Hey, guys. So in this video, we're going to take a look at a specific type of inductor circuit called an LC circuit because you might need to know how they work. Now, we're going to see that it's very, very similar to another kind of motion that we've seen in an earlier chapter in physics. Let's check it out.

An LC circuit is made up of an inductor, which remember is the "L." So that's the "L" right here. It's also a capacitor. So these are combined together, and that's the "C" in LC circuit. It actually follows an 8-step cycle. We're going to take a look at each one of the steps very, very carefully. Now for the sake of argument, just say that we have an inductor. So we have an inductor right here and a capacitor, and these things are combined together without a battery. Let's just say that one of the capacitors, the capacitor is charged initially. So we have a maximum amount of charge that's on the plates. You have all these electrons that are sort of locked up together on these plates. The green lines represent the electric fields that go through this spot. What happens is that without a battery or anything like that or a resistor, all these electrons want to start flowing out like this. Right? They want to start discharging from the capacitor.

Now, initially, the current is zero. There are no charges anywhere in the circuit, but immediately after this step right here, what happens is that this charge will start leaking out of the positive plates. Right? So it wants to go around the circuit like this, and you're going to generate some kind of current. Now what happens is this current wants to go from zero to a100. Right? It wants to basically discharge immediately, but it can't do it because remember that the function of this inductor is that inductors always resist any changes in current. They resist Δi's. So this inductor makes it impossible for the current to go from zero to a hundred, or for it to change very rapidly. So what happens is that some of this charge is still locked up in the capacitor, but the current is going to be increasing at this point right here. Right? So it's still going to be increasing. You have these charges that start to basically pile up on the other side of the capacitor like this. So now what happens is eventually, in the next step, we're over here, all the charges have finished discharging from the capacitor. At this point, all the electrons are basically going through the circuit, which means that the current at this point is actually maximum right here. So, when it's maximum, there is no more charge left on the capacitor. We see that initially, the current was zero, the capacitance, or the charge, was maximum, and the current was zero. Now it's the opposite. The charge is zero, but the current is maximum. Now, what ends up happening is that you have all these positive charges that start to pile up on the other side of the plate. Initially, whereas this was positive on the left side, now, actually, this is going to be the positive on the right side, because all the positive charges from the current are basically piling up on this side.

Now, what ends up happening is that the current in this case is decreasing in this phase because it's sort of like running out of steam, like all the charges are piling up on this side. But remember, just as in the second step, where this inductor was resisting any large changes in increasing current, it does the same exact thing here. It resists any changes in current, even if they are decreasing. So this inductor is still making it impossible for the current to basically go down to zero immediately. Anyway, so then what happens is that all of the charge finishes piling up on the other side. And now you basically have a reversal of what of the first sort of half of this step right here. So now you have all the charges that are on the right side, whereas the charges are on the left side. But now there's no more current, and the charge is maximum here. So you have maximum charge and zero current. And basically from here, the entire process just goes in reverse. So now, instead of the charges wanting to leave to the left, they want to leave to the right. That's what they're going to start doing. So you're going to have a current right here that's going to increase, but remember that this inductor is going to resist any changes in current.

And then what happens is, so whereas there's still some amount of charge right here that's still left on the capacitor, eventually, all of that is released from the capacitor and the current is maximum here at this step right here. And the charge is zero. And then what happens is that now, the current is going to be decreasing here as all the charges start to pile up again on this side, so where it originally started on. So the current's going to be decreasing there, but remember that this is going to be resisting any changes in the current. And then basically, you just go back to the way it began. Right? So now you basically start starting the whole entire cycle over again. Okay. Got it. We've seen that this system is oscillating. This system goes back and forth. The current, basically, I like to think about it like a pendulum. The current is going back on one side and then back through the other, and the inductor is always preventing any changes in current like that. So this system is oscillating, and it actually behaves very similar to another kind of motion that we've seen called simple harmonic motion.

Simple harmonic motion is when we had a block attached to a spring. We had a block like this, and we had if we pull it back, there was some force that was acting on it. So the force was here, and the velocity was equal to zero. And if we just let it go, then this block wanted to basically oscillate. There was an equilibrium position like this, and it would start to speed up during this phase right here, then it would go past its equilibrium position, and then it would slow down over here on this phase. And then it would basically get to the other side, and then the whole entire thing would go back, it would be going in reverse order. So in other words, the force would be now this way, and the velocity would be zero in this case. It was basically just doing the exact opposite of what it just did. And this thing would just do this forever. Right? It would just go back and forth and back and forth. So, this system oscillates the same way that a simple harmonic motion oscillates. And because this system is oscillating, the formulas for the charge and the current are actually represented by sinusoidal functions. So, these sinusoidal functions just they don't necessarily mean they're both sine functions, it's just the ones that oscillate, so sines and cosines. Now we said that at the beginning of the cycle, all the charge was maximum on the capacitors. So, the function that starts off as maximum is the cosine function. So the way that the you're going to remember this is that the charge is always going to be maximum at first. So that means that this function right here, q of t, is going to be q max times the cosine of omega t plus phi. So it looks very very similar to how simple harmonic motion equations used to work. So you have the maximum amount of charge on the plates on the capacitor and then you have this omega term right here. Now, when we studied omega for simple harmonic motion, it depended on things like the stiffness of the spring, and the mass, and things like that. Well, here it just depends on the inductor and the capacitor. So it's the square root of 1 over LC.

And remember, that is the angular frequency. So we can always relate the angular frequency to the linear frequency by 2 pi times f. So this f represented the number of cycles that happened per second. So they're not quite the same thing, but you always have that relationship right there. And you could also relate that to the period as well. Okay. So, and then this, phi term right here, so this little, angle right here phi, is called the phase angle. And it basically just determines the starting point of your oscillation. It's just a constant that goes out there, just in case you start at some other point in the cycle. Alright. So that's the charge, and the current is slightly similar. It's similar to that, but it's slightly different. It's actually negative omega times q max, and then you have instead of cosine, you have sine, and that's going to be omega t plus phi. Now, just a heads up for those of you who are taking calculus, you actually might recognize this as the derivative of the QT function. But if you're not in a calculus course, you don't have to worry about that. Alright. So these are the two functions. The last thing I want to point out is that make sure that your calculators are in radians mode, because we're working with these cosine and sine functions with radians. Right? So, we have angular frequencies. So just go ahead and make sure that your calculator is set to radians mode, and then we're going to go ahead and check out this examplr