### 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?

Letโs highlight some of the important information weโve been given. Weโre told that our diffraction grating has 2000 lines per centimeter; weโll call that ๐ฟ. Weโre also told that the light weโre considering has a wavelength of 520 nanometers; weโll call that value ๐. We want to solve for an angle, weโll call it ๐, where the first-order diffraction maximum appears.

First letโs draw a sketch of a diffraction grating. A diffraction grating is an extension of the double-slit set-up that weโve 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, where ๐ is the angle between the diffracted ray and a horizontal line in this case, is equal to the order number ๐ times the wavelength ๐. In our case, weโre considering the first-order maximum, so ๐ is one.

We want to solve for ๐. And when we rearrange this equation, we see that the angle ๐ is equal to the arcsine of the wavelength ๐ divided by ๐. Weโre given ๐ in the problem statement and can solve for ๐ based on ๐ฟ, 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 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 equals 5.97 degrees. Thatโs the angle of the first- order maximum for this diffraction grating.