 Hi, I'm Zor. Welcome to Unizor Education. We will continue talking about electromagnetic induction, but in this case about some construction which is much more practical than the one which we were using in the previous lecture to generate electricity. Now, this lecture is part of the course called Physics for Teens, presented on Unizor.com. I do suggest you to watch the lecture from the website. You have to go to Physics for Teens course from the main page of the site and then choose the electromagnetism under the Physics for Teens and choose the electromagnetic induction topic on the next screen. The whole thing in electromagnetic induction is based on the Lorentz force. Let me just remind you a couple of things which we were talking about in the previous lectures, but I think it's very important to understand the fundamentals of this because everything else can be logically derived from this one fundamental fact. So the fundamental fact I'm talking about is if you have a magnetic field which is directed perpendicular to the board towards, behind the board, so we are looking at the end of the magnetic field lines. So these are magnetic field lines which go inside the board. And there is a charged particle. Let's use the electron in this case. So this is an electron, minus. So it goes this way. Now, as soon as it goes into the space where this particular kind we are considering uniform magnetic field is, it deviates from the trajectory this way. This is the manifestation of the Lorentz force. Now everything else, as I was saying, just follows from this. Now next experiment was with the rod. So instead of this, consider we have a rod, metal rod, which is moving the same this way. What happens in this particular case? Well, in metals, atoms are more or less stable at their position because of interatomic forces, links, whatever. Electrons are rotating on certain orbits around nuclei in each atom. Now there are so-called free electrons, like one or two electrons from the outermost orbit around each atom. They're relatively free, which means they can jump from one atom to another. They're relatively far from the nucleus, so the connection between the positive nucleus and the negative electron is a little weaker. And then electrons are basically very active particles and they can jump from one atom to another, basically acting like free agents. The whole body of the metal remains electrically neutral, but there is some kind of a movement of electrons in between. So these free electrons are very easy to basically bump from wherever they are if there is some kind of a force. Now, as this particular rod is moving across the magnetic field lines, it pulls all the free electrons, right? So each electron, each free electron inside this rod is pulled towards this trajectory. Now, you remember what happened with electron if it moves this way. There is a force, Lorentz force, which moves it vertically down on this picture, right? Which means that these electrons will be forced towards the bottom of this rod. So as the rod moves, and that's very important, it's moving, so electrons are moving. And as they are moving, there is a force which pushes them down, so each electron has this orbit. So these electrons are connecting, are gathering at the bottom of this rod. Now, are all of them gathering there? No, obviously not, because they are repelling each other. So there is certain density, if you wish, of the free electrons, which are accumulated on this end, and therefore there is a lack of electrons on the opposite side, and that what makes the voltage between these two points. Now, the electrons, as I was saying, are repelling each other. So if we are moving slowly, the Lorentz force is not as strong, which pushes the electrons down. So the density of the electrons is not as big, so the voltage is not as big. But the faster we are moving the rod this direction, the stronger Lorentz force pushes the electrons down, and that's why they are more dense, and their difference in potential between top and bottom is greater, so there is a greater voltage. Okay, so we understand that. Now, if you are moving with a certain constant speed, well, there is certain density of the electrons which can be accumulated at the bottom part of this rod, and there is certain fixed voltage. It will not go any further because repelling force from the electrons is neutralized by the Lorentz force which pushes them down. And there is this dynamic equilibrium, there is this maximum density at this particular speed of the rod. Great. Now, what happens if we are connecting in a circuit? Well, there is a difference in potential here, right? So electrons are here, minus. This is positive. So what happens? Well, obviously there is an electric current in this case. Current goes from plus to minus, so current goes this way, and certain number of atoms or absence, yes, atoms with absent electrons are supposed to move this direction. Now, what does it actually mean that the positive is moving here? It basically means that the free electrons from here will replace these guys, which seems like they are moving basically, electrons are here and lack of electrons is moving here. So gradually lack of electrons is reaching this point and the electrons which are accumulated here will replace those holes which are here, which means there is less density and the Lorentz force is constant, right? These are rails. You remember you taught me about the experiment. So Lorentz force is constant, density is less, so certain new electrons are pushed by the Lorentz force down to basically replace whatever has been lost here. So electrons goes this way, the electric current goes opposite direction, and so we do have a loop which is the electricity between this electric current generated by moving the rod is constant with a constant speed. Now, as far as the formulas are concerned, now if you remember the Lorentz force, if there is one particular particle goes this way, it's Q times V times vector product B. So if Q is negative, then the whole thing becomes negative. Q is positive, it's positive, so that's why it's vector quality. Okay, so this is the Lorentz force on the charge Q. We know that from this we have derived another formula in this particular case. We have derived the formula that the... Well, if you consider I to be the current in this particular case and U to be the voltage between these things, which is maintained constant, then the I times U is the power, if you remember, power of the current running, and the work during some time dt would be this. That's the work which is performed by this. Now, at the same time, we are moving our rod with the constant speed to the right. Now, what happens in this case? Since we have a current, we have another manifestation of the Lorentz force. Remember that if there is a current running in the rod, then the rod itself is pushed in one particular direction. So if this rod is moving this way, the current moving this way, and this is basically exactly the same thing. When the charge is moving in the field, there is a force which moves it towards a direction which is perpendicular to its speed and the magnetic field lines. Same as in the beginning, each electron as it moves here is pushed down. Now, since we have this current, we have electrons are moving this direction, right? And if electrons are moving this direction, so it's a combination. Electrons on one hand are moving this way because the rod pushes them. On another hand, electrons are pushed down the road because the Lorentz force. But as they are pushed down the road, there is another magnetic effect of Lorentz force in the magnetic field. As they are going down, then there is a force which is supposed to be perpendicular to this, which is their movement, and direction of the magnetic field lines, which is this. So we are moving to the right with a constant speed by overcoming another manifestation of the Lorentz force. So Lorentz force moves our electrons down, but at the same time this causes another Lorentz force to move against our rod. And to move it with a constant speed, we have to really spend a certain amount of work to overcome this force, right? The same force. Okay, so to overcome this force, we have to move it with a certain constant speed, and the force, therefore, is constant. So if we will multiply this by the length of the trajectory of the rod, we will get also work. So this is a work by electric current, which is performed. Let's say it's converted into heat here in a resistor. On another hand, we have to exhort exactly the same thing, the same amount of work to move it with a constant speed to the right. Well, the s is obviously speed times time, right? Now, the force which is acting on the current is equal to, as we know, current times L times B, right? That's something which we have covered in one of the previous lectures. So if there is a rod with a current I, it has lengths L, and the B is magnetic field, this is the force which is acting on this piece of wire, this metal rod or whatever. From which we see this is cancelled out, and we have is equal to L times B times B, which is equal to, what is V? Well, if this line is, let's say, B and this is L, right? Now, V is actually V is equal to dB for dt, right? Because B is lengthening, because we are moving the rod, B is lengthening, and its derivative is the speed, right? And so we have equals to L times dB by dt times B. Now, what is this? This is, this is dB. This is the error, so L times dB is this error. It's increment of the differential of the increment of the error, and so we can say that this is dA by dt. This is increment of error times B. Well, B is a constant, right? A times B is called flux, magnetic flux, and that's what we have ended actually, the previous lecture. So this is equal to d phi for dt. Well, usually they put minus sign here because it's directed against the movement. But as far as the absolute layer of the scalar is, the derivative of the flux is equal to voltage or electromotive force. Now, this is something which is a repetition, basically, of whatever was before, and I was trying to come up with something which is more practical as far as generating the electricity. Because to generate the electricity using the moving rod is kind of technically difficult. How long can you move it, right? There is a better way, and that's what we will very briefly talk about right now. So what's important here is the very last formula that the EMF or voltage is equal to minus d phi by dt, a change of magnetic flux which goes through this frame. You have this frame, and you have magnetic field lines going through it, and as we are increasing this distance, the flux is increasing. So we can do exactly the same type of logical manipulations to find out the following. If you have magnetic field lines, let's put them this way, and the frame, the wire frame, rotating along the vertical axis on this particular drawing. What happens? These two lines are always crossing the magnetic field lines. So because they are crossing, it's exactly the same as before we had this picture, and we had the rod which moves this way across. So magnetic field lines this way, rod goes this way, and that's why there is a generated electricity. Same thing here, just a different view. Magnetic field lines this way, but from this position if we start rotating, so this rod goes behind the board, and this rod goes in front of the board. So they are moving this way, which means they are actually crossing the magnetic field lines. They are crossing the magnetic field lines, which means there must be electricity generated in this particular rod and in this particular rod. Because of the same consideration we were using before, we were talking about Lorentz force, et cetera. Electrons are pushing this way, that's why Lorentz force pushes the electrons vertically, down or up depending on the direction of rotation. So this is the rotation, and that's why there is an electric current. What's important is that this part of the wire loop is going into one direction. This one goes to another direction, which means that electricity should be generated, electric current should be generated in opposite directions. So let's consider this one goes this way, then this one will be this way, which means we have a loop. Now, if we don't have any connection here, then what happens is basically the certain amount of electrons will be generated and stop generated, because they will repel each other, as this thing moves this way and this thing moves this way. What happens if we will continue the rotation? Well, if before this particular thing moves towards us, that's one direction of crossing magnetic field lines. After it moves 180 degrees, it will go back behind the board, which means an opposite direction of crossing the magnetic field lines. And obviously, all depends on how we cross magnetic field lines, because that's the direction of the current changes. So as soon as we do, as soon as we basically cross the middle point, electrons, which used to be accumulated in one end, will start basically flowing back to another end. So the electrons will just move back and forth, back and forth on rotating wire frame. Now, if we have some kind of electricity here, some kind of a consumer of electricity, then as soon as they will accumulate here, they will flow this way, and that will be the direction of the flow. But then when we change, the direction of the flow will be different. So the electricity will go all the way this way and then all the way this way. So it's an alternate current. That's how alternate current is generated. Now, let's talk about this quantitatively. I would actually like to come up with the same formula more or less. And that's quite frankly not very difficult at all, because we can always consider only the moment from time t to time t plus dt, infinitesimal increment. To make it a little bit more understandable, let me just do the top view. So how does the sink look from the top? Well, magnetic field lines are exactly the same way, but as we are looking at the top, let me just put the... If this is my frame, this is A, this is B. If I look from the top, and these are magnetic field lines, the magnetic field lines are the same, but from the top I see only line B, right? So I have only B line, and as it rotates, my B line basically rotates at certain angle phi as a function of t. This is my B, from this point to this point. And my A and its opposite line go behind the board, right? So all this construction is behind the board. So we can definitely say that if this thing is rotating with a constant angular speed omega, then this particular point, P, has coordinates. This is B, so this is B over 2, right? B over 2 times x is equal to cosine. Angle phi, angle phi is omega t. And y-coordinate, this is x, and this is y. And the y-coordinate is equal to B over 2 sin omega t, right? That's simple. Now what we are actually paying attention to is only the vertical component, only y component, because x component is not against, crossing the magnetic field lines. It's only the y component is crossing. So we're paying attention to this, to y component. This is a point where my, this or this side of this wire frame are behind. So this one is behind P, and this one is behind Q. So if we know this, we know the speed. So V as a function of t, as a function of time, is the derivative of this, right? What's the derivative? It's B over 2, derivative of sin is a cosine, and then there is inner function omega, so it will be cosine omega t. So we have a speed. Now if you have a speed, remember, I times L times B, which is I times V times t, whatever times B. So during time dt, during time dt, what happens? We have to overcome the force, the Lorentz force, which acts on this particular rod, which is behind point P, which acts against its movement. So we are moving my point along the circle, but again it's only the vertical component. How it moves across the magnetic field lines is important. And we have the speed as it goes in the vertical direction. So now we can also assume that there is some kind of a current generated by this. We will be able to calculate the amount of work, which we need, remember? So in this particular case, and if we will multiply F times, we don't really need this. If we will multiply F by the certain lengths during which we are moving this thing, it would be I times L times B times dS. Now what is dS? dS is d times dT. Okay? So that's what we have. And on the other hand, this is I times U times dT. That was our main formula. Now, in this case, the L is the length of this rod, which is A. So we have I of T, because obviously the current is changing since we are changing the speed. L would be A, B would be as it is, V would be over 2 omega cos omega T and dT. And that should be equal to I of T times U of T times dT. So this is the power of the electric current, which is generated times time. That's the work. And obviously we have the same thing here. dT also goes here. And what do we have now? We have that our voltage, which depends on the time now obviously, is equal to A times B over 2 times B times omega times cos omega T. So this is the voltage generated by one rod behind this one. Another one does exactly the same thing. And so one is generated a certain amount of voltage in the other and they are connected sequentially to each other, so voltage is added together. So the total voltage will be equal to double, basically. It's A B times B times omega times cos omega T. This is the total voltage generated by the wire. By the wire frame. Now, does it look like this? Well, yes. And it's very actually simple to make sure. What is the area of this? The area of this is A times B. But we are not talking about total area of the wire frame. We are talking about only the area which is crossed by magnetic field lines. So we need actually only this area, the projection, if you wish, of the area onto this vertical plane. Which means it would be what? If this is completely this way, the projection is zero, and as we are going, this thing is increasing as a sign of this angle, phi of T. So basically A times B times sin phi is the area which is visible from this direction from this point to this point. And so this is basically a flux. If you multiply it by B, right? By the magnetic, intensity of magnetic field. And the derivative of this is equal to the derivative of sin of, this is the function of T, right? So it would be A times B times phi of T is omega T. So derivative from the cosine is, from sin is a cosine. And in the function omega T, so it's omega times cosine omega T times B, which is exactly this one. So we can say that U is equal to, well in this case U is electromotive force, which depends on the time, is equal to, I will use this minus again to signify that it's against the movement. So as we are moving our wireframe, as we are rotating it, there is a resistance which is caused by Lorentz force basically. But as a result of our movement, our doing some work actually, there is electric current which is generated, which can in turn produce its own power and work. And that's what we are using on the consumer side. Obviously if we are spending some energy by rotating this energy is converted into the energy of the electric current and that's what we are consuming. Well, basically we are coming up with exactly the same, this is T, magnetic flux. So magnetic flux, change of magnetic flux, speed of change if you wish. Well, the first derivative, speed of change, rate of change, whatever. This is the source of electromotive force, voltage which is generated and can be used. Okay, that's it for today. Thank you very much. I do suggest you to read the notes for this lecture. And good luck.