Video Transcript
The diagrams (a), (b), and (c) show
the possible resonant cavity modes for electromagnetic waves that are emitted from a
point in a cavity. Considering the wavelength of a
wave that can form a resonant cavity mode and the number of modes with that
wavelength that can exist in a particular cavity, would increasing the wavelength
increase, decrease, or not affect the number of modes?
To answer this question, we need to
determine how the wavelength of a wave relates to the number of resonant cavity
modes that can exist for that wave. Let’s first explore how these
cavities and the waves inside them work. Cavities such as these are
idealized objects that are useful for modeling properties of blackbodies. Blackbodies are also idealized
objects, meaning perfect blackbodies don’t exist in nature. But still, understanding
blackbodies helps us better understand properties of some real-life phenomena.
Recall that a blackbody is a
perfect absorber, meaning it totally absorbs all electromagnetic radiation incident
upon it, but a blackbody can also emit radiation. It’s important to remember that
this is true for any electromagnetic radiation, not just visible light. But for ease of communication, we
can simply refer to electromagnetic radiation in general as light. To better understand this concept,
let’s clear some room on screen and model a blackbody as a cavity that can let light
in but not let light out.
As such, let’s consider light
entering this cavity through a small hole. Once light enters, it cannot leave,
so it continually reflects off the interior walls of the cavity. At each reflection, some of the
light’s energy is absorbed by the walls of the cavity. As the cavity absorbs more energy,
it increases in temperature and emits radiation. In this example diagram, we’ve just
drawn straight lines to help show the path taken by light in the cavity. However, we need to remember that
light is a wave.
So, let’s draw another diagram
representing light in the cavity as a waveform. When we do this, it’s important
that the wave’s displacement must be zero at the boundaries of the cavity walls
because light cannot pass through the walls. Such a requirement is called a
boundary condition. Because of this boundary condition,
only certain waveforms can fit inside the cavity, and waves that don’t meet this
boundary condition are not allowed to exist within the cavity. So, for instance, a wave like this
cannot exist in the cavity as it breaks the boundary condition. That’s because the wave’s
displacement at this boundary is nonzero. The only waves that can exist in
the cavity are those that have a displacement of zero at the cavity walls.
With all this in mind, let’s return
to the three diagrams given to us in the question. We were told that in each diagram,
electromagnetic waves are emitted from a point in the cavity. We can see that the waves in the
cavities all obey the boundary condition we just discussed, as they all have zero
displacement at the cavity walls. Notice, though, that the waves
shown in the different cavities all have different wavelengths. Let’s compare their
wavelengths.
It’s easy to see that the waves
shown in diagram (a) have relatively short wavelengths, here, a full wave cycle, so
both positive and negative displacement can fit within the cavity. For the waves shown in diagrams (c)
and (b), only one-half of a wave cycle can fit, so they’re clearly of greater
wavelengths than the waves in diagram (a). The waves in these two diagrams
have pretty similar-looking wavelengths, but they’re not the same.
Notice that all the waves in all
the diagrams are emitted from a point in the cavity. So they each have one end fixed at
this point here on the left wall of the cavity. Knowing this, we can tell that the
wave in diagram (c), which has another end all the way at the bottom right corner of
the cavity, has a greater wavelength than the waves shown in diagram (b), since
those ones don’t extend as far. Thus, the wavelength of the given
waves increases as we look at the diagrams from right to left, from (a) to (b) to
(c). We were told that these diagrams
show the possible resonant cavity modes. The resonant modes are those waves
that can fit in the cavity according to the boundary condition so that the wave’s
initial and final displacement is zero at the walls.
Now, this question is asking us
whether increasing the wavelength of a wave increases, decreases, or doesn’t affect
the number of modes possible in the cavity. The given diagrams can help us
figure this out. Looking at the wave in diagram (c),
which we know has the largest wavelength shown, it can only fit one mode in the
cavity. This means that with one end fixed
at this point like we noted earlier, there is no other way to draw light of the same
wavelength so that it fits it in the cavity without violating the boundary
condition. For diagram (b), two modes at this
wavelength can fit in the cavity. For diagram (a), which shows the
smallest wavelength of the three diagrams, the number of modes that can fit into the
cavity is three.
So we have seen that with
increasing wavelength, there is a decreasing number of allowed modes. Therefore, when asked whether
increasing the wavelength would increase, decrease, or not affect the number of
modes, we know that the answer is decrease.
