Question

A diffraction grating has 2000 lines per centimeter. At what angle will the first-order maximum be for 520-nm- wavelength green light?

$5.97^\circ$

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This is College Physics Answers with Shaun Dychko. This formula will help us find out where is the first order maximum for a diffraction grating. So

*d*is the separation between lines in the direction grating,*m*is the order that we're interested in, lambda is the wavelength of light incident on the diffraction grating and we'll solve this for theta to figure out the angle to the first order maximum. And notice that the question says first order maximum, it does not say first maximum. And that's a distinction that's important because the first maximum occurs when the order is zero. And the second maximum occurs when the order is one. So it said first order maximum, so that means*m*is one. The wavelength is 520 nanometers and*d*the separation between the slits and the diffraction grating and we have to figure it out based on knowing that there are 2,000 lines per centimeter in which case you use one centimeter for every 2,000 lines. We want to know how many centimeters there are between each line and so we'll divide these and we get one times ten to the minus two meters divided by 2,000 lines and that's 5.00 times ten to the minus six meters per line. you could say, or between each line. Good. So that's what*d*is. And so we'll solve this for sine theta by dividing both sides by*d*. And then take the inverse sine of both sides and we get theta is inverse sine of the order times a wavelength divided by the separation between the slits. So the inverse sign of one times 520 times ten to the minus nine meters divided by five times ten to the minus six meters give an angle of 5.97 degrees.