 Hi, I'm Zor. Welcome to a new Zor Education. We are talking about energy of waves, and we have already considered energy of, let's say, spring or a rope, when you are making waves with a rope. Well, right now we will talk about the energy of electric field. Well, we know that the electric field being basically part of electromagnetic field and it has oscillations. Now, in theory, we will actually consider right now a static electric field. The most simple one seems to be electric field between two plates of a capacitor, because it's a uniform field within it and it has a finite value. Obviously, we can consider some other fields, but this is the simplest one and that's why this is the subject of this particular lecture. This lecture is part of the course. It's called Physics for Teens, presented on Unisor.com, together with a prerequisite course, which is Mass for Teens. Mass is very important for physics. Nothing is basically done in physics without mass, in particular calculus, vector algebra, and some other things. Now, the whole website, all the courses on the website are completely free and I do recommend you to watch this lecture from the menu which is presented on Unisor.com, because it's part of the course, so you know the logical sequence of lectures. So, if you found this lecture, let's say on YouTube, by doing some search, I still recommend you to go back to Unisor.com. Again, it's a free, there are no advertisements, so it's basically the same thing, but it will give you a global picture on the whole course and the place of a particular lecture in the course. Alright, so, back to our today's theme. Now, as I said, the electric field between two plates of a capacitor, well, ideal plates, obviously, is the simplest because it's uniform. Now, another very important thing is that I'm not talking about energy of an entire field. We will be concentrating on energy as a local property of the field, which is energy density. Now, it's very important to concentrate on energy density because even if you consider a static field and you find the dependency between energy density of this field and intensity of the electric field in that particular point, in that particular local point of the field, then we can actually transfer, logically transfer the same result to any other electrical field because it doesn't really matter how the field is created. Even whether it's variable or static, dynamic or static field, if we are concentrating on a particular point in that field and we are talking about energy density in this particular point, then even the results of this particular lecture, which was basically about a uniform field between the plates of a capacitor, these results in this formula can be actually applied to any other. If at the end this formula will not depend on the general shape of the field, whether it's uniform or not, whether it's produced by capacitor or not. But with capacitors it's simpler and that's why I present it in this particular way. So let's say you have a capacitor which we would like to charge. So this is my battery and it has certain capacity, obviously. And this is a capacitor which I would like to charge from the battery, which means, let's say, I have a switch here. So what happens if I close the switch? Well, since my battery is capable of chemical reaction of separating electrons from the nucleus of the atoms, my electrons from one plate will start going, considering this battery has certain power. Now it will go to another plate and there will be a difference in potentials. So one plate will be plus and another will be minus. Now it's a process, so in the beginning we have no difference in potentials, but with certain time our charge on these plates will grow, positive grow and negative grow. So it will be plus Q as some kind of a time-dependent variable and will be minus Q. Okay, so let's consider that during certain time my charge grows from zero to some kind of Q maximum, whatever the maximum. It doesn't really matter what is this maximum. It's some number. Not necessarily the real maximum this particular capacitor can be charged. It all depends on many different factors. So it grows and Q max is something which where we would like to end. Now, obviously whenever we have plus and minus two charged opposite charge plates, there will be a certain voltage between them. So the voltage also will grow from zero to some kind of V max. Alright, now what is voltage? The definition of the voltage is its amount of work, a voltage between two points in electric field, let's say, or between two plates or between two any points within the electric field. The voltage is amount of work which is needed to transfer unit of charge from one place to another. Now if it goes against the field, let's say you have plus and minus and you would like actually to separate plus from plus to go to minus. I mean that means that we have to do the job. Or for instance if you would like a positive charge at some point, a point charge and we have another positive charge and we would like to move it closer. So it means we have to do the job. If it's a negative charge then there is an attraction so the field will go, basically will make this particular work. But in any case there is some work involved. Now in this particular case to separate these two charges, in the beginning it was zero which means it was the same number of electrons and nuclei on the same plate. And now we're separating, we're ripping the electrons from the nuclei from this particular plate and move it to here. That's what battery does. So we have to do some work. Now what actually this work depends on? Well if there is already some kind of a voltage between these, some kind of voltage V, then to bring a unit of charge from here to here the battery should do some work which is actually is amount quantitatively equal to V. So it's V times one Coulomb if you wish. Coulomb is measure of the charge. Now, but V is changing. So what we were doing is we will have an infinitesimal amount of charge and transfer it to this. So what happens then? Well this is amount of work. Now V is actually function of Q. So it's probably better to write it this way. Obviously V is function of how much electricity we have already separated. And this is amount of work differential. Differential amount of work. Infinitesimal amount of work we need to bring infinitesimal amount of charge when already there is a voltage between these two. Now, what I would like to do next is I would like to calculate the total amount of work I have to spend to charge it to Q max or to reach the voltage of V max. How can I do it? Well if I have an infinitesimal amount of work as a function of charge I just have to integrate it from zero to Q max this dW of Q which is equal to integral of zero to Q max of V of Q times dQ. Okay. So Q is accumulated by certain moment in time electricity. The charge between these two, V is voltage. But we know from the theory of capacitors how they are related. V is equal to Q over C. C is capacitance of the capacitor. So the voltage is proportional to the charge between the plates. Now, if you don't remember this, go to the corresponding lecture. I didn't mention it but every lecture on this website has a textual part where everything whatever I'm doing right now is written basically. And there is a direct reference to capacitor lecture in this course. I don't remember exactly the names of the chapter and part of the course etc. It's part of the electromagnetism of this course. So this is the theory of capacitors and we know that there is something if every capacitor has certain characteristic, physical characteristic which is called capacitance. And this is basically a coefficient of proportionality between amount of charge accumulated on the plates and the voltage. Alright, so I'll just use it over there. It's equal to, so I have Q divided by C dQ. Okay, this is a simple integral. And it's equal to 1 half Q square over C. The formula Newton-Leibniz. Again, if you forgot your calculus and you forgot that integral of Q dQ is actually Q square in the limits of integration, you have to go back to math, to calculus and review that particular section. So I substitute the top which is Q max. So the result is 1 half Q max square divided by C minus bottom, bottom is 0, this is 0. Okay, so this is total amount of work, output of W max, which we have to spend to accumulate this particular amount of charge on the plates of the capacitor. Okay, but I was talking about energy density. So this is an energy which is basically accumulated inside the whole electric field between the plates. And again, in ideal case, there is no electric field outside, it's also in between, and there is also an assumption that this field is uniform between the plates. It's not exactly true, but whenever the distance between the plates is relatively small and the area is relatively large, it's basically approximately can be considered as a true statement. So how can I determine the density of the energy which is actually the characteristic of every point inside? Well, if everything is ideal as I said and the field is completely uniform and it doesn't really spread around just only in between the plates, all we have to do is divide this by the volume between the plates. Okay, the volume between the plates is equal to area of the plates times distance between the plates, right? So area is area of the plates, d is distance between the plates. Okay, that's value, fine. What is C? This is capacitance. Now capacitance, and again it goes to the logic presented in that corresponding lecture about capacitors, it can be measured and it's basically proportional to the area. So the greater the area of these plates, the more electricity it can store basically. It's inversely proportional to the distance which means the smaller the distance between the plates the greater electricity can be accumulated because plates are actually attract each other. So electrons are attracted so the greater energy is concentrated in this particular case. What I didn't really write yet is it also depends on what's in between the plates. Vacuum is one thing, glass is another thing, paper is still another thing and they actually affect the capacitance of the capacitor. The more insulated, so to speak, this medium between the plates is the greater amount of electricity you can accumulate without basically having some kind of sparkle between them which basically discharge all the electrons. So we need the greater insulation, ability of the medium, the better and what's the best is, vacuum is the best. Okay, so in any case there is something which is called epsilon here. The epsilon is called permittivity of the medium. How permittivity it is for electrons to go through and the best is the vacuum. Okay, now, let's just substitute these things to this. So let me just skip the suffix max because it doesn't really matter, it depends on where exactly we stop when we charge. Not necessarily to maximum, it can be charged to half of the maximum capacity. It doesn't really matter. So W is one-half Q squared divided by C equals to one-half Q squared divided by C which is epsilon A and G goes to the top. Okay, at the same time it's equal to, again, if V is equal to Q over C, Q equals V times C, right? So it would be V times C, V squared, one-half, V squared, C squared, D divided by epsilon A. And C, again, can be substituted in this. It would be one-half B squared, epsilon squared, A squared divided by D squared, epsilon squared. That's C squared, okay. And then we do have epsilon A and D, right? I think that's it. I probably should have started in the beginning because substitute Q here, let me start here. It would be V squared here. It would be this one. It would be one-half V squared, C squared divided by C which is one-half V squared times C. And now substitute C. It would be one-half V squared and C which is A, epsilon A divided by D. Right. Is it the same thing? A squared, D squared, epsilon squared. Yes, it's the same thing. So this is my formula. Okay, let's write it down somewhere here and then we will talk about density. So this is the total amount of work. W equals one-half V squared, epsilon A divided by D. So this is the total amount of work which we have spent separating the charges between these two plates. Well, that's basically the total amount of potential energy which is concentrated between these plates. I mean, where the energy actually did go. Well, we have to conserve the energy, right? So we have to spend this energy in the battery which means right now it's supposed to be a potential energy between the plates. Okay. Now again, the field within the plates, between the plates, is uniform. So if I will be able to divide this by volume, volume is equal to A times D, I will have the density. But the problem is that this density depends on the voltage between the two plates which is not a local factor. What is the local characteristic of the field? The local characteristic of any electric field is intensity of the field in this particular field, in this particular point, right? Okay. How related are intensity of the field and voltage? Well, let's just think about what is intensity of the field? It's the force which is exhausted by the field to a unit charge. Okay, so this is the force. Now, what is the voltage? Voltage is between two different points. The voltage is amount of work which we have to perform moving a unit charge from one point to another. So again, if you consider this capacitor, the voltage is amount of work which is needed to bring a point, a unit charge from one plate to another, amount of work. Now E is the force which is acting on that particular charge, that particular point, any point inside. And again, I assume the field is uniform, so E is the same between the plates everywhere. So obviously, if I will multiply the force by distance between the plates, that would be the work which is needed to move unit charge from a plate to a plate, which is actually the definition of the voltage. So that's a very important part. This is amount of work. This is the force and this is the distance. Force times distance equals work for a unit charge. Great, so I will do this and I will substitute instead of V, I will substitute E. What will I have then? I will have amount of work is equal to one-half, V square which is E square D square times epsilon times A divided by D is equal to one-half, epsilon E square A times D. D square and D are cancelling each other. And what is this? That's volume. So if I will divide W by volume, I will have one square epsilon E square. So a local characteristic of the field, intensity of the field, which is the force acting in that particular point of the field onto a unit charge, E is the only participating, epsilon is a constant, right? So it's the only participating variable which characterizes a local property of the electric field because this E is a function of every point because it's a vector field. E is vector field. At any point it's a vector force acting on a unit charge. So that's how we have calculated the density of the energy between two plates and expressed it through a local characteristic of the electric field. Now, whether this particular field, in this case it's between the plates, is performed, is created by the capacitor, or maybe the electric field is created by some kind of a point charge and we are on some distance from it. This is an electric field. There is a Coulomb's law which basically guides how it's done. It's basically, again, since everything is dependent, energy density is dependent only on the local characteristic which is intensity of the field in that particular point, it doesn't matter how we created that field with a capacitor or with a point charge or with something else. Now, if it's a vacuum, then it's epsilon zero. If it's some kind of a medium, it's just epsilon which is usually measured as epsilon relative to vacuum. So this is the vacuum and this is a multiplier by which we multiply the vacuum's permittivity to get the particular medium permittivity. But in any case, that's what it is here. And if the field is in the vacuum, electric field is in the vacuum, then this is just epsilon zero. For vacuum, epsilon r relative is equal to one of this. Okay, that's it. I recommend you to read the notes for this lecture. They are basically the same thing which I did right now, but once more kind of a repetition is always helpful. Well, other than that, good luck. Thank you. Bye-bye.