Video Transcript
Which of the following formulas
correctly relates the reactance 𝑋 of a capacitor to its capacitance 𝐶 when
connected to an alternating voltage source with a frequency 𝑓? (A) 𝑋 equals two 𝜋𝐶 divided by
𝑓. (B) 𝑋 equals two 𝜋𝑓 divided by
𝐶. (C) 𝑋 equals 𝐶 divided by two
𝜋𝑓. (D) 𝑋 equals two times 𝜋 times 𝑓
times 𝐶. (E) 𝑋 equals one over two 𝜋𝑓
times 𝐶.
Let’s begin by thinking about a
capacitor that is connected to an alternating voltage source. This capacitor has some
capacitance, we’ll call it 𝐶. And we’ll say that the source
oscillates at some frequency 𝑓. We want to know how changes in
these two variables, 𝑓 and 𝐶, affect what’s called the reactance of the
capacitor. Reactance is like resistance; it’s
the measure of a component’s opposition to the flow of charge. Reactance comes up when we have an
alternating current circuit with other components in it.
To figure out which of our five
answer options is correct, let’s think about what will happen in our circuit as we
change the frequency 𝑓 and capacitance 𝐶. First, imagine that we increase the
frequency of oscillation 𝑓. The higher 𝑓 goes, the more likely
it is that electric charge that accumulates on the plates of our capacitor will be
able to travel across this gap. In other words, an increase in 𝑓
will lead to a relative decrease in the opposition of the capacitor to the flow of
charge. As 𝑓 increases, the capacitor’s
reactance decreases. This means that for whichever of
these answer options is correct, there must be an inverse relationship between the
reactance 𝑋 and the frequency 𝑓.
Note that in answer options (B) as
well as (D), there is not an inverse but rather a direct relationship between 𝑋 and
𝑓. This would mean that as frequency
in the circuit increases, so does its reactance 𝑋. But we know this not to be the
case. Therefore, we can cross out answer
options (B) and (D).
Next, let’s think about what will
happen in our circuit if we increase the capacitance of our capacitor. Doing so means it’s possible for
more electric charge to build up on the plates of the capacitor. Perhaps surprisingly, increasing
capacitance then increases the ability of charge to flow in a circuit. This is the opposite of the effect
of reactance on a circuit. As the capacitance of our capacitor
increases, allowing more charge to move onto the plates of the capacitor, the
opposition to the flow of charge in the circuit decreases. Therefore, in whatever formula
correctly shows the relationship between 𝑋, 𝑓, and 𝐶, we expect an inverse
relationship between reactance 𝑋 and capacitance 𝐶.
Of our remaining answer options,
the only one demonstrating this relationship is option (E). Here, reactance is inversely
proportional both to frequency 𝑓 and capacitance 𝐶. We choose this as our answer.
Just as a side note, when we’re
talking about the reactance of a capacitor, often this is represented by 𝑋 sub
C. We might see an equation then that
reads 𝑋 sub C equals one over two 𝜋 times 𝑓 times 𝐶.
