A diffraction grating has 2000 lines per centimeter. At what angle will the first-order maximum be for 520-nanometer wavelength green light?
Let’s highlight some of the important information we’ve been given. We’re told that our diffraction grating has 2000 lines per centimeter; we’ll call that 𝐿. We’re also told that the light we’re considering has a wavelength of 520 nanometers; we’ll call that value 𝜆. We want to solve for an angle, we’ll call it 𝜃, where the first-order diffraction maximum appears.
First let’s draw a sketch of a diffraction grating. A diffraction grating is an extension of the double-slit set-up that we’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 𝑑 is the distance between adjacent slits, where that distance is uniform across the grating. That distance multiplied by the sine of 𝜃, where 𝜃 is the angle between the diffracted ray and a horizontal line in this case, is equal to the order number 𝑚 times the wavelength 𝜆. In our case, we’re considering the first-order maximum, so 𝑚 is one.
We want to solve for 𝜃. And when we rearrange this equation, we see that the angle 𝜃 is equal to the arcsine of the wavelength 𝜆 divided by 𝑑. We’re given 𝜆 in the problem statement and can solve for 𝑑 based on 𝐿, the number of lines per centimeter. 2000 lines per centimeter means that one 2000th of a centimeter is equal to 𝑑, the distance between adjacent lines.
In meters, that equals 5.00 times 10 to the negative sixth. Now that we know both 𝜆 and 𝑑, 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 𝜃 equals 5.97 degrees. That’s the angle of the first- order maximum for this diffraction grating.