 Hi, I'm Zor. Welcome to Unisor Education. We have finished talking about mechanical oscillations. And it's a very interesting, actually, part of the course. Now we will continue talking about oscillations, but not quite mechanical. However, there is a very strong resemblance. So today we will talk about oscillations of electric current in the circuit. Well, we have already started something like this when we were talking about alternating current, if you remember. Basically, whenever you are rotating a wireframe in the magnetic field, for example, in a constant magnetic field, if you are rotating this wireframe, you will generate, because of magnetic induction, you will generate an alternating current, which is basically sinusoidally changed within this wireframe. Well, that's how motors actually are done. Now, in some way, using the terminology of the mechanical oscillations, we can say that these oscillations of electric current are induced, or using this terminology which we used in mechanical oscillation, it's forced oscillations. Today we will talk about oscillations which are basically resemble alternating current without external force. Again, the similarity with mechanical oscillations, if you have, let's say, an object in the spring, well, first you initially, let's say, displace it, you stretch the spring, and then you let it go, and if there is no resistance, then it will just indefinitely oscillate back and forth. Here, we will talk about something very similar in this lecture, but with electric current. Okay, this lecture is part of the course. Physics 14 is presented in unison.com. I suggest you to watch this lecture from the website, because it's part of the course. Basically, everything is logically related. All lectures are combined in some kind of a menu. Every lecture has very detailed notes, which basically, like your textbook, there are problem solving and exams on the web page. The web page is completely free. There are no advertising, just pure knowledge. Okay, and there is a prerequisite course called mass routines on the same website, and mass is a must. You have to know mass to study physics. Okay, so we are talking about some kind of forced, no, in this case, it's free oscillations of electric current. How can we basically obtain it? What is the electric equivalent of spring in mechanical oscillations? Okay, here is what I suggest as a circuit. First, let's make a circuit which contains inductor and capacitor. Now, no sources of electricity yet. Now, capacitor is not charged, right? So we have to really charge the capacitor to start something. So how can we arrange it? Very simply, we will do it this way. We will have another circuit here with battery and a switch here, and we will switch in this position. In this position, this part of the circuit is basically not involved, because there is no connection here. So this is the electric circuit which will have electric current generated by battery. So the battery, what will do the battery if you connect the capacitor? Well, the plus will go to plus, the minus will go to minus, and our capacitor is charged. That's it. Very, very fast charging. Nothing more happens. Nothing circulates. Just charge goes from plus of the battery to plus of the capacitor's plate, and the minus to the minus plate. When this is done, we switch this switch, switch the switch into this position. Now, the battery is gone. Nothing is happening on this side. And now we have this circuit. So this doesn't exist anymore, basically. So what happens? Well, we have a closed circuit, right? So plus and minus electrons from minus go to plus. Well, by definition, we are saying that electric current goes into opposite direction, unfortunately, but in any case, what happens, happens. There is electric current which goes. Okay, now let's just think about this way. First, your capacitor is charged at maximum. In theory, let's forget about this inductor, which is just a straight line, straight wire. Well, in theory, the plus and minus will start circulating electric current. Electrons will go from minus to plus. And that's it. The capacitor will discharge, basically, because the plates are connected. It's shorted, basically. So the charge will basically disappear. The deficiency of electrons on the plus and access of electrons on the minus will neutralize each other. And it will be the same number of neutral kind of, neutral number of electrons in both cases. Capacitor will discharge, and that's it. But inductor is a very important thing in this particular case. So let's just think about how the electric current will go if there is no inductor. Well, first electrons are at maximum difference between plus and minus, right? Maximum voltage. And obviously, the electric current will be at maximum. Then, as the number of electrons is changing, the flow of electrons will slow down. Now, let's just reasonably, very reasonably assume, and we will prove that this is the correct assumption. Let's reasonably assume that our current in this particular case will change as cosine of some angle. So a negative will be angular speed and t is time. And this is some kind of a maximum. At t is equal to zero, that's the maximum. And then the cosine of zero is one, right? And then as the t goes, this thing is diminishing. Well, because the difference of potential is diminishing, obviously. So it's a reasonable assumption. And again, I will prove that this is the correct assumption. But what happens in this case? Let's just think about it. What happens with the speed of change? Not the change itself, but the speed of change, which is the first derivative. Well, the first derivative is minus i zero times omega times sine of omega t. Now, that's very important. Cosine goes like this, right? From maximum to zero. The sine, minus sine in this case, is something like this. So this is i zero, and this is i zero times omega. So in the beginning, the speed of change, which is basically an angle of the tangential line, is zero. So in the very beginning, when our voltage is at maximum, the speed of change of the electric current is at minimum. And as soon as our voltage goes down, our speed of change is increasing by absolute value. So the current is slower, is changing slower in the beginning when this is at maximum. And then when we are approaching zero, the current goes to zero fast. So that's very important. So the speed is increasing as the voltage is, the speed is increasing as the voltage is decreasing. Okay? Okay. Let's just remember this, because it will be very important in the future. Now I made this assumption. So let's just make a little bit more theoretical research as far as we are concerned here. Assumption is assumption, but we have to really prove it. And here is what we can do. Let's assume that we have inductor has inductance l, capacitor has capacitance c. And I will go back to definition of these capacitors, capacitance and then inductance, because it will be very, very necessary for our calculations. So let's assume also that the voltage between these two is changing with time. So it's v and index c. The voltage between these two would be l of t. What's interesting right now is that their sum should be equal to zero, because there is no source of electricities. I mean if there was a source here, then the voltage between these would be some of the voltages there. Voltage drops, right? From, if you have a source here, it will be, the difference between these will be the sum of the drops of electricity, of voltage, of electrical voltage around the, around the whole circuit. If there is no source of electricity and the current is still running, so the sum is supposed to be equal to zero. So that's something which is basically known as Kirchhoff's law, but in this particular case doesn't really matter how it's called. What's important is if there is a circuit and there is no source of electricity, well if there is no current, obviously all the voltages on every piece will be equal to zero. But if there is some current and we know that there is a current because this thing is discharging, right? So the current goes. So at any given moment of time, the sum of the voltage drops between these two things which participate in the circuit must be equal to zero. Okay, that's good. Now let's talk about definition of capacitance and inductance, capacitance first. Now we know that from purely practical experimental facts, if you charge the capacitor with certain electric charge, well put some number of electrons on one of the plates, let's say, and reduce the number of electrons on another, for instance, using the battery. So this is the total amount of charge in coulombs, and this is the voltage observed between the plates. So experimentally it was basically established that there ratio is a constant thing which is a characteristic of the capacitor itself. And that's what it's called capacitance of the capacitor. So we increase the number of electrons by the factor of two, and that means the voltage will increase by the factor of two. Okay, remember this. From this we can actually have the vc of t is equal to q of t divided by c. Okay, now let's talk about inductance. So we know first that whenever you have electric current there is a magnetic field around it. Okay, we know that. Again, that's experimental, etc. Now we also know that if you have something like this inductor, the magnetic field which is basically produced by moving electrons, it has certain characteristic which is called magnetic field flux. Okay, so magnetic field flux. And again, it depends on the current. Okay, magnetic field flux is equal to l times i of t. So if this is the current, the magnetic flux will be proportional to the current. And l is this coefficient of proportionality. And again, it's a characteristic, physical characteristic of inductor. Depends on how many loops we have, what kind of a wire this is, etc. etc. But again, the stronger our current, the stronger the magnetic field flux we can observe. But now we have a very interesting thing. If magnetic field is changing, and there is a wire near it, the magnetic changing magnetic field generates the electricity. More precisely, again we were talking about before, it's called self induction basically. So whenever you have a changing current, it results in changing magnetic field flux. And changing magnetic field flux results in changing the voltage. So the voltage which is generated by moving electrons here is actually equal to the derivative of the flux by time. Now this thing was actually discussed in details in the electromagnetism part of this course. So I'm just doing it very briefly. And I refer you to those lectures which explain the details. But in any case, the electromotive force, which is basically the voltage between these two things, will be generated by changing flux. And why flux is changing? Well, because our electric current is changing. Our electric current will change because we are changing the amount of electricity accumulated in the capacitor. So everything is changing. And the change of one is producing change of another. Discharge of the capacitor causes the change of the current. Change of the current is causing the change of magnetic field. Change of magnetic field is causing the electromotive force generated by the inductor. Now what this electromotive force does actually. Again, we were talking about this in the electromagnetism part of this course. It supports actually the current. If current is going down, then the electromotive force will try to extend it and support it so it doesn't go down as fast. Or goes down, yes, goes down not very fast. And what happens after the initial current almost dies when it's really around zero, the speed of the change would be at maximum, as we were talking before. And the voltage would be correspondingly at maximum, because it depends on the speed of the change of the flow of the flux. So here we can actually say that this is equal to L and the derivative of T, right, using this. So what's interesting is we have expression for this and we have expression of this and we have an equation where we can connect them together. So let's just connect them together. The only thing is I would like to extend this thing a little bit, but not here right now. You see the problem is it's kind of difficult to deal with amount of electricity in coulombs. We would like everything to be related to the current. Now what is the current? Current is speed of the change, derivative speed of the change of the charge, right? So how can we bring everything to terms of IT? Well you see if I will differentiate this, I will have vc of T is equal to 1 over ci of T, right? If I differentiate this, this will be first derivative and the first derivative of charge would be my current. Now since we are talking about derivatives, let's have a derivative of this one and vL derivative would be L i of T to the derivatives, right? It was one and I take derivative again so it would be the second derivative. And now I will substitute it to here. Obviously I can differentiate this because I have everything in derivatives and now I have an equation. C is 1 over ci of T plus this derivative L i second derivative of T is equal to 0 or more traditionally fT plus 1 over LC i of T equals to 0. Great, we have an equation. Differential equation from which we can derive i of T, derive the current. Now I would like to refer you back to mechanical oscillations. Remember when the spring and an object and the object was freely oscillating, we had an equation. For free oscillations we had this equation. Again if you don't remember it go to mechanical oscillation topic of this part of the course. Now you see the resemblance? Just different coefficient. Now in this case our solution was where omega is square root of k over m. So again this was a general solution. C1 and C2 are just some constants which are defined by initial conditions, initial displacement of the object and initial speed whenever we are just or let it go after we stretch the spring or we push it in some direction with some speed. Okay, so C1 and C2 are constants defined by initial conditions. Omega is called angular speed, sometimes it's called angular frequency which is not exactly correct, but in any case that's what we have here. And this is exactly the same equation. So the solution of this equation would be i of T is equal to C1 cosine omega T plus C2 sine omega T, where omega is equal to square root of 1 over LC, where L is inductance and C is capacitance of our components. Well from this you can see that i is really sinusoidal. Now if you remember we can always change this into equals g cosine of omega T plus phi. Again very simple trigonometric trick will switch from this to this instead of constants C1 and C2 will have constants d and phi. But in any case this is a cosine in this particular case which means we have sinusoidal oscillations of electric current. And again if there is no resistance basically it will be indefinitely circulating back and forth, back and forth. So in this particular case whenever this charge goes from some maximum to zero the electric current will change from maximum to zero, but the speed of change would be the faster when this closes to zero. So whenever we are almost exhausted in the force of the capacitor which pushes the electrons in some direction our inductor because of self-inducting process will support this dying electric current and will continue it will prolong its existence and it will go even after this thing reaches zero and this thing will reach zero and it will actually charge in reverse. So these electrons first go this direction almost died when it's close to zero but the speed of this dying of electric current would be faster at the end and that means that the supporting force of the inductor will be the biggest the strongest and it will continue and it will charge the another plate with electrons. So the charge will reverse that's what basically this this thing means the charge will be reversed and and then the story will repeat again and again. So this why electrons will go this way and then this way whenever electrons are almost gone from here the continuation of the flow of electrons and it will be deficiency here and exist here it will be supported by the inductor. So inductor and capacitor are actually together are helping this oscillation first inductor sorry for first capacitor starts because there is a difference in electric potential it will force electrons to move but whenever this force is getting weaker the force of inductor which supports the move of the electrons will be stronger and then then the situation will be repeated again and again in opposite direction and that's what and that's what makes our electric current oscillating sinusoidally. I kind of simplified the whole thing I did not really take into account resistance that will be the subject of the next lecture but so far this is called LC circuit and next lecture we will talk about RLC circuits where resistance will be involved as well. Meanwhile I just want to mention that it's very interesting and it's almost like philosophical you see it's completely different arrangement here and mechanical oscillations but oscillations are oscillations and the oscillations are here and there are happening according to well relatively speaking the same formula which is kind of amazing I mean physically they're completely different however there are certain fundamental laws probably of the of the nature which actually brings the whole thing into sinusoidal oscillations in both cases. I think it's kind of amazing that we have exactly the same formula and by the way whenever we will have the resistance introduced if without resistance it resembles the free oscillations of the mechanical oscillations with the resistance it will be damped oscillations because resistance will consume certain amount of energy and these oscillations will be damping it will be less and less by amplitude but we will see it in the next lecture so again very very strong resemblance between oscillations of the electric circuit and oscillations mechanical ones. Okay that's it for today I suggested to read the notes for this lecture on Unisor.com so you go to physics 14 course the part is called waves and inside the waves it's electrical oscillations that's it thank you very much good luck