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Energy in an LC Circuit

Patrick Ford
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Hey, guys. So now that we know how to use Elsie circuits, we're gonna take a look at how the energy sort of changes between the capacitor and the induct. Er, because sometimes you're gonna need to know that. Let's check it out. So whenever we looked at Elsie circuits, we just assume that there was no resistance. And because there is no resistance and NLC circuit, the energy is conserved. It basically just bounces between the capacitor and the induct er in the L C circuit. Now the energy is gonna oscillate because we know that Elsie circuits oscillate between electrical energy in the capacitor and this electrical energy is actually stored inside of the electric field that exists between the plates. So we have this electrical field right here, and that's actually where this electoral energy is stored. And then what happens is the magnetic energy comes from the fact that you have current going through an induct er so this magnetic field sort of get 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 that we were able to use well for magnetic energy oven induct er we have another equation that's gonna be one half l times I squares just at one equation that relates the induct INTs and the currents. Now, basically, we know that in an L C circuit, the charge and the current are constantly sort of 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 L C 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 currents 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 is no current going through this circuit right now, um, so that we have the maximum amount of charged 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 induct er energy or the magnetic energy is going to be zero 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 eight steps and here where the charge is zero, the current is gonna 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, um, 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 throughout, this induct er 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, which 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 gonna 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 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 zero magnetic energy and then over here there is going to be maximum. There's going to be maximum in magnetic energy because you have IMAX like this. And then there is going to be zero electric energy because there is no more charge across the plates. All right, so basically 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 could sort of draw again. So remember when we had simple harmonic motion where we basically just had a ah spring that was attached to a ah, block or 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, um, so we had Delta X 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 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, All right. So let's check out this equation or let's check out this practice problem. Or in this example, now that we have another equation for the magnetic energy. Cool. So we have an L C circuit, and we're told that it has a 0.1 Henry induct er 15 Nana Farid capacitor. And we're supposed to find out at after one second or after a 10.1 seconds. How much energy is stored by the induct? Er so for part A, let's see, we're gonna be looking at what is the energy? So that's gonna be U of L. Right. So what is the magnetic energy right here? So let's say we have a conductor. That is 0.1. We have a capacitor that is 15 times 10 to the minus nine. Remember that that prefix nano and we have Q max, which is our maximum charge is going to be 50 Milica cool. Um, so that 0.5 right there. So the equation that is going to tie all of these things together is going to be the magnetic energy equation. So that's gonna be you. L equals one half L I squared. So let's see, We have the induct, insp. That's gonna be our l. All we have to do is figure out the current, but this is actually the current at a specific time. So what happens is that this current that we're gonna plug in actually comes from the current, often Elsie circuit. So we're gonna have to use the equation for I, which is that I have Tea is equal to negative. Omega que max times sign of omega T plus five. Okay, so this is actually where we're gonna get our current equation from where we need the current at a specific point. So let's see. We can figure out what are Omega is that's gonna be the square root of one over l c. So let's see. I get, uh, square roots of 1/0 10.1 times times 10 to the minus nine. That's the capacitance. And so we get that omega is equal to 25,820. So that's one variable And so we want to evaluate with the current is when t is equal to 0.1. So let's see. I have everything else. So I have omega. I have Q max. Uh, 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 sights. The system begins with the capacitor that is maximally charged. So what that means is, whenever that happens, um, so if he cycle begins with que Max, then that means that Phi is equal to zero. That means that your phase angle is just nothing. It's just starting. Cosine Omega T right. So remember that that phasing will just determined the starting points. And if you're starting with the capacitor maximally charged, then that starting point is just zero. So that fight is just equal to zero. Cool. So I'm just gonna move this around somewhere else that's gonna go over here Cool. Uh, eso we could just go on with our equation. So we have I don t is equal to 0.1. We have negative omega. So that's negative. 25 8 to 0 and then we have Q max, which is 0.5 And now we have this sign off. Then we have 25 8 20 because that's Omega and then T, which is 0.1. And remember that you have to have your calculator in radiance mode and your current should be. I get 490 amps. So now we're just gonna plug that into 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 anyways, so let's see. Our magnetic energy is gonna be one half now. There are inducted 0.1, and now our current is 490 squared. So that means that our magnetic energy is going to be I actually get 12,000 jewels. So that's our answer for that first part of the question. And now the second part says, What is the maximum current throughout the induct? Er, So let's see the equation for maximum current. You might have seen in another previous video that the maximum current the magnitude of the maximum current is going to be Omega Times Q. Max Because it's going to be wherever the sine function just equals one or negative one doesn't really matter. So let's see. Imax is just going to be Omega, which is 25 8 20 then Q max, which is 0.5 right. 0.5 So what I get is, I get, um so he gets 1291 So I get 12 1291 amps. Alright, guys. So that is it for this problem? Let me know if you have any questions I'll see in the next one.