Question Video: Describing the Relationship between Two Destructively Interfering Light Waves Physics • 9th Grade

The diagram shows the paths followed by coherent light waves of wavelength πœ†, emitted from two slits in an opaque screen that are separated by a distance 𝑑. The waves emitted from the slits at angle πœƒβ‚‚ from a line normal to the screen intersect at a point on a line parallel to the screen, where they interfere destructively. Which of the following must be a value of the length, 𝐿? [A] π‘›πœ†, where 𝑛 is an integer [B] π‘›πœ†, where 𝑛 is an odd integer [C] π‘›πœ†/2, where 𝑛 is an integer [D] π‘›πœ†/2, where 𝑛 is an odd integer [E] 2π‘›πœ†/𝑑, where 𝑛 is an odd integer Which of the following correctly describes the relationship between the paths that light waves follow that make the angle πœƒβ‚‚ with a line normal to the screen containing the slits? [A] The lines are parallel. [B] The lines differ in length by πœ†. [C] The lines are the same length. [D] The lines are almost parallel but diverge gradually. [E] The lines are almost parallel but converge gradually.

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Video Transcript

The diagram shows the paths followed by coherent light waves of wavelength πœ† emitted from two slits in an opaque screen that are separated by a distance 𝑑. The waves emitted from the slits at angle πœƒ two from a line normal to the screen intersect at a point on a line parallel to the screen, where they interfere destructively. Which of the following must be a value of the length 𝐿? (A) π‘›πœ†, where 𝑛 is an integer. (B) π‘›πœ†, where 𝑛 is an odd integer. (C) π‘›πœ† over two, where 𝑛 is an integer. (D) π‘›πœ† over two, where 𝑛 is an odd integer. Or (E) two π‘›πœ† over 𝑑, where 𝑛 is an odd integer.

The length 𝐿 is the difference in the path lengths between the light waves emitted from the two slits in the screen. But before we can determine this value, we have to clear some space. So let’s start writing out what the question tells us. So we have coherent light waves of wavelength πœ† emitted from two slits in an opaque screen that are separated by a distance 𝑑. The waves emitted from the slits at angle πœƒ two from a line normal to the screen intersect at a point on a line parallel to the screen. This essentially means that the light waves produced from the two slits eventually converge at a single point on a screen parallel to the one with the two slits in it, where they then interfere destructively.

This means at the point where the two waves meet on the opposite screen, it forms a dark spot. To see how this can be, remember that when we send light through two slits, it forms an interference pattern, which is a pattern of bright spots and dark spots all along the opposite screen. Constructive interference forms bright spots represented by the yellow dots, while destructive interference forms dark spots, the areas in between the bright spots. When the question tells us that the light from these two slits interferes destructively, it is telling us that it is forming a completely dark spot. This destructive interference occurs with light because light is a wave.

The light waves from the two slits meet up at this dark spot, and at that particular dark spot, the waves are out of phase. This does not mean that the waves have different wavelengths. The wavelength of the original wave does not change as it splits when it goes through the two slits. Both of these light waves have the same wavelength. What matters here is the phase difference or where along in the wave the waves are when they both touch at the same point on the opposite screen. For instance, at this bright spot, the waves may look like this when they come together, or interfere, at that point.

When waves interfere, it means that every point along both waves is added together, which when you have two waves that are in phase with each other, which means that the waves are the same at the same point in time, for instance, having a peak at the same point or having a valley at the same point, then they can constructively interfere. The points are added together and they create one big wave, creating a bright spot.

On the other hand, if we were to look back at our dark spot, we would see that at the point where the two waves meet, the waves are exactly opposite to each other, which is to say wherever there is a peak of one wave, there is a valley of another and vice versa. When these waves interfere, all of the points are still being added together. But because they’re opposite to each other, they destructively interfere and create a wave that looks something like this, a straight flat line, which is of course not a wave at all. The wave has been destroyed.

So in order for these two light waves to interfere destructively at a point, they must be completely out of phase. When you have two waves of the same wavelength, then it turns out that shifting one over an amount equal to the wavelength of the waves over two causes destructive interference. If we were to then shift this wave over again by the same amount, the wavelength over two, which would make the total distance just πœ†, then we would find that the waves would no longer destructively but rather constructively interfere with each other. They would match up. If we then shift this wave over one more time again by that same amount of the wavelength over two, making the total shifted distance three πœ† over two, then the parts of the waves that overlap would be destructively interfering again.

So when the distance between the waves is three πœ† over two, there is destructive interference, and there’s also destructive interference when it’s just πœ† over two as well. Now light waves are actually quite long and you can shift them back and forth a lot. And we found that whenever the distance between these waves is equal to π‘›πœ† over two, where 𝑛 is an odd integer, then there is complete destructive interference. So then this relates to the length 𝐿 in the diagram here because the length 𝐿 is actually the difference in path length between the two waves coming from the slits. And whenever the path length difference between two waves is π‘›πœ† over two, where 𝑛 is an odd integer, there is destructive interference, which is what we want.

If 𝑛 was not odd, then we would sometimes not get destructive interference since constructive interference occurs when the length is equal to π‘›πœ†, where 𝑛 is an integer. So if 𝑛 happened to be even, like say four, then it would divide by the two in the denominator, making it just two πœ†, which is constructive, not destructive. This means that it is certainly not answers (A), (B), or (C). However, when we look at answer (E), we see that it contains the variable 𝑑, the distance between the two slits. And while this distance can influence the length of 𝐿, what we care about is the proportion between the two light waves, which doesn’t change with 𝑑. So when there is destructive interference, the value of length 𝐿 must be π‘›πœ† over two, where 𝑛 is an odd integer, making the correct answer (D).

Now we can move on to the second part of this question.

Which of the following correctly describes the relationship between the paths that light waves follow that make the angle πœƒ two with a line normal to the screen containing the slits?

Looking at answer (A), we know the lines are not parallel because they have to meet at a certain point. We just assume that they’re parallel to simplify our trigonometry here. In reality, these two angles are actually very very slightly different. But the difference is so small that it’s safe to assume that they’re parallel, but in actuality they are not. (A) is not it, nor is (B) it because the difference between the length of the lines 𝐿 is not πœ†, but rather π‘›πœ† over two as we already discovered. As well for (C), when we have a length of π‘›πœ† over two, where 𝑛 is odd, we have to have a length difference because 𝑛 cannot be zero, which is an even number.

Looking at answer (D), β€œthe lines are almost parallel but diverge gradually” is not quite it because diverging means that the lines would go off in different directions when they actually meet at the same point, or converge. So when we look at the paths that these light waves follow that make an angle πœƒ two with a line normal to the screen containing the slits, we see that the lines are almost parallel but do eventually converge gradually at a point. The correct answer is (E).

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