 Hi, I'm Zor. Welcome to Unizor Education. I would like to continue talking about energy of electromagnetic field and in this particular case it will be magnetic component. The previous lecture was about electric component and we have derived the formula and this lecture is about magnetic component. This lecture is part of the course called physics 14's presented on unizor.com website. I do suggest you to watch this lecture from the website rather than from, let's say, YouTube and somewhere else because on the website it's part of the course, which means there is a menu, there is a sequence, logical dependencies between the lectures and like today, for instance, lecture is definitely based on whatever I was talking about before. Now there is also a math 14's course on the same website, which is kind of prerequisite because you have to know math. I'm using cold time, the calculus, the vector, algebra, etc. Okay, so let's go to magnetic field energy. Now, if you remember from the previous lecture, I have derived the energy density of electric field based on a local characteristic of the field, which is intensity of the field. Sometimes it's called just field. I prefer to call it field intensity. Now, in this lecture I will do exactly the same. I will derive the formula for density of the energy of magnetic field based on its intensity, magnetic field intensity. Why it's important to mention this word local? Because we are talking about not energy as an entire amount of energy in some big space, maybe infinite space. That's kind of difficult. So we are talking about density, energy density, which means basically infinitesimal piece of that field, volume of that field, space where the field actually exists. What's the, well, if it's an infinitesimal, you can say average amount of energy per unit of volume. This is something like this. So it's always beneficial to deal with this because obviously everything is changing. The field is most cases variable. So to talk about the total energy actually is kind of difficult. So we prefer to have it at energy density at particular point in space, at particular moment in time. So whenever everything is changing, it's okay. But basically this density function is a function of time and point in space. And that's why to relate it to local characteristic, which is intensity of the field exactly at that point, at that moment in time, makes a lot of sense. Okay. That's preamble. Now how can we do it? Again, back to the previous lecture, I was using a capacitor as a model of some kind of a device where inside it, the electric field is relatively uniform. Again, everything in physics is approximate. So if plates are close enough of this capacitor and they are large enough as far as the area is concerned, then inside more or less we can just assume that the field is uniform. Same thing with magnetic field. We would like to come up with a device where the field is uniform. Then we will talk about the entire energy which we have spent to create that field in that particular volume of space. And since it's uniform, I can divide it by volume and that's my density. And that density would be expressed in, obviously, the local characteristic which is field intensity. Okay. So what is the device which can give us the uniform magnetic field? Well, in absolute terms there is none, but approximately we can think about one. Now, if you refer back to electromagnetism part of this course, I was talking about solenoids. Remember the solenoid is this amount of wire which looks like this. Now, I was talking about infinitely long solenoid because infinitely long solenoid, in theory, can give you really a true uniform magnetic field inside if there is an electric current going through it. So we need a battery and if the battery goes through this, let's say voltage is R, resistance is voltage is V, resistance is R, then there is a current which is equal to, this is ohm's law, right? That everybody knows. So if this solenoid is infinite, then I did calculate a very important formula that magnetic field inside has intensity V which is equal to mu times n times i, where i is obviously the electric current and is density of the wire loops. So how many loops per unit of lengths? And mu is magnetic, how is it called? Permeability. I always mix together permittivity for electric and permeability for magnetic. Crazy words, but in any case that's what it is. Of the inside, whatever the inside of that solenoid is, because there might be something like ferromagnetic core and in this case it will be a very magnetically capable device or it can be vacuum, etc. So whatever is inside, it's magnetic characteristic of the medium inside. And obviously everything depends on this particular qualities of this medium. The magnetic field, the intensity of the magnetic field depends on what's inside, not only on current and density of the wire loops. Now obviously if current is changing, intensity is obviously changing as well. As it depends on T. So that's the preamble. So this is something which has been covered before. If you don't remember this formula and to tell you the truth I didn't remember this formula, I went back to my lectures. So this describes the uniform magnetic field inside. So now let's say I have a switch here. So there is no current in the beginning, there is no magnetic field because there is no current. Now I flip the switch. Well, let's just wait a little bit. After I wait a little bit, my current will be D divided by R. That's kind of a maximum current which can be achieved, but it was zero in the beginning. So which means that during this time I was waiting, my current was increasing from zero to this value, right? Now that's a very interesting process. Now we have a variable I. During this very short period of time while the current was rising from zero to this maximum value, my current was changing, obviously. Now changing current, changing magnetic field inside the solenoid. What happens if magnetic field is changing? Well, again, back to previous lectures, the Faraday's law, changing magnetic field generates electric field. That's how the whole electromagnetic field is propagating, right? So changing magnetic field generates EMF. And this EMF works against increasing the current. It's against the voltage whatever we are supplying. Again, that's something which has been previously discussed with Faraday's law, etc. Now that's very important because we have to raise the electric current to this value. We have to overcome resistance of the rising magnetic field. So magnetic field is rising, but it resists. Which means it slows down basically the increase of the current from zero to whatever the maximum is. So you remember this, it's very important whenever you're turning on and turning off the switch, the electric voltage can actually be very dangerous sometimes. That's why, especially when you are cutting down electricity from some maximum to zero, because at that particular time the current goes down, which means the electromotive force tries to increase it. And that might be dangerous for certain devices. It was all covered in the electromagnetism part of this course. So we have to overcome the resistance of the solenoid, so to speak. It doesn't want this i to increase as fast as we would like, not momentarily, of course. So what kind of work we spend to overcome the resistance of this particular electromotive force generated by the solenoid? Well, this is the work which we have to spend and it should go somewhere. Where will it go? Well, it will go to a potential energy of magnetic field. So when everything is finished, we have spent this energy to rise the electric current to this value. And where this energy goes, we have the conservation of energy law, right? So it goes to a potential energy of the magnetic field. Same thing as before in the previous lecture, whenever we are rising the voltage on the capacitor again, there is some resistance which we have to overcome and eventually it goes into a potential energy of electric field between the plates of the capacitor. And here this energy which we are spending to overcome goes to a potential energy of the magnetic field. So what is exactly this particular energy? Okay, what we know about the EMF electromotive force generated against increasing current is, again, that's the farthest law. This u, this electromotive force, I think I used the letter u, yes. It's equal to minus d phi of t, obviously, by dg, where phi is the magnetic flux. So whenever magnetic flux which goes through this is changing, so b is changing because i is changing, b is changing. That's why magnetic flux is changing and magnetic flux rate of change generates my electromotive force which works against, this is obviously the function of t. And minus means that it goes against it, so that's very important. Now you remember that magnetic flux is basically a product of magnetic force and the area of this loop of the solenoid. So for one loop is this, but we have more than one loop, right? So for one loop, if I know the b, so v1 of t is equal to b times area, this is area of the circle of the loop, of the wire loop, one loop. Now, if we have many loops here, each one is generated with this piece. So basically I have to put u n of t is equal to minus d phi phi n of t by dg. And what is phi n of t is phi 1 of t times n, where n is number of loops. So far so good. So one loop generates this emf and n loops generate this emf, which goes against, which we have to overcome to drive our current, electric current to this value. Okay, so this is something which we have to basically spend extra. And what I would like to know is how much work we have to really spend on this. Well, work is in electricity, the power is voltage times current, right? And work, obviously, you have to multiply it by time. So basically, let's put it this way. Work is equal to u times i times t. That's a general formula. Again, back to electromagnetism if you don't remember. So this is the work which battery, for instance, does or whatever. Okay, so let's just calculate it in this particular case. So this is the voltage which we have to spend extra. Current, I kind of know. So if I will multiply them, I will have the work by the unit of time, right? Which is called power, right? Power is equal to dw by dt. It's a rate of, rate of spending some work. Okay, so let's just do this. I mean, it's kind of a simple procedure. I don't need phi 1 anymore. So phi n of t is equal to n times b, which is this, times the area, right? So n, because n loops in my solenoid, mu is permeability of the medium inside, n is density of the loops per unit of lengths, i is current, and a is area loop. So I don't need this anymore. Now let's assume our solenoid has lengths l, in which case n is equal to l times n, right? l is lengths, n is density, which is number of wire loops per unit of time. So that's why the total number is this. So I will put equal to mu l n square i, t, and a. Okay, I substitute it instead of n, l times lower case n. So that's my formula for phi. Okay, now whenever I want to find u, that's electromotive force, which basically prevents us from its dFn by dt, right? Now this is constant, constant, constant. So I will put mu. Now l times a, this is actually a volume. Remember, we need a volume of the space inside, if we calculate the total amount of energy, to divide it by, to get the density. So I'll put the volume, I'll put wall. So it's distinguished from the voltage. Okay, n square and di by dt. So that's u. Okay, so the power, which is power is u times i, I have to multiply it by i. So I will have basically dw by dt, which is the power. It's multiplication of this electromotive force by i. So it's mu times volume times n square times i and times di dt. So that's the very important formula from which I will get the work. Well, obviously I can get rid of this and I can integrate this from zero to, let's call it i max, to i max. To find out the total amount of energy I have to spend. Now this is the amount of energy to basically rise it from i to i plus plus di. The total amount is, you have to integrate it to get all the way up to i max. Okay, that's simple. So my total amount of energy, total amount of energy is equal to, well this is a simple integral. I mean I have to really, I can disregard the t because everything is differential of t, right? And everything is differential of i, so everything depends on i right now. i is no new variable, substituted variable. So it's integral of i di is one half i square, right? So I have one square mu volume n square and i square max, yeah, one half is already there, right? That's this integral. I mean obviously there is a new term, Leibniz. You have to have indefinite integral, indefinite integral from this, forget about these constants, from this is one half i square in the limits from zero to i max. So that's what you have with i max and for zero it will be zero. So that's the answer from which density of the magnetic energy would be this work which I have spent divided by volume which is one half mu n square i max. Okay, so there is only one step left. I will use this formula. Instead of i max I will put B max, so the maximum magnetic field intensity which we have achieved by connecting this thing. So B square would be what, mu square n square and i square. So the difference between this and this is one half mu B square max, right? B square max is mu square n square i square and there is only one extra mu here so I divided it by mu. And this is my total formula. Now if you remember for electricity energy density was one half epsilon e max. Very similar. So the total energy is there some. If it's an electromagnetic field, again don't forget that this is a local characteristic because I took something which has a uniform magnetic field. And divided by volume so it's per unit of volume. It's a local characteristic. If this is a very small one then it would be almost down to a point. Obviously the whole logic of being a small one contradicts to something which I started with. I started with the formula for magnetic field intensity which is true for infinitely long solenoid. But look that's what physicists do. They approximate to something to the level, mathematicians would prefer not to. But look whatever it is it is, this is the contemporary kind of, no not contemporary. This is the end of 19th century approach to this particular problem. Okay so one more thing. We are spending energy and where is it going? It goes to a potential energy of magnetic field in this particular case. Now what happens if this is an oscillating electromagnetic field? We have generated it once. Let's say sun. It produces a lot of electromagnetic field oscillations around it including the visible oscillations which is the light. So what's going on with energy? Well very simple. Sun generates energy because electrons are moving there. That creates a variable electric field. And it creates not only just a variable electric field. The energy of initial energy which sun spends goes to potential energy of electric field it generates. But since it's variable it generates magnetic field. To do that electric field needs to spend that energy. So it transfers the potential energy which it accumulated from the sun to the next in-chain magnetic field. So energy is transferred from electric field to magnetic field. Now what happens next? Well magnetic field accumulates while it's being created. It accumulates the potential energy. And when the accumulation is finished electric energy which was before lost basically. So the wave is no longer until sun will generate new one. But if sun doesn't generate anything else, if sun goes down then that's it. I mean this energy like a wave, one wave goes and there is no subsequent waves. Okay so let's talk about one wave. So one wave electric field goes potential energy it transfers to magnetic field. Then magnetic field is also variable and that's why it creates the electric field and transfers the potential energy further. And then again and again this exchange between electric and magnetic field. That's how energy is moving along the ray of oscillations of the electromagnetic field. If sun is still functioning then the new wave and new wave and new wave that's how it goes all the time. Well that's it. I suggest you to read the notes for this lecture. They are arranged in a little bit better kind of geometry and real estate than I have on this white board. And again don't forget that there are certain references in the text to earlier electromagnetic for instance electromagnetism for instance lectures. I do suggest you to read all the referenced lectures again to refresh your memory. Other than that that's it. Thank you very much and good luck.