Pearson+ LogoPearson+ Logo
Start typing, then use the up and down arrows to select an option from the list.

LC Circuits

Patrick Ford
Was this helpful?
Hey, guys. So in this video, we're gonna take a look at a specific type of induct er circuit called Elsie 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 earlier Chapter in physics. So let's check it out. And l c circuit is made up of an induct er which remember is the L S O. That's the L right here, and it's also a capacitor. So these sort of our combines together, and that's the C and L C circuit, and it actually follows an eight steps cycle. So we're gonna take a look at each one of the steps very, very carefully. Now, for the sake of argument, let's just say that we have an induct, er so we have a conductor right here and a capacitor, and these things are combined together without a battery. And let's just say that one of the capacitors of the capacitor is charged initially, so we have a maximum amount of charge that's on the plates. So you have all these electrons that are sort of locked up together on these plates. The green lines represent the electric fields that goes to the spot. And what happens is that without a battery or anything like that or a resistor, all these electrons wants to start flowing out like this, right? They want to start discharging from the capacitor. Now, Initially, the current is zero. There's 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 wants to go around the circuit like this, and you're gonna generate some kind of current. Now, what happens is this current wants to go from 0 100 right? He wants to basically discharge immediately, but it can't do it because remember that the function of this induct er is that conductors always resist any changes in current, they resists delta eyes. So this induct er makes it impossible for the for the current to go from 0 to 100 or for 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 in this point, right? here, right? So it's still going to be increasing. So 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 over here, all of the charges have finished discharging from the capacitor. So that means at this point all of 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. So we see that the initially the current zero, the capacitance are the charge was maximum and the current was zero. Now it's the opposite. The charges zero. But the current is maximum. So know what ends up happening is that you have all these positive charges that assert to piling up on the other side of the plate. So were 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 currents in this case is decreasing in this phase because it's sort of like running out of steam, like all the charges air piling up on this side. But remember that just as in the second step, where this induct er 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 induct er 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, um, all of the charges that are on the right side, whereas the charges on the left side. But now there's no more currents, and the charge is maximum here. So you have maximum charge and zero currents, and basically from here, the entire process just goes in reverse. So now, instead of the charges wanna leave to the left, they wanna leave to the right, So that's what they're going to start doing? So you're gonna have a current right here that's going to increase. But remember that this induct er is going to resist any changes in current. And then what happens is, um, 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 charges 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 whereas originally started on, so the current gonna be decreasing there. But remember that this is going to be resisting any changes in the currents and then basically, you just go back to the way it began. Right? So now you basically stir starting the whole entire cycle over again. Okay, got it. So we've seen that this system is oscillating, so this system goes back and forth. The current basically I like to think about like a pendulum. The current is going back on one side and then back through the other. And the induct er is always preventing any changes in current like that. So this system is oscillating. It actually behaves very, very similar to another kind of motion that we've seen called simple harmonic motion. So simple harmonic motion is when we had a block attached to a spring. So 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. So 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. Um, 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 on 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 the system is oscillating, the formulas for the charge and the current are actually represented by Sinus soil functions. So the spinal fluid or functions just they don't necessarily mean they're both sign functions. It's just the ones that oscillates. So signs and co signs. Now, we said that in the beginning of the cycle, all the charge was maximum on the capacitors. So the function that starts office maximum is the co sign function. So the way that the do you remember this is that the charge is always gonna be maximum at first. So that means that this function right here, q of t is gonna be Q max times the cosine of omega T plus five. 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 this stiffness of that spring and the mass and things like that. Well, here it just depends on the induct er and the capacitor. So it's the square root of one over l. C. And remember, that is the angular frequency. So you can always relate the angular frequency to the linear frequency by two pi times f. So this f represented the number of cycles that happen per second. Eso they're not quite the same thing, But you always have that relationship right there. Then you could also relate that to the period as well. Okay, so and then this fi term right here. So this little angle right here five is called the phase angle, and it basically just determines the starting point of your oscillation. It's just a constant goes out there, just in case you started some other point in the cycle. All right, 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 cosign you have sign, and that's gonna be Omega T plus five. Now, just a heads up. For those of you who are taking calculus. You actually might recognize this as the derivative of the Q T function. But if you don't, if you're not in the calculus course, you don't have to worry about that. All right, so that's the two functions. The last thing I wanna point out is that make sure that your calculators are in radiance mode because we're working with these co sign and sign functions with radiance. Right, So we have angular frequencies, so just go ahead and make sure that your calculators sets radiance mode and we're gonna go ahead and check out this example. So we have a capacitor with a capacitance like this and has initial charge of magnitude. This and we're supposed to be figuring out that during the current oscillations that that occur after the circuit is completed, what's the maximum current in the induct? Er, so let's take a look. We're looking for the maximum current, so that means that we're gonna be looking for I Max right here. So we're gonna have to relate this to the current function often Elsie Circuit. So remember that the current function I of T is equal to negative omega que max times the sign of Omega T plus five. Right, So this current is going to oscillate. It's gonna go up and down like this. If you were to plotted on a graph, right? So if you were to plot the charge versus time, it would start to look like this. So, do you have some kind of oscillating function now, whenever the current is maximum, whatever it occurs at these points or these points, what this really means? Is that the sign? So let's let's write this out because it's actually really important. So I is equal to I. Max. Yeah, when the sine function, the sign term sine omega T plus fi, whatever those variables are is equal to one or negative one, right? So whenever this whole entire equation is equal to one, that's when the current is going to be maximum. So in other words, you just have that the absolute value of I Max is just when you have Omega Times Q. Max, right, because when this is equal to one or negative one, it doesn't really matter. Then that current is going to be maximum, so let's see. So basically, we're gonna use be using this equation. We actually know what the maximum charge on the capacitors are. Now we just have to figure out what this Omega term is so remember that omega is the angular frequency and we can relate it to the induct INTs and the capacitance. By the equation, Omega equals square root off one over L C. So Omega is gonna be the square root of one over the induct. Its was four. Henry's in the capacitance is Be careful with this because you see, See here, Not by confusing, that's capacity to remember that school loans, that's for charge. So this is actually que Max and this is actually the capacitance. And then this is the induct install, right, So don't get those confused. So we've got four times, five times 10 to the minus nine and you work this out, you're gonna get 77071 And that we can do is we can plug this back and figure out what IMAX is. So we just get that IMAX is equal to 7071 times Q max, which is equal to two times 10 to the minus four, and you work this out. You should get a maximum currents that's equal to 1.41 amps. Alright, guys. So that's it for this? That's the maximum current in the induct. Er, and we're gonna take a look at a couple more practice problems. Let me know if you have any questions.