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).
