An experimenter detects 251 fringes when the movable mirror in a Michelson interferometer is displaced. The light source used is a sodium lamp which emits light with a characteristic wavelength of 589 nanometers. By what distance did the movable mirror move?
Let’s start by highlighting some of the vital information given to us. We’re told that 251 fringes were detected when the mirror moved; we’ll call that number 𝑁. We’re also told that the wavelength of light used in this interferometer is 589 nanometers, which we’ll call 𝜆. We want to know the distance that the moveable mirror moved, which we’ll call 𝑑.
Let’s start by drawing a diagram of the scenario. In a Michelson interferometer, what we start with is a light source which emits coherent light and sends that light into a beam splitter. The beam splitter reflects half of the light and lets the other half of the light through. Both of those separate rays hit mirrors and bounce back to return to the beam splitter.
When they recombine on a single path, they then travel to hit a screen. And the screen shows a series of fringes. Really its alternating bright and dark areas caused by constructive and destructive interference of the two rays as they recombine and reach the screen. We’re told that in this experiment, one of the mirrors was moved some distance 𝑑, which changes the overall path length of that ray of light.
As a consequence of that path length change, the observer noticed 251 fringes pass on the screen. There is a mathematical relationship between the distance a mirror is moved and the number of fringes that shift on the screen. 𝑑, the linear distance moved by the movable mirror, is equal to the number of fringes observed times the wavelength of the light divided by two.
When we apply this relationship to our scenario, we can plug in for 𝑁 and 𝜆 to solve for 𝑑. 𝑁 is 251. And in units of meters, 𝜆, the wavelength, is 5.89 times 10 to the negative ninth. When we calculate this value, we find that, to three significant figures, 𝑑 is 73.9 micrometers or 73.9 microns.