Hey guys. In this video, we're going to talk about inductors and the role that they play in AC circuits. Alright? Let's get to it. Now remember that the current in an AC circuit at any time is going to be given by the equation that by now we've seen a bunch of times. Okay? Okay, this is going to tell us that the current is simply oscillating with an angular frequency of omega between some value positive i_{max} and some value negative i_{max}. Now the question is, how does the voltage across the inductor look? Well, remember that the voltage across an inductor, which we saw during our discussion of Faraday's law, is the inductance times delta I over delta t, which is the rate at which the current is changing. Okay. Now I can't show you how but using calculus you guys can arrive at the answer that the rate at which the current is changing looks like this.

So if I multiply this by the inductance, then I get the voltage across an inductor in an AC circuit at any time. This is going to be i_{max} times omega times L times cosine of omega t + π/2. Okay, so once again, remember the voltage across a resistor. The voltage across a resistor looks like i_{max} times R times cosine of omega t. So the angle that it operates at omega t is different than the angle that the voltage across the inductor operates at. This is some other angle theta prime which is omega t + π/2. Okay? So the current and the voltage across the resistor are in phase, right, their plots line up, but the current and the voltage across an inductor are not going to line up. They're going to be out of phase. If we plot the current across an inductor and the voltage across an inductor, you can see that the voltage across an inductor actually leads the current by 90 degrees. Okay? What's happening here is that the voltage is deciding to go up at a time when the current is 0. Okay? Then at a future time, the current starts to go up. But at this point, the voltage has already peaked. Then at a future time, the current peaks. Right? It's trying to match what the voltage is doing. But at this point, the voltage is already decreasing. So then at a future time, the current decreases but the voltage has already bottomed out. So you see the current is trying and trying and trying to match the voltage, but it's lagging behind, or we can say that the voltage leads the current. Either one is fine. Okay? Now, the maximum voltage across an inductor is going to look like i_{max} times omega L. This looks a lot like Ohm's Law where we said that the voltage across the resistor was I times the resistance. There appears to be a resistance-like quantity of omega L. That resistance-like quantity for capacitors we called the capacitive reactance. Now we are calling it the inductive reactance for inductors. The units are still ohms. Right? The same unit as for resistance. Alright. Let's do an example. An AC power source delivers a maximum voltage of 120 volts at 60 hertz. If an unknown inductor is connected to the source and the maximum current in the circuit is found to be 5 amps, what is the inductance of the inductor? Okay? What is the maximum current in an inductor circuit? This is just going to be the maximum voltage across the inductor divided by what they both have to share the maximum voltage. That's what Kirchhoff's loop rule says. So this is just going to be V_{max}, the maximum voltage by the source, divided by XL. And what is that inductive capacitance? Well, that is just going to be omega L. Okay so plugging this into our equation, we can say that i_{max} is simply V_{max}/omega L. And our unknown is the inductance. So what I want to do is multiply the inductance up and divide i_{max} over, and then I have inductance as V_{max} / omega i_{max}. Before I can continue though, we need to know what omega is. We're told the linear frequency is 60 hertz. Okay. Remember, this is linear frequency because the units are hertz. So the angular frequency, which is 2πf, is going to be 2π times 60 hertz, which is going to be about 377 inverse seconds. Now we can solve for the inductance. And the inductance is just going to be V_{max} / omega i_{max}. V_{max} is 120, right? Always look at and make sure that's a maximum voltage not an RMS voltage, divided by 377, which was our angular frequency. The maximum current in the circuit was 5 amps. So that's 5 and this whole thing is 0.064 henrys, the unit of inductance. Alright guys, that wraps up our discussion on inductors and AC circuits. Thanks for watching.