Video Transcript
Which of the electromagnetic waves
shown in the diagram of a cavity corresponds to a possible resonant mode of the
cavity?
In these three options, we see
three different waves inside of our cavity represented by the black box. We want to identify which one
corresponds to a possible resonant mode of the cavity. A resonant mode of the cavity means
we’re talking about an electromagnetic wave that satisfies what are called the
boundary conditions of the cavity. Any cavity, any shape, has boundary
conditions. Say that we had a cavity with this
shape, and we wanted to set up an electromagnetic wave between, say, this point and
this point in the cavity walls.
The requirement for that, the
boundary condition, is that the displacement of whatever wave exists between these
two points at those points in the wall is zero. So, for example, because a wave
like this satisfies the conditions of having a displacement of zero at the two
walls, this is a possible resonant mode of the cavity. And there are other waves possible
between these two points that satisfy those same boundary conditions. For example, a wave like this meets
those conditions of having a displacement of zero at the walls, and so does this one
and so on.
In looking over our answer options
then, we’ll be looking for a wave that satisfies the boundary condition that its
displacement is zero at the two points where it meets the cavity wall. In option (A), those two points are
here and here. We see that the wave’s displacement
at the first of those points is zero. But that’s not true at the second
point. Because this wave does not meet the
boundary conditions of the cavity, it cannot be a possible resonant mode.
In answer option (B), the two
points where the wave meets the wall are here and here at the ends of the dashed
line. And once again, the displacement of
the wave at the first point is zero, but not at the second point. This wave too fails to meet the
cavity’s boundary conditions. In option (C) though, at the ends
of the dashed line where the wave interacts with the cavity wall, we see the wave’s
displacement at both of these points is zero.
Our answer then is option (C). The wave shown in this choice does
correspond to a possible resonant mode of the cavity.
