Video Transcript
Which of the electromagnetic waves shown in the diagram of a cavity corresponds to a possible resonant mode of the cavity?
Here then, we’re working with a cavity. A cavity is any bounded interior space. In this case, the black box represents the cavity. In a cavity, it’s possible for electromagnetic waves to exist in what are called resonant modes. We can come to understand these modes by thinking of, for example, a rope that’s fastened between two walls. Under these conditions, it’s possible for certain standing waves to exist in the rope. For example, a standing wave that looks like this is possible. Another possibility is one that looks like this, or like this.
All of these waves, we could say, are resonant modes of the system. And notice that all of them obey what are called boundary conditions. That phrase just means that there are requirements for how the wave, in this case the one that moves along the rope, meets up with its boundaries, in this case the two walls. Because the ends of this rope are fixed in place, that determines what kind of waves, what sort of resonant modes, can exist on the rope.
The same thing applies for electromagnetic waves in a cavity. A cavity too has boundary conditions. The requirement is that at whatever two points on a cavity wall a wave reaches — for example, in answer option (A), those two points are here and here, and in option (B), they’re here and here — at those two points, the wave must have a displacement of zero. Essentially, the wave must act like a rope did over here. The ends of the wave must be fixed in place, so to speak.
Looking at our answer options, we see that one of them satisfies this condition and one does not. In option (A), the wave drawn in blue has a displacement of zero at this point here, and it also has a displacement of zero here at the other point where it reaches the wall of the cavity. Therefore, this is a possible resonant mode of the cavity. The boundary conditions are satisfied. On the other hand, in option (B), that wave does have a displacement of zero at this point. But over here, at the other wall of the cavity, we can see its displacement is nonzero. Returning to our system of the rope between two walls, this would be as though the end of the rope somehow moved up the wall, even though it was required to be fixed in place. That can’t happen, and therefore the wave shown in option (B) is not a possible resonant mode of the cavity.
For our answer then, we choose option (A). This wave corresponds to a possible resonant mode of the cavity.
