Video Transcript
Which of the following graphs
correctly shows how the reactance of a capacitor varies with the frequency of the
alternating voltage source that the capacitor is connected to?
For our answer choices, we have
four different graphs. Each graph has frequency in hertz
on the horizontal axis and reactance in ohms on the vertical axis. The blue curve on each graph shows
a possible way that reactance might change with frequency. Our task for this question is to
choose the graph that correctly shows the relationship between reactance and
frequency for a capacitor hooked up to an alternating voltage source.
Recall that for a capacitor
connected to an alternating voltage source, the capacitor first charges with, say,
positive charge on the top plate and negative charge on the bottom plate. Then, after the emf reaches a
maximum value, the capacitor begins to discharge and then charge again, but this
time with negative charges on the top plate and positive charges on the bottom
plate. After the electromotive force
reaches its maximum value in the other direction, the capacitor again discharges and
the cycle repeats itself. The more charged the capacitor is,
the more it opposes current, so the larger its reactance. However, the faster the
electromotive force changes directions, the less time the capacitor has to charge
before it begins to discharge. This means that the capacitive
reactance gets smaller as 𝜔, the angular frequency of the voltage source, gets
larger.
The converse is also true. The capacitive reactance is larger
for smaller angular frequencies. If we write this as a formula, we
can write that the capacitive reactance is equal to one divided by the angular
frequency of the voltage source times the capacitance of the capacitor. To use this formula, we’ll need to
convert between 𝜔, the angular frequency, and 𝐹, the regular frequency. This is easily accomplished with
the simple relationship 𝜔 equals two 𝜋𝐹. So our relationship between
reactance and frequency is that capacitive reactance is equal to one divided by two
times 𝜋 times the frequency of the alternating voltage source times the capacitance
of the capacitor.
Since our formula with frequency
and our formula with angular frequency both have the same basic form with frequency
and angular frequency both appearing in the denominator, we can clearly see that the
capacitive reactance will get smaller as the regular frequency gets larger and vice
versa, just like with angular frequency. All of the graphs pictured show a
larger reactance at lower frequencies and a smaller reactance at higher
frequencies. So we need some other way to
distinguish between these graphs. If we look back at our formula, we
can see that the reactance will always be greater than zero as long as the frequency
is finite. The reactance may get very, very,
very small, but the right-hand side of this formula is never zero.
Looking back at our graphs, we can
eliminate choices (b) and (c) because both of these show a reactance that reaches
zero at some frequency. Both graphs (a) and (d) show a
reactance that gets smaller with increasing frequency but never actually reaches
zero. Since these two graphs have the
same behavior at high frequency, let’s see what happens at low frequency. As the frequency gets lower and
lower, the denominator of this fraction gets smaller and smaller, so the overall
fraction gets larger and larger. And since we can make the frequency
as close as we want to zero, the capacitive reactance should increase without limit
as the frequency gets closer and closer to zero.
Of choices (a) and (d), only choice
(a) shows a reactance that appears to be increasing without limit for very low
frequencies. On the other hand, the graph in
choice (d) appears to be tapering off and will not increase without limit. So the correct answer is the graph
shown in choice (a). It turns out we actually didn’t
need to analyze all these graphs in detail to get this answer. Both our qualitative and
quantitative relationships between reactance and frequency tell us that reactance
and frequency are inversely proportional. This means that as one of them gets
larger, the other gets smaller and as one gets smaller, the other gets larger. A graph showing the relationship
between two inversely proportional quantities always has the same basic shape, and
that’s the shape of the graph in (a).
