The Michelson interferometer can be used to measure the index of refraction of a gas by placing an evacuated transparent tube in the light path along one arm of the device. Fringe shifts occur as the gas is slowly added to the tube. Assume 600.-nm light is used, the tube is 5.00 cm long, and 160 fringe shifts occur as the pressure of the gas in the tube increases to atmospheric pressure. What is the index of refraction of the gas? Hint: The fringe shifts occur because the wave-length of the light changes inside the gas-filled tube.
So in this problem, we're trying to find the refractive index of a certain gas using Michael's an interferometer. At first, an empty two off length five centimeter is shine with the light off wavering 500 nanometer. Now the tube is actually slowly started to feel with the gas and brought to the atmospheric pressure. So at this point, the total of 160 friendship happens. Okay, so let's recall that the pad difference of 1/4 Web link causes the friend shift off one. This is true for either gas. Like either cases, whether the tube is empty or the tour is filled with gas, so elicits Write it down when the two is empty. The friendship for total friendship for empty case is going to be from the problems we did before. If you remember, is going to be four el over Lambda and for the situation when the tube is filled with gas, it's going to be for L over lambda Gas. All right, so from here we can find out the total change in friend French between the empty and gas feel tive. Okay, so that actually iss n, which is 160. And that's the difference between the guess and and empty, which is going to be for L one over. Lambda minus one over Lambda Prime. So we don't know the Sorry. We call this Lambda. I guess so. We don't know the wavelength of the light when the tube is filled with the gas. Okay, so we don't know the web, Little of light in the gas. So to find that out, we used the definition of the refractive index. So let's call this as equation one and less defined the refractive index of the guests and gas he calls velocity off light in, vacuum over. Velocity off. Last in gas, which is going to be frequency. Times were blink over frequency times were blanked in gas. You see here I wrote down F on both sides. It's because the frequency actually doesn't change. It is the Web link that changes the velocity off light in different mediums. Okay, so if I cancel this out, I'm going to get Lambda over land. I guess I'll just read this down in the next page so we can write down and gas. Sorry again. So it's gonna be and gas equals Lambda over. Let me guess. It's called his equation too. And if we saw a prison one and two, we will actually get from one and two. We can actually write down, and he calls four hell over Lambda and gas minus one. And from here, you consult this to see and gas is going to be called to 1 60 lambda over for L plus one. Okay. And we have the valley of the Web linked and the length of the tube. So we'll just plug this in here and we will get 1.5 So this is the refractive index of the light of the gas.