Video Transcript
The diagram shows the paths
followed by coherent light waves of wavelength 𝜆 emitted from two slits in an
opaque screen that are 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 destructively. Which of the following must be a
value of the length 𝐿? (A) 𝑛𝜆, where 𝑛 is an
integer. (B) 𝑛𝜆, where 𝑛 is an odd
integer. (C) 𝑛𝜆 over two, where 𝑛 is an
integer. (D) 𝑛𝜆 over two, where 𝑛 is an
odd integer. Or (E) two 𝑛𝜆 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 𝜆 emitted from two slits in an opaque screen that are 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. 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 𝜆, 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 𝜆 over two, then the parts of the waves that overlap would
be destructively interfering again.
So when the distance between the
waves is three 𝜆 over two, there is destructive interference, and there’s also
destructive interference when it’s just 𝜆 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 𝑛𝜆 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 𝑛𝜆 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 𝑛𝜆, 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 𝜆,
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 𝑛𝜆 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 𝜃 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 𝜆, but rather 𝑛𝜆
over two as we already discovered. As well for (C), when we have a
length of 𝑛𝜆 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), “the 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 𝜃 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).
