Video Transcript
The diagram shows some apparatus
used in holography, including a cylindrical object. Which of the following is true of
the phase difference between light waves that travel the path 𝐴 capital 𝐵 capital
𝐶 and those that travel the path 𝐴 capital 𝐵 capital 𝐷? (A) There is a nonzero phase
difference between the waves. (B) The phase difference is
zero.
Let’s start off by identifying
these two paths 𝐴 capital 𝐵 capital 𝐶 and 𝐴 capital 𝐵 capital 𝐷 on our
diagram. We see there that light from a
laser is incident on this component. We see that some of the light
passes straight through this component, while other light is reflected. This reflection happens at the
point labeled 𝐴. And that beam then travels down to
the point labeled capital 𝐵. Here, the light bounces off a
mirror and then passes to a prism, which spreads out this narrow beam of light. The very top of this now spread-out
beam of light goes to point capital 𝐶, while the very bottom part goes to point
capital 𝐷.
Between the prism then, this
component, which, as we said, spreads out an incoming beam of light, and the final
component in our diagram where all the light ends up, the path of the light to point
capital 𝐶 is what we can call the top of our newly expanded beam. While the path of light to point
capital 𝐷 is the bottom of that expanded beam.
Our question asks us about a phase
difference between the light waves that travel these two different paths. Because the waves that follow path
𝐴 capital 𝐵 capital 𝐶 and those that follow the path 𝐴 capital 𝐵 capital 𝐷
come from the same laser source, that tells us that the only way there can be a
phase difference between the light waves that follow these paths is if there is a
path length difference between them. In other words, if the length of
path 𝐴 capital 𝐵 capital 𝐶 is equal to the length of path 𝐴 capital 𝐵 capital
𝐷, then there will be no phase difference between these light waves.
We can see that from point 𝐴 to
point capital 𝐵 up to the prism, these two paths overlap. We can then notice that the
distance from the prism to point capital 𝐶 is the same as the distance from the
prism to point capital 𝐷. Therefore, the total path length of
path 𝐴 capital 𝐵 capital 𝐶 is equal to the path length of 𝐴 capital 𝐵 capital
𝐷. Since the waves that follow these
two paths started out in phase with one another, that is, with the phase difference
of zero, that phase difference is maintained over these two equal path lengths, up
until the waves reach this final component in the diagram. We can conclude then that the phase
difference between the light waves that travel these two paths is zero. We choose option (B).
Let’s look now at part two of this
question.
Which of The following is true of
the phase difference 𝜙 between light waves that travel the path 𝐴 lowercase 𝑏
lowercase 𝑐 uppercase 𝐶 and those that travel the path 𝐴 lowercase 𝑏 lowercase
𝑑 uppercase 𝐷? (A) Zero is less than or equal to
𝜙, which is less than or equal to two 𝜋. (B) Zero is less than 𝜙, which is
less than two 𝜋.
We see that the only difference
between our answer options is that option (A) says 𝜙 could be equal to zero or
equal to two 𝜋, while option (B) says it must be greater than zero and less than
two 𝜋. Let’s consider our diagram and the
two paths indicated. First, we have a path that goes
from 𝐴 to lowercase 𝑏 to the point lowercase 𝑐 and then to uppercase 𝐶; that’s
one pathway. While the second path also goes
from 𝐴 to lowercase 𝑏. But from there, when the light on
this path reaches the prism, it instead goes to point lowercase 𝑑 and then from
lowercase 𝑑 to uppercase 𝐷.
Our question asks about a phase
difference called 𝜙 between the light waves that travel these two paths. Since the light waves start from
the same coherent source, a laser here, we know that initially they have a phase
difference of zero. Therefore, the only way that these
light waves could have a nonzero phase difference once they reach this final
component on our diagram is if there is a difference between the length of these two
paths indicated, 𝐴 lowercase 𝑏 lowercase 𝑐 capital 𝐶 and 𝐴 lowercase 𝑏
lowercase 𝑑 capital 𝐷.
Beginning at point 𝐴, we can see
that these two pathways are identical; they overlap up until the point where they
reach this prism. The prism then spreads out the
incoming beam of light so that some of that light goes to point lowercase 𝑐 and
some of that light follows a path to point lowercase 𝑑. Judging by eye, we can say that
this path length here, to point lowercase 𝑐, is shorter than this path length here
to lowercase 𝑑.
And then even more so, the distance
between point lowercase 𝑐 to point uppercase 𝐶, here, is smaller than the distance
between point lowercase 𝑑 to uppercase 𝐷 here. All this tells us that the paths
our light waves follow are not the same length in this case. Rather, the length of the path 𝐴
lowercase 𝑏 lowercase 𝑑 uppercase 𝐷 is greater by some amount we don’t know than
the length of the path 𝐴 lowercase 𝑏 lowercase 𝑐 uppercase 𝐶.
The fact that we don’t know by how
much this one path is longer than the other is important. Consider two waves of light that
are emitted by our coherent laser source. These waves are in phase; they have
a phase difference of zero. And let’s say that this wave on top
follows our first path, 𝐴 lowercase 𝑏 lowercase 𝑐 uppercase 𝐶, and the wave on
bottom follows the second path. If we try to imagine the phase
relationship between these two waves once they reached the final component of our
diagram, we actually have trouble doing this because we don’t know how much farther
this second wave traveled than the first one.
Let’s imagine that the path length
difference between these two paths is some multiple of two 𝜋. In that case, when the two waves do
reach that final component, they’ll be in phase. On the other hand, if the
difference between the two path lengths is some odd multiple of 𝜋, then on reaching
that final component in the diagram, waves from the two paths will be 180 degrees
out of phase.
And note that we’re only able to
draw the phase relationship between these two waves at our final component in the
circuit as we guess different path length differences between the two paths. That path length difference, we
know, is called 𝜙. And really, 𝜙 could be anything,
from zero up to and including two 𝜋 radians. That is, the actual phase
relationship, and therefore the phase difference between these two waves, depends on
the path length difference between our two paths identified.
Considering our two answer options
then, we can’t say for certain that 𝜙 will be greater than zero. Likewise, we can’t be certain that
it will be less than two 𝜋. Note by the way that because of the
cyclical nature of waves, a phase difference of zero is equivalent to a phase
difference of two 𝜋. So really 𝜙 could be equal to
zero, or it could be equal to two 𝜋 or any value in between. We know then which answer we will
choose. The phase difference 𝜙 between
light waves that travel the two paths indicated will be greater than or equal to
zero and less than or equal to two 𝜋. We choose option (A).
