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Concept #1: LC Circuits

**Transcript**

Hey guys, in this video we're going to talk about a type of inductor circuits called LC circuits, let's get to it. LC circuits are made up of two things as the name implies, they're made up of inductors and capacitors, right? L for inductor, C for capacitors, alright? Very simple. Now, LC circuits, we're going to assume have no resistance, okay? Never assume that any wires carry resistance, unless you're specifically told that they do. What does the current look like in an LC circuit? Well, I have these diagrams right here to help us out. We're going to start with the capacitor initially charged with the left side positive, what that means is it's going to build up some current as charges leech off of the positive plate trying to run around to the negative plate, okay? So, this produces a current in the circuit and then eventually all of those positive charges have neutralized the negative charges on one plate and both plates are neutral, so the charge on the capacitor is 0, but thereÕs still a current in this inductor. Now, for reasons that will be explained in another video the current through this inductor is actually at its maximum value. So, even though the capacitor has no charge, current is still going to be flowing through the circuit and it's actually going to flow at its maximum value. So, what happens is, positive charges start accumulating on the opposite plate, now itÕs on the right plate and negative charges on the left plate until the capacitor becomes fully charged again, then what happens is once it's fully charged for an instant, there's no current, then the current switches direction and starts kicking out to the right from the positive plate, that current starts running around in the opposite direction, right? Instead of clockwise, which is what we had initially, now we have current going counterclockwise until all the positive terms is from the right plate neutralize the negative charges on the left plate and now, the capacitor is neutral again. Once again, just because the capacitor is neutral, doesn't mean that there's no current in the circuit, in fact, the current is at a maximum as I pointed out, okay? And then that current carries all the positive charges, back to the left plate and we're back where we started, okay? This is kind of like a block undergoing simple harmonic motion on a spring, when the block is initially compressed as a lot of potential energy store but no kinetic energy, no motion, and it has a lot of force propelling it forward, just like when the capacitor is charged there's a lot of voltage propelling the current forward, then the spring moves forward and it reaches the equilibrium point, once that spring is at the equilibrium point, there's no longer any motivation for it to keep moving, right? That force is gone, that's exactly what this looks like here, once the charge is gone, there doesn't appear to be any more voltage, there doesn't appear to be any more motivation, but that block has kinetic energy so it is, in fact, going to keep going, alright? This is exactly like what's happening with the current, initially there's a motivation for the current, just like, when the block is compressed there's a force that motivates it forward, but when the charge is 0 there doesn't appear to be any apparent motivation, just like when the block reaches the equilibrium point and thereÕs no longer a force, however that block has kinetic energy that carries it forward and the inductor has this current that carries the process forward, okay? So, because of those similarities, that's why you get this oscillatory motion, these oscillations just like you would get in simple harmonic motion, okay? Now, we're going to describe this mathematically and these equations are actually going to look identical to the equations for simple harmonic motion as they should. Mathematically, the current is going to look like some value Omega times Q, although as a capital Q, this is representative of the maximum charge on the capacitor, times cosine of Omega t plus Phi, Phi is just some number, Omega is just some number and we'll get to both in a second. Now, the charge on the capacitor is going to be Q, sorry, this is not cosine, this is sine, that is my bad, Q is cosine. So, Q is capital Q, once you get capital Q represents the largest possible value of the charge times cosine of Omega t plus Phi. So, first of all, what is Omega? Well, Omega mathematically, is one divide-, itÕs a square root of 1 divided by L times C, okay? So, thatÕs how you find the number for Omega, but what Omega is, is it's the angular frequency of the oscillation. Now, remember, angular frequency is related to linear frequency by 2 PI f, and the linear frequency is how many times this oscillation cycles. These images right here, represent one complete cycle, starting at some initial state, so the left plate charge positive and returning to that initial state, the left plate being charge positive, so the frequency tells us how many times this occurs per second, okay? So, just remember these things because these are from much, much earlier in your physics courses, alright? Now, what Phi is, is it's known as the phase angle, and it determines what part of that oscillatory process you begin in, we started with the capacitor fully charged, let's do an example to illustrate the phase angle could be different. An LC circuit with an inductor of 0.05 Henrys and a capacitor of 35 microfarads begins with the current at half its maximum value. What is the phase angle of this oscillation? So, the current equation we know is Omega times Q sine of Omega t plus Phi. We want to know it half the maximum value? Well, looking at this equation right here, what's the maximum that current could ever get? That depends upon sine, what's the max value that sine can never get? The largest sine can ever get is 1, right? It can go from negative 1 to positive 1. So, when sine is 1, that's the largest value. So, we know that iMAX is Omega Q, very simply when sine is 1. So, I can take this and I can say i of T is actually iMAX times sine of Omega t, sorry, little technical difficulties, there it go, plus Phi. Now, I can solve it. ItÕs asking what's the phase angle? If we start with the current at half the maximum value, okay? So, start means t is 0, I'm going to divide this maximum value over and that means that sine of Omega times 0 plus Phi equals i over i MAX and the current is half the maximum value, so that's holding equal to 0.5, right? If, this is 0, Omega times 0 is just 0. So, what we have is sine of Phi equals 1/2? Well, at what angle design equal 1/2? 30 degrees, or more properly PI over 6 radians, alright? The radians are important because when you plug it into this equation, right? When you multiply Omega times t, that value is going to be unitless, let's say it's 2. What's 2 plus 30 degrees? It's not 32 degrees, because that 2 is unitless, but that 30 carry the units of degrees, you need to represent your answer in terms of radians, because radians are unitless, you can add 2 to PI over 6. Alright guys, that wraps up our discussion on these LC circuits. Thanks for watching.

Example #1: Oscillations in an LC Circuit

**Transcript**

Hey guys, let's do an example. An LC circuit with L equals 0.05 Henrys and C equals 50 millifarads, begins with the capacitor fully charged. After 0.1 seconds the current in 0.2 Amps. Under these conditions, how many seconds does it take for a fully charged plate the transfer all of its charge to the other plate? Okay, so first of all, what is the physical process that's happening here? Alright, we have a capacitor that has one plate fully charged, the other place fully charged negatively and we are talking about how long it takes for all of this positive charge to go over to the other side and for this to end up being what? What would happen, if all of the positive charge moved to the other plate, this capacitor would be neutral, okay? How we're going to solve this is we're going to take advantage of fact that this process is cyclic like simple harmonic motion, okay? It's oscillatory whatever word you want to describe it, there is a pattern, right? The next step is for currents to keep flowing, don't forget that there's current continuing to flow for this to be negative and for this to be positive, then for current to flow in the opposite directions are missing an inductor here, for constant flow in the opposite direction neutralizing it again, and ending with, because they're still current here, ending back at our initial point, okay? Remember that this whole process is cyclic and this whole process takes one period to complete. Remember, that the period T is by definition the amount of time to complete 1 cycle. This whole process is 1 cycle and this cycle repeats itself and repeats itself and repeats itself and repeats itself. So, how are we going to find the period? Because that will tell us how long it takes to jump from this step to this step okay? Well, in order to find the period, we're going to think about some associated quantities of period. We know that frequency is 1 over the period but we have no easy way of finding frequency, however there is another type of frequency that we have an easy way of finding, we know that the angular frequency is 2 PI f and we do have an easy equation to find the angular frequency of an LC circuit that angular frequency is just 1 over LC. So, let's find that first, this is going to be 1 over 0.05 Henrys, which we are told and 50 millifarads, which is also 0.05 all of this together is going to be 20 in per second or 20 radians per second but radians or unitless, okay? Now, we can take this value to find the frequency, once we know the frequency you can take that value to find the period, okay? So, the frequency is going to be Omega over 2 PI, all I have to do is divide the 2 PI over, this is going to be 20 divided by 2 PI, which is 3.18 Hertz, okay? And now I can find the period, which is 1 over f, which is 1 over 3.18, which is just going to be a 0.079 seconds, that is the period, okay? How does the period help to solve this problem? Don't forget what we're looking for, we're looking for this step. Well, a period describes a full cycle, what fraction of a cycle is this step? Well, this is one step, right? This is two steps, this is three steps, this is four steps, there are four steps to complete this period, right? So, the first step represents 1/4 of the total cycle, so the amount of time is going to take is 1/4 of the period, so the amount of time is going to be t over 4, which is 0.079 seconds divided by 4, shoot I made a mistake here, this is not 0.079, I misread my notes, this is 0.314 four seconds and that equals 0.079 seconds, sorry about that guys, okay? But the process is pretty straightforward, we're talking about time durations during these cyclic processes, these cyclic processes are called symmetric, meaning each step takes the same amount of time, okay? Thanks for watching guys.

Practice: An LR circuit has a 0.5 mF capacitor initially charged to 1 mC. If it is connected to a 0.04 H inductor, what is the maximum current in the circuit?

Concept #2: Energy in an LC Circuit

**Transcript**

Hey guys, in this video we're going to be talking about what happens to the energy inside of an LC circuit, alright? Let's get to it. Now, an inductor is just a coil of wire, we don't want to assume it has any resistance, unless we're specifically told that it does, alright? Now, in an LC circuit the capacitor is giving up its charge to produce a current in the circuit, as it's losing its charge, it's losing its electric energy stored, where does that energy go? That's going to be the big question. That energy is not lost because there's no resistance in the circuit, if there was a resistance, if there was some resistance in the circuit, then it would be radiated off its heat, but since there isn't, what happens to it, it has to be converted in something else. An inductor can sort magnetic energy and that's exactly where the electrical energy stored by the conduc-, sorry, stored by the capacitor goes, becomes magnetic energy, and the magnetic energy stored by an inductor is 1/2 times the inductance times the current squared, okay? So, now we want to focus on is our cycle. Remember, that the current oscillates back and forth in this circuit and we want to focus on one cycle of this oscillation, we want to track, what happens to the energy during this cycle, so initially our left plate is charged to a maximum value, alright? So, all of the possible energy that we have is potential energy stored in the capacitor, it's electric energy. Now, because of that the inductor has no energy and therefore there's no current in the circuit. Notice that I did draw a little current here, that's only to indicate the direction the current will go in, there's only a split second that capacitor can maintain all of that charge without letting some of it go as current. So, immediately, it's going to produce a current and that current is going to go in the direction indicate, alright? So eventually, charges are flowing, charges are flowing and the charges on the capacitor balance each other out and now the capacitor has no charge, when the capacitor has no charge, it has no electrical energy stored, it has to have some sort of charge to have an electrical energy. So, all of that energy has been converted into magnetic energy stored by the inductor, since this is the largest amount of magnetic energy the inductor is ever going to store, this is the largest current, that's going to pass through the inductor, okay? So, this current is a maximum. I have indicated that this was true a little while ago, but I didn't elaborate on why it's true, this is why. Once the conductor, sorry, once the capacitor has no more charge stored on it, all of that electrical energy has been converted into magnetic energy stored by the inductor and since that's the largest amount of magnetic energy, it has have the largest current, okay? So, that current keeps flowing charges and then eventually the opposite plate becomes positively charged, once that reaches the maximum positive charge, all of that magnetic energy is converted back into electric energy and our magnetic energy is 0, so there's no current in the circuit, once again this indicated current is just the direction the current is going to flow, currents are flowing in that direction, eventually the plates neutralize again, no more electrical energy, the magnetic energy stored by the inducer is a maximum and consequently the current is a maximum, exactly like our second step and then finally that current carries positive charges back to our original plate and once the charges are built up on that plate to the maximum, all of the energy is returned back to electrical energy, there is no more magnetic energy and we are exactly right back where we started, okay? So, this is how the energy changes from electrical energy stored in the capacitor to magnetic energy stored on the inductor, okay? Throughout the process of an LC circuit, alright? Let's do an example. An LC circuit has a 0.1 Henry inductor and a 15 nanofarad capacitor, and begins at the capacitor maximally charged. After a 0.1 seconds, how much energy is stored by the inductor? If the initial charge on the capacitor were 50 milicoulombs, what is the maximum current in the circuit? So, we have two questions we need to answer at some time, how much energy stored by the inductor? And then what is the maximum current in the circuit? At any particular time, the current in the circuit is given by Omega the angular frequency of this oscillation, Q the maximum charge stored, time sine of Omega t plus Phi, where Phi is the phase angle. Now, we're starting with the capacitor maximally charged, this is something I've already addressed, but the phase angle is 0, if the beginning, sorry, if in the beginning the capacitor is maximally charged and a very quick way to see that is if you plug 0, and for t, then all you're left with this sine of Phi, right? That term Omega C goes away. So, sine of Phi is what you have. Initially, if the capacitor is fully charged the current is 0, so the current equals 0, what is Phi half to equal? It also has to equal 0, okay? So, this equation simply becomes Omega Q sine of Omega t, we know what Q is, right? 50 millicoulombs, but we do need to find Omega, for an LC circuit Omega is 1 over the square root of L times C, which is 1 over the square root of 0.1 times 15 times 10 to the negative 9, right? That's what Nano is 10 to the negative 9, plugging all of this into a calculator you are going to get 25,820 radians per second as the angular frequency. Now we can continue. So, at a particular time given, the current is going to be 25,820 millicoulombs toward 0.05 coulombs, sine of 25,820 and the time is 0.1 seconds, okay? Plugging all that in, we get negative 490 Amps, alright? Really quickly, what does the negative sine mean? It just indicates the direction that the current is going, you can ignore it, also in magnetic energy, that value is squared, so the negative time goes away anyway, okay? The magnetic energy stored is 1/2 Li squared, which is going to be 1/2 times at 0.1 right? The inductors at 0.1 Henry inductor time, the current 490, I dropped the negative just because it's going to cancel since I'm squaring and all of this equals 12,005 joules or if you want 12 kilojoules, either one of those answers is correct, okay? So, that's, how much energy is stored, how much magnetic energy is stored on the inductor after 0.1 seconds, alright? Now, let's look at what the maximum current is. The maximum current is going to be simply the maximum value that this function can take, alright? The maximum value we can take is when sine is 1, because sine canÕt be bigger than 1. So, iMAX is just Omega Q, which is 25,820 times 0.05, which is 50 millicoulombs and that equals 1,291 and, alright? That wraps of our discussion on how the energy changes within an LC circuit. Thanks for watching guys.

Practice: Let’s say an LC circuit begins with the capacitors carrying a maximum charge of 10 mC. After the capacitor has lost half of its charge, what is the current in the circuit if L = 0.01 H and C = 50 mF? If during the time for the capacitor to lose half its charge, resistance within the circuit dissipated 0.2 mJ, what then would the current in the circuit be?

A charged capacitor with C = 8.00 x 10-9 F is connected to an inductor with L = 5.00 H. The resistance of the circuit can be neglected. At the instant when the current in the inductor is increasing at a rate of 70.0 A/s, what is the charge on the capacitor?

A charged capacitor with C = 4.00 x 10 -12 F and initial charge Q = 8.00 x 10 -6 C is connected to an inductor that has L = 4.0 H. During the subsequent current oscillations, what is the maximum current in the inductor?
A) 2.0 A
B) 4.0 A
C) 6.0 A
D) 8.0 A
E) None of the above answers

Which LC circuit shown has the largest period of oscillation?

The circuit on the left has an oscillation frequency of 400 Hz. The frequency of the circuit on the right is?
a. 200 Hz
b. 282.8 Hz
c. 400 Hz
d. 565.7 Hz
e. 800 Hz

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