An alternating voltage source with a frequency of 120 hertz is connected to an inductor, producing a reactance of 24 ohms. What is the inductance of the inductor? Give your answer to the nearest millihenry.
First, it’s worth noting that the henry is the SI standard unit of inductance. Now, in this example, we have an alternating voltage source. And we’re told that it’s connected to an inductor. We don’t know the inductance of the inductor 𝐿, but we are told something called its reactance.
Inductive reactance is symbolized 𝑋 sub 𝐿. And we see that this is a quantity with units of ohms. Indeed, reactance is a measure of the opposition to the flow of charge of a component in a circuit. In this sense, it’s like but not identical to resistance. Anyway, we know the frequency of oscillation of voltage in this circuit. It’s 120 hertz. And knowing all this, we want to solve for the inductance of the inductor.
We can begin doing this by recalling the mathematical expression for inductive reactance. We can solve for inductive reactance if we know the frequency of oscillation of a circuit as well as its inductance. Here, of course, it’s 𝑓 and 𝑋 sub 𝐿 that we know and 𝐿 that we want to solve for. Let’s do that then by rearranging this expression. We’ll divide both sides by two times 𝜋 times 𝑓, canceling those factors on the right. We now have this expression where 𝐿 is the subject. And we can substitute in the known values for 𝑋 sub 𝐿, that’s 24 ohms, and 𝑓, that’s 120 hertz. Writing this fraction as a decimal, we have 0.031830 and so on henries.
We want to give our answer though to the nearest millihenry. The prefix milli- tells us that one henry is made up of 1,000 millihenries. This means to write our result in millihenries, we would shift the decimal place one, two, three spots to the right. That gives us 31.830 and so on millihenries. Rounding this to the nearest millihenry, we see that the number after the decimal, eight, is greater than or equal to five. And so we will round up. The inductance of the inductor is approximately 32 millihenries.