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I am trying to determine the total number of primary maxima that can be observed when light of wavelength 500 nm is incident normally on a diffraction grating, with the third-order maximum of the diffraction pattern observed at 32.0 degrees.


Rearranging the diffraction grating formula for maxima number ( $m$ ):

$$ m= \frac{d \space \sin \space\theta_\text{bright}}{\lambda} \, . $$

I can get the right answer if I let $$\theta = 90 ^\circ \, .$$ However, I do not understand why this angle value is used.

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any angle over theta 90 will mean that the diffraction will be going behind the diffraction gratings which is impossible. so 90 is the maximum that you can get this is why you have to round down the decimal answer you will get.

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The diffraction pattern is essentially infinite on the screen on which it appears. We regard the location on the screen with a single coordinate, $\theta$, which is the angle between the perpendicular line stretching from the center of the grating to the prime maximum (we can call this point the center of the diffraction pattern). If you imagine opening this angle slowly, you can see that the point in which the line touches the screen moves further and further away from the center of the pattern. The more you move towards $\theta=90^o$ the point on the screen gets infinitely farther from the center of the pattern. So, $\theta=90^o$ is the upper bound for the angle $\theta$. I wrote upper bound and not the maximal value since reaching $90^o$ in $\theta$ is equivalent to reaching infinite distance from the center of the pattern on the screen, which is of course not possible.

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