Determine the angles between diffraction maxima

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Homework Statement

microwave of wavelength 3 are incident to a row of parallel metal rods. The separation of the rods is 10cm. The first order diffraction is observed at 25° to the direction of the incident wave. Determine the angle between the first and the second order of diffraction maxima.

Homework Equations

I suppose that the maxima is the strip of bright fringe and this is the diffraction of single strip diffraction. I think that this picture is suitable to describe the problem,

assume that the wave 3 and 4 intersect a point P at a screen and form Bright Fringe(picture shows dark fringe, but i just assume it is bridge fringe, couldn't find a better picture).

First assumption : W4 - W3 = λ...at first maxima
and solving the triangle like how shown in this image

I get θ = mλ/a ...m is 1,2,3...and a is the size of the metal rod separation, 10cm
θ = 1 x 0.03m / 0.1m = 0.3 rad = 17.2°...but the problems say it is 25°but when I assume W4-W3 = 1.5 λ ... for first maxima
i get
θ = 1.5 x 0.03m / 0.1m = .45 rad = 25.78°...closer to the 25° stated in the question

Now my main confusion is W4-W3 for the first bright fringe λ or 1.5λ, what i am thinking now is the bright fringe is the superposition of the W4 and W3, W4 is longer than W3 by 1λ, but the question seems telling it is 1.5λ and so do some of the books and site...

 

[haruspex]

microwave of wavelength 3

I gather from what you wrote later that should be 0.03m, right?
The symbol after '25' for the angle is not coming out on my browser. Is it degrees? If so, I have to agree that the information given appears inconsistent. That angle would correspond to the second minimum.
Is it possible that the incoming rays are at an oblique angle to the grating?

 

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Yeah that 25 is degree, it was in the correct symbol at the time i submit the post... The question didn't state how the microwave hit the grating, i think it should be perpendicular, so what do u think, finding the angle θ for the first bright fringe is:

θ = mλ/a...a = the gap between the rod and m is either positive or negative integers

or

θ = (m + 1/2)λ/a

 

[haruspex]

The top picture in the OP is for a single slit, not a grating. That will give the first minimum at (w/2) sin(θ) = λ/2, and the first non-central maximum at w sin(θ) = 1.5 λ. See: http://web.utk.edu/~cnattras/Phys250Fall2012/modules/module%201/diffraction_and_interference.htm .

 

[ehild]

If it is a grating, d sinθ = kλ is the condition for the maxima. sinθ can be approximated by θ for angles less than 10 degrees or so, so you can not use it here.
If it is normal incidence, the data are inconsistent.

ehild

 

[ehild]

The top picture in the OP is for a single slit, not a grating. That will give the first minimum at (w/2) sin(θ) = λ/2, and the first non-central maximum at w sin(θ) = 1.5 λ. See: http://web.utk.edu/~cnattras/Phys250Fall2012/modules/module%201/diffraction_and_interference.htm .

Well, it is said that the light arrives at a row of parallel metal rods, separated by 10 cm. That is a grating.

In case of a single slit of width W, the minima appear where Wsinθ = mλ.
ehild

 

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Based on the image given in the website,
http://web.utk.edu/~cnattras/Phys250Fall2012/modules/module%201/images/diffra12.gif

If i am to find the first constructive fringe, the w1 and w7 difference by 1λ right?
if that true, the construction of wave happen when w1 and w7 difference by mλ
so mλ = (W/2)sin θ

using sinθ approximation to θ,

θ = (2λm)/W

So going back to my question, based on the information given, ignoring the 25 degree...

θ = 2(0.03)(1) / 0.1 = 0.6 rad = 34.4 degree, so that 25 degree is an error?

 

[haruspex]

Based on the image given in the website,
http://web.utk.edu/~cnattras/Phys250Fall2012/modules/module%201/images/diffra12.gif

That image is for a single slit. The dots are not grating lines, they're just to illustrate the argument. Likewise, the first image in the OP is for a single slit. The question is a for a grating, which is different.

Well, it is said that the light arrives at a row of parallel metal rods, separated by 10 cm. That is a grating.

Yes, that's my point. But the 25 degrees (1.5λ) is consistent with its being a single slit.

 

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Oh really sorry, my bad, I thought diffraction grating is same as diffraction of single slit...but they re more related to the change of wavelength. This question never state it is single slit problem or grating problem, but I guess it is more to single slit problem, because it is under the chapter "nature of wave".

Yes, that's my point. But the 25 degrees (1.5λ) is consistent with its being a single slit.

I really don't get it where the 1.5 comes from, in case of constructive wave, the w1 and w7 difference is 1λ or mλ isn't it?

then i get
(W/2)sin θ = mλ

θ = 2mλ/W

but the question 20 degree seems telling me it is 1.5, something must have been wrong with my concept...

Likewise, the first image in the OP is for a single slit.

You mean the diagram with read waveform? But how do you differentiate it is single slit or grating by looking at the diagram?

 

[null void]

After drawing out the diagram on my own, I finally realize how the 1.5 come, thanks very much ^^

 

[haruspex]

This question never state it is single slit problem or grating problem, but I guess it is more to single slit problem, because it is under the chapter "nature of wave".

The description mentions metal rods. That makes it a grating.

You mean the diagram with red waveform? But how do you differentiate it is single slit or grating by looking at the diagram?

By the fact that it looks exactly like the diagram for a single slit in the link I posted.