Hey, guys. So now that we know how to use LC circuits, we're going to take a look at how the energy changes between the capacitor and the inductor, because sometimes you're going to need to know that. Let's check it out. So whenever we looked at LC circuits, we just assumed that there was no resistance. And because there is no resistance in an LC circuit, the energy is conserved. It basically just bounces between the capacitor and the inductor in the LC circuit. Now, the energy is going to oscillate because we know the LC circuits oscillate between electrical energy in the capacitor, and this electrical energy is actually stored inside the electric field that exists between the plates. So we have this electric field right here, and that's actually where this electrical energy is stored. And then what happens is the magnetic energy comes from the fact that you have current going through an inductor. So this magnetic field sort of gets set up because you have current that goes through a coil of wire. And we know that coils of wire will generate sort of these loops, and you have a magnetic field that points in this direction. That magnetic field actually stores energy, and so this is actually where that magnetic energy comes from. Right? So it's constantly going between the electrical and the magnetic energy. We actually have the equations that give us the electrical energy for a capacitor. It's those three equations we were able to use. Well, for magnetic energy of an inductor, we have another equation that's going to be 1/2LI2. So just have one equation that relates circuit, out of phase and one's going up while the other one's going down. So let's take a look at the steps of an LC circuit and see what the energy is doing. We know that there's a relationship between the charge and the electrical energy, and then the current and the magnetic energy. So let's take a look at that. Right. So, in this first step here, where you have no current and the capacitor is fully charged, that actually means that there's no current going through this circuit right now. So that we have the maximum amount of charge that is between the plates of the capacitor, so the Q is max over here. So that tells us from our equations that the electrical energy is going to be maximum here, while the inductor energy or the magnetic energy is going to be 0 because there is no current. So that means that there is no magnetic energy here, and it's all electrical. So I'm just gonna draw a little bar graph like this. So now we know that there is a second step that is sort of in the middle between these. We know that there's 8 steps. And here, where the charge is 0, the current is going to be maximum right here. So that means that throughout the sort of middle step in this process, we know that the charge is going to go down, which means that the energy that's stored in the capacitor is going to go down, whereas the current is increasing. So that means that the magnetic field energy is going to go up. So what happens here is that now we have no more charge left on the capacitor, and then when the current reaches its maximum point right here, the magnetic field it threw out this inductor is going to be at its strongest. So we have a magnetic field that points in this direction like this. And so from our equations, it's We can see that if the current is going to be maximum, then that means that the magnetic field energy is going to be strongest here. So we're going to have a bar graph like this. And if there is no charge across the capacitor, then there is going to be zero energy over here. And now what happens is that we know that the cycle just repeats itself in reverse. So that means that the electrical energy is going to be maximum here because you have the maximum amount of charge. It's just going in the opposite direction, and there is 0 magnetic energy. And then over here, there is going to be maximum, there's going to be maximum in magnetic energy, because you have I_max like this, and then there is going to be 0 electric energy, because there is no more charge across the plates. Alright? So, basically, it just oscillates back and forth, but we can see that at any point in time, what happens is that the total amount of energy, that's e, is going to remain the same across all of them. So what happens is you have this relationship between the electric and the magnetic energies, such that as one goes up, the other one goes down. It's kind of like how potential and kinetic energies worked in simple harmonic motion. So there's that analogy that we can sort of draw again. So remember, when we had simple harmonic motion, where we basically just had a spring that was attached to a block or a block on a spring, then there was the equilibrium points right here, and what happens is that you have all this potential energy. So here you had so we had Δx was equal to its maximum right here. So that means that you had the maximum amount of potential energy. And then when it came through the middle like this, either in this direction or that direction, we know that the velocity was maximum here. So that means that the kinetic energy was maximum at this point. And then as this thing went back and forth, that energy would change between the potential and the kinetic throughout that cycle. It's kind of like the same thing here. Okay. Alright. So let's check out this equation or let's check out this practice problem or this example, now that we have another equation for the magnetic energy. Cool. So we have an LC circuit and we're told that it has a 0.1 Henry inductor, 15 nano farad capacitor, and we're supposed to find out after one second or after 0.1 seconds, how much energy is stored by the inductor. So for part a, let's see. We're going to be looking at what is the energy, so that's going to be u of l. Right? So what is the magnetic energy right here? So let's see. We have an inductor that is 0.1. We have a capacitor that is 15 times 10 to the minus 9. Remember that that prefix nano. And we have q max, which is our maximum charge, is going to be 50 milli Coulombs. So that's 0.05 right there. So the equation that's going to tie all of these things together is going to be the magnetic energy equation. So that's going to be UL equals 1/2LI2. So let's see. We have the inductance, that's going to be our L. All we have to do is figure out the current, but this is actually the current at a specific point. So we're going to have to use the equation for I, which is that I of t is equal to negative ω q max times sin of ω t Left Button+ φ. Okay? So this is actually where we're going to get our current equation from, because we need the current at a specific point. So let's see. We can figure out what our ω is. That's going to be the square root of 1 over LC. So let's see. I get the square root of 1 over 0.1 times 15 times 10 to the minus 9. That's the capacitance. And so we get that ω is equal to 25,820. So that's one variable. And so we want to evaluate what the current is when t is equal to 0.1. So let's see. I have everything else, so I have ω, I have q max. The one thing I don't know is I actually don't know what the phase angle is. But we actually do know that the capacitor begins, or the states the system begins with the capacitor that is maximally charged. So what that means is whenever that happens, so if the cycle begins with Qmax, then that means that φ is equal to 0. That means that your phase angle is just nothing. It's just starting cosine ωt. Right? So remember that, that phase angle just determines the starting points. And if you're starting with the capacitor maximally charged, then that starting point is just 0. So that φ is just equal to 0. Cool. So I'm just going to move this around somewhere else. That's going to go over here. Cool. So we can just go on with our equation. So we have I when t is equal to 0.1. We have negative ω, so that's negative 25820, and then we have q max, which is 0.05. And now we have the sign of then we have 25820, because that's ω, and then t, which is 0.1. And remember that you have to have your calculator in radians mode, and your current should be, I get 490 amps. So now we're just going to plug that in to the equation. By the way, if you got a negative sign, it might have been something there. But the thing is that it's going to go away when you square it anyway. So let's see. Our magnetic energy is going