Video: Angle for First-Order Intensity Maximum Using a Diffraction Grating

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

02:47

Video Transcript

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

Letโ€™s highlight some of the important information weโ€™ve been given. Weโ€™re told that our diffraction grating has 2000 lines per centimeter; weโ€™ll call that ๐ฟ. Weโ€™re also told that the light weโ€™re considering has a wavelength of 520 nanometers; weโ€™ll call that value ๐œ†. We want to solve for an angle, weโ€™ll call it ๐œƒ, where the first-order diffraction maximum appears.

First letโ€™s draw a sketch of a diffraction grating. A diffraction grating is an extension of the double-slit set-up that weโ€™ve seen before, where a diffraction grating may have dozens, hundreds, or thousands of slits. The equation for interference maxima for a diffraction grating is the exact same as for a double-slit experiment.

Just like for the double slit set-up, here ๐‘‘ is the distance between adjacent slits, where that distance is uniform across the grating. That distance multiplied by the sine of ๐œƒ, where ๐œƒ is the angle between the diffracted ray and a horizontal line in this case, is equal to the order number ๐‘š times the wavelength ๐œ†. In our case, weโ€™re considering the first-order maximum, so ๐‘š is one.

We want to solve for ๐œƒ. And when we rearrange this equation, we see that the angle ๐œƒ is equal to the arcsine of the wavelength ๐œ† divided by ๐‘‘. Weโ€™re given ๐œ† in the problem statement and can solve for ๐‘‘ based on ๐ฟ, the number of lines per centimeter. 2000 lines per centimeter means that one 2000th of a centimeter is equal to ๐‘‘, the distance between adjacent lines.

In meters, that equals 5.00 times 10 to the negative sixth. Now that we know both ๐œ† and ๐‘‘, we can plug in those values. When we do so, being careful to use units of meters for both values, we find when we calculate this arcsine that ๐œƒ equals 5.97 degrees. Thatโ€™s the angle of the first- order maximum for this diffraction grating.

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