GR9277 #35


Problem


\prob{35}
Light of wavelength 5200 A is incident normally on a transmission diffraction grating with 2000 lines per cm. The firstorder diffraction maximum is at an angle, with respect to the incident beam, that is most nearly
 3 degrees
 6 degrees
 9 degrees
 12 degrees
 15 degrees

Optics}Diffraction Grating
Diffraction gratings have the same formula as 2slit interference, except each slit is (obviously) much smaller. The condition for maximum is given by , relating the width of the slit to the wavelength and angle and order m.
The width of each slit is given by the grating . Thus, plugging in the wavelength one has .
Now, the approximations to get rid of the trig function. Since , one can approximate , where the angle is in radians. Now, convert the angle from radians to degrees. , as in choice (B).


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Comments 
mvgnzls 20110915 15:46:39  why is d= to the inverse of (number of slits per unit length)?
pam d 20110928 18:15:35 
If there are 2000 slits per cm, then every slit has to be spaced 1/2000 cm apart. Think about it this way, if you line up 2000 of them in a row (which is what a diffraction grating is) then you'll have a 1 cm long diffraction grating with 2000 slits.

Quark 20111026 13:28:09 
Makes sense, nice Pam!

  chrisfizzix 20081006 12:24:21  In the double/Nslit diffraction equation, d is the slit spacing, not the slit width. This problem implicitly assumes that the slits are extremely narrow, so that even though there are 2000 of them in a cm they are much more narrow than they are far apart. Given this interpretation, d = slit spacing = (number of slits in unit length) = (1 / 2000) cm. The solution is correct besides that.   hungrychemist 20070922 18:58:38  the diffraction formula, w *sin(angle) = m*wavelength gives minima not the maxima. (See pg 893 Halliday)
to approximate the maxima, one can set m = 1.5 (since the maximum occurs in between the two minima).(See pag 894 Halliday)
following yosun's calculation but with m = 1.5, I found the answer to be nearly 9 degrees.
What's going on?
hamood 20071003 17:40:09 
The minima is for a single slit diffraction pattern. In this question we have a diffraction grating, and the same formula would give the maxima.

alemsalem 20100925 06:06:53 
grating maxima are at the same angle as interference maxima, this is because the pattern is a multiplication of the interference term and the diffraction term.
but because the slit width is extremely small the first diffraction minima would be at an angle more than 90 degrees which means all the pattern is within the central diffraction maximum and the rest is interference maxima and minima.

 

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