Calculate the wavelength of light that produces its first minimum at an angle of 36.9 degrees when falling on a single slit of width 1.00 microns.
In this statement, we’re told that the angle we’ll use is 36.9 degrees, which we’ll call 𝜙. We’re also told we have a setup with a single slit which has a width of 1.00 micrometers, which we’ll call 𝑤. We want to calculate the wavelength of light that produces its first minimum at this angle when falling on a slit of this width; we’ll call this wavelength 𝜆.
Let’s start by drawing a diagram of the scenario. In our scenario, we have waves approaching a single slit. When they reach that slit, they then bend around the edges, or diffract, and the wavelets produced there begin to interfere with one another, called interference. At the screen on the far right side, an intensity pattern is created with maxima and minima.
Knowing that the angle between the first minimum and the central maximum is 36.9 degrees and also knowing the width of the slit, we want to solve for the wavelength of the incoming light. To do that, let’s recall a relationship for destructive interference for a light coming through a single slit.
That relationships says that 𝑤, the slit width, times the sine of 𝜙, the angle between the central maximum and the particular minimum being studied, is equal to 𝑚 times 𝜆 where 𝑚 is an integer value and 𝜆 is the wavelength of the incoming light. Applying this relationship to our scenario, we’re told that were considering the first minimum, therefore 𝑚 is equal to one in our case.
So 𝜆, the wavelength of the light, is equal to 𝑤 sin 𝜙 or, plugging in for the given values we have, 1.00 times 10 to the sixth meters times the sine of 36.9 degrees. Plugging this into our calculator, we find 𝜆 is equal to 600 nanometers. That’s the wavelength of light that creates a first minimum at this angle, 𝜙, for this given slit width, 𝑤.