Question Video: Impedance of Alternating Current Circuits

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? [A] Graph a [B] Graph b [C] Graph c [D] Graph d

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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 ๐œ”, 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, weโ€™ll need to convert between ๐œ”, the angular frequency, and ๐น, the regular frequency. This is easily accomplished with the simple relationship ๐œ” equals two ๐œ‹๐น. So our relationship between reactance and frequency is that capacitive reactance is equal to one divided by two times ๐œ‹ 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, letโ€™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 didnโ€™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 thatโ€™s the shape of the graph in (a).

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