of the ranges by calculating:

sin(90°) = 1
d·sin(θ) = d × 1 = d = n·λ
d = 6 × 550-nm = 3,300 nm = 3.3 microns = 3.3 × 10⁻⁶ m

But, how can I get the other range of 2.2 × 10⁻⁶ m

Could anyone explain it to me? Thanks
 

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  • #2
Steve4Physics
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Homework Statement:: If 550-nm light is incident normally on a diffraction grating and exactly 6 lines are produced, the ruling separation must be:
There can’t be ‘exactly 6 lines’ because there is a single 0-th order central line and pairs of higher order lines (symmetrical about centre). So there is an odd number of lines in total. For example, if the highest order line is 6 (n=6), there are exactly 2*6+1 = 13 lines visible.

Have you stated the question correctly?
 
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  • #3
Charles Link
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In general, the number of primary maxima will be odd, because of the ## m=0 ##, with ## m=\pm 1 ##, and ## m=\pm 2 ##, etc. I think the question needs additional clarification. Perhaps someone else has an idea of what it is referring to.
 
  • #4
hidemi
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There can’t be ‘exactly 6 lines’ because there is a single 0-th order central line and pairs of higher order lines (symmetrical about centre). So there is an odd number of lines in total. For example, if the highest order line is 6 (n=6), there are exactly 2*6+1 = 13 lines visible.

Have you stated the question correctly?
There isn't any typo. The picture of the original question is as attached.
 

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  • #5
Charles Link
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I tried to solve it assuming there are orders up to and including ## m=5 ##. I can also see how they got the 3.3 but not the 2.2. I think they need to do their homework more carefully. The question is flawed.
 
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  • #6
Steve4Physics
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There isn't any typo. The picture of the original question is as attached.
The question is wrong (as already noted by @Charles Link and myself). And the so-called correct answer can't be correct because the limits of the ruling-separation range are 4λ and 6λ, which makes no sense.

I believe the question should be:

Light of wavelngth 550nm is incident normally on a diffraction grating. The highest order lines visible are the 6th-order. What is the possible range of the ruling-separations?

To answer this we consider the 2 limiting conditions.

a)The ruling-separation gives the 6th order line at θ = 89.9999999999º (but call it 90º!) which is just visible.

b)The ruling-separation gives the 7th order line at θ = 90º (you can’t actually see a line if θ = 90º). Note that the 6th order lines would now be visible at some angle smaller than 90º.

Using the amended question I get the answer C) in your list.

From what text-book (or other source) is the question?
 
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  • #7
hidemi
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The question is wrong (as already noted by @Charles Link and myself). And the so-called correct answer can't be correct because the limits of the ruling-separation range are 4λ and 6λ, which makes no sense.

I believe the question should be:

Light of wavelngth 550nm is incident normally on a diffraction grating. The highest order lines visible are the 6th-order. What is the possible range of the ruling-separations?

To answer this we consider the 2 limiting conditions.

a)The ruling-separation gives the 6th order line at θ = 89.9999999999º (but call it 90º!) which is just visible.

b)The ruling-separation gives the 7th order line at θ = 90º (you can’t actually see a line if θ = 90º). Note that the 6th order lines would now be visible at some angle smaller than 90º.

Using the amended question I get the answer C) in your list.

From what text-book (or other source) is the question?
It's from class, and I'm not sure where the professor extracted from.
One of the answer C is above the limit of 3.3e-6, so I wonder if I misunderstood what you meant.
 
  • #8
Steve4Physics
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One of the answer C is above the limit of 3.3e-6, so I wonder if I misunderstood what you meant.
There is no upper limit of 3.3e-6 m. Remember the original question is completel wrong - it is nonsense.

I’m saying answer C) is the answer to a different question:

Light of wavelength 550nm is incident normally on a diffraction grating. The highest order lines visible are the 6th-order. What is the possible range of the ruling-separations?

##n\lambda= dsin\thetaθ## gives ##d = \frac{n\lambda}{sin\theta}##

If the 6th order is just visible, that means ##\theta = 89.99999999º##.
This corresponds to a line-spacing ##d = \frac {6\lambda}{sin(89.99999999º)} = 6\lambda##.

If the 7th order is just invisible that means ##\theta = 90º##.
This corresponds to a line-spacing ##d = \frac {7\lambda}{sin(90º)} = 7\lambda##.
(In this case the 6th order would easily be visible – you can work out, it is at 59º.)

Therefore the line-spacing (d) must be between ##6\lambda## and ##7\lambda##.

##d = 6 \times 550\times 10^{-9} m = 3.30\times 10^-6 m##.
##d = 7 \times 550\times 10^{-9} m = 3.85\times 10^-6 m##.
 
  • #9
hidemi
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There is no upper limit of 3.3e-6 m. Remember the original question is completel wrong - it is nonsense.

I’m saying answer C) is the answer to a different question:

Light of wavelength 550nm is incident normally on a diffraction grating. The highest order lines visible are the 6th-order. What is the possible range of the ruling-separations?

##n\lambda= dsin\thetaθ## gives ##d = \frac{n\lambda}{sin\theta}##

If the 6th order is just visible, that means ##\theta = 89.99999999º##.
This corresponds to a line-spacing ##d = \frac {6\lambda}{sin(89.99999999º)} = 6\lambda##.

If the 7th order is just invisible that means ##\theta = 90º##.
This corresponds to a line-spacing ##d = \frac {7\lambda}{sin(90º)} = 7\lambda##.
(In this case the 6th order would easily be visible – you can work out, it is at 59º.)

Therefore the line-spacing (d) must be between ##6\lambda## and ##7\lambda##.

##d = 6 \times 550\times 10^{-9} m = 3.30\times 10^-6 m##.
##d = 7 \times 550\times 10^{-9} m = 3.85\times 10^-6 m##.
Thanks for the response.
I wonder how about the orders lie in between 0 < theta < 90?
 
  • #10
Steve4Physics
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I wonder how about the orders lie in between 0 < theta < 90?
I don't understand the question. What you can try is this:

Pick a value for d.
Work out the angle for each order using
##\theta = sin^{-1}(\frac {n\lambda }{d})## for n=0, 1, 2, 3, ...

You will then see exactly where each order lies in the range 0≤θ<90º.

You can repeat this for a different value of d and see what difference it makes,

(If you can use a spreadsheet, this is very quick/easy to do.)
 
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  • #11
hidemi
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I
I don't understand the question. What you can try is this:

Pick a value for d.
Work out the angle for each order using
##\theta = sin^{-1}(\frac {n\lambda }{d})## for n=0, 1, 2, 3, ...

You will then see exactly where each order lies in the range 0≤θ<90º.

You can repeat this for a different value of d and see what difference it makes,

(If you can use a spreadsheet, this is very quick/easy to do.)
I see, Thank you/.
 
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