Question Video: Recalling the Relationship between Reactance and Inductance Physics

Video Transcript

Which of the following formulas correctly relates the reactance 𝑋 of an inductor to its inductance 𝐿 when connected to an alternating-voltage source with a frequency 𝑓? (A) 𝑋 equals one over two 𝜋𝑓 times 𝐿. (B) 𝑋 equals 𝐿 over two 𝜋𝑓. (C) 𝑋 equals two 𝜋𝑓 times 𝐿. (D) 𝑋 equals two 𝜋 𝑓 over 𝐿. And (E) 𝑋 equals two 𝜋𝐿 over 𝑓.

We’re looking here for the correct formula for the reactance of an inductor. Let’s say that this is our inductor and it’s connected, we’re told, to an alternating-voltage source. Reactance is a term we often encounter when we’re working with alternating-voltage circuits. Reactance is like resistance. It’s the measure of a given component’s opposition to the flow of charge. In our circuit then, which has an inductor of inductance 𝐿 and is operating at a frequency 𝑓, we want to write an equation for how much this inductor opposes charge flow. That’s its reactance. To begin figuring out which of our answer options is correct, let’s imagine what will happen as 𝐿 and 𝑓 in the circuit are varied.

First, let’s think about changing the frequency 𝑓. We know that a general property of inductors is that they resist changes to current through them. This means that if we increase 𝑓, the frequency at which voltage as well as current oscillates in the circuit, then that will also lead to an increase in the opposition of the inductor to the flow of charge. In other words, increasing 𝑓 will make the inductor’s reactance 𝑋 go up. Knowing this, we can now eliminate any answer options where reactance 𝑋 and frequency 𝑓 are inversely related. We see such an inverse relationship in answer option (A) as well as answer option (B) and answer option (E). All of these choices claim that, for example, if 𝑓 increased, then 𝑋, the reactance, would decrease. We know that’s not true though. So we’ll cross out these answer choices.

Next, let’s imagine in our circuit what will happen if the inductance 𝐿 of the inductor goes up. In this case, the inductor’s capacity for opposing changes in current will be increased, therefore, so will its capacity to oppose the flow of charge. Its reactance will increase as inductance increases. This means we can eliminate any answer options where 𝑋, the reactance, and 𝐿, the inductance, are inversely related. We see that answer choice (D) claims that they are. This though would mean that as inductance increases, reactance decreases. And we know that that’s not the case.

This leaves us with answer choice (C). This choice says that reactance is directly proportional to frequency as well as inductance. We found this to be an accurate description of what happens physically in an alternating-current circuit with an inductor. Inductive reactance equals two times 𝜋 times 𝑓 times 𝐿. And just as a side note, the reactance of an inductor is often represented 𝑋 sub L. This lets us know specifically which circuit component we’re considering. In any case, in answer to this question, we choose option (C).