Video Transcript
The diagram shows the paths followed by coherent light waves of wavelength 𝜆 that are emitted from two slits in an opaque screen separated by a distance 𝑑. The waves emitted from the slits at angle 𝜃 two from a line normal to the screen intersect at a point on a line parallel to the screen where they interfere constructively. How does 𝜃 one compare with 𝜃 two? And which of the following must be a value of the length 𝐿? (A) 𝜆, (B) 𝑛𝜆, (C) 𝑛 over 𝜆, (D) 𝑛 over 𝜆, where 𝑛 is an integer, or (E) 𝑛𝜆, where 𝑛 is an integer.
Let’s start by looking at how 𝜃 one compares with 𝜃 two. 𝜃 one, which is here, is the angle between the distance between the two slits 𝑑 and the perpendicular distance between the two waves emitted from the slits. 𝜃 two, which is present both up here and down here on the diagram, is the angle between the path of the light waves and a line normal to the opaque screen. When we say that a line is normal to something else, it just means that it forms a 90-degree angle with it. Now then, we want to see how the angle 𝜃 one compares with the angle 𝜃 two.
The triangles that contain 𝜃 two up here and down here are the same. So we can focus on either one. But the triangle done below will be more useful since it is directly touching the triangle that contains 𝜃 one, which together forms a third triangle. Looking at just this third triangle, we see that, like all triangles, it has three angles, a 90-degree angle with a distance 𝑑 and a line normal to the opaque screen, the angle 𝜃 one at the top, and an unlabeled angle down here. We can find a way to find this angle though because of this line 𝐿 right here, which forms the angle 𝜃 two. The triangle that is formed from this line also has a right angle, meaning that both the larger triangle and the smaller triangle that is a part of it share two angles, the right angles and the mystery angle.
The only angles they don’t have in common are 𝜃 one and 𝜃 two. But all triangles have the same number of degrees 180. 180 degrees minus the 90-degree angles in both of these triangles leaves us with just 90 for both triangles. So when we subtract the mystery angle from this remaining 90 degrees, we should expect to get the last angle, which will be 𝜃 one or 𝜃 two. But again, the mystery angle is the same for both triangles, meaning that the angle produced from 90 degrees minus this mystery angle has to be the same. So 𝜃 one has to be equal to 𝜃 two. So the way in which 𝜃 one compares with 𝜃 two is that 𝜃 one is equal to 𝜃 two.
Now then, let’s look at the second part of our question which looks at the length 𝐿. To do this, we’re going to clear some space to make another diagram. But let’s not lose any of the original question text that we might need. We know that there are light waves of wavelength 𝜆 emitted from two slits in an opaque screen separated by a distance 𝑑. The waves emitted from the slits at angle 𝜃 two from a line normal to the screen intersect at a point on the line parallel to the screen, where they interfere constructively. When the question says they interfere constructively, it means that there is constructive interference at the point where the two light waves meet on the opposite screen.
Constructive interference in light waves means that it creates a bright spot at that point. And to see why this bright spot occurs, let’s recall that light waves are waves. And waves can constructively interfere when they have the same amplitude at the same moment in time, which is to say their peaks and valleys match up, causing them to add together and create an even bigger wave. In order for these waves to match up though, they have to have the same wavelength and they have to be in phase. And luckily, the wavelengths do match up for both of these waves. They’re the same as the original progenitor light wave before it passes through the slits. So we just have to worry about the phase of the waves. And this can be done because waves repeat.
Even though these two waves don’t line up exactly, they’re still constructively interfering because the parts that do overlap do line up. However, if we were to move this bottom wave forward just a little bit — meaning that these waves would actually destructively interfere, meaning that one of them has peaks where the other one has valleys and vice versa — what determines whether the waves constructively or destructively interfere is their difference in path length or the path length difference. And this path length difference always causes constructive interference when it is at specific values, which turn out to be 𝑛𝜆, where 𝑛 is an integer.
This means that it can differ by one whole wavelength, like here, zero whole wavelengths, like this, meaning 𝑛 is equal to zero and zero is an integer, or any other whole number so long as it satisfies this relation. And this relation still holds whenever we look at the difference in path lengths between the light waves, which means that it holds for 𝐿, which is the path difference between the light waves going through the two slits.
Looking at our other answers first though, it is true that (A) is a suitable answer for determining where there is constructive interference. But the reason that it is not correct is because it is not the only answer. And (B) 𝑛𝜆 seems like it could be correct, but it doesn’t specify that the value of 𝑛 has to be an integer, meaning that the path length difference can really be any number and potentially cause destructive, not constructive, interference. The only correct answer for, “Which of the following must be the value of the length 𝐿?” is (E), 𝑛𝜆, where 𝑛 is an integer.
