 Let's talk about field energy. How much energy is stored in an electric field and derive an expression for that. And I'll also tell you why this concept truly blows my mind away. Now, because we want to keep the math simple, we'll derive this expression using a familiar setup. So we'll take a charged capacitor. Imagine we have some capacitor which is charged to some voltage. Let's start by asking ourselves, what is the energy that is stored in this capacitor? We've already derived that expression in previous video. So the energy stored in that capacitor, let me use green for energy. U is there are multiple equations. One of the equation happens to be, the one that I want to use over here, is half Cv squared. Now, because I want to talk about energy stored in electric field, what we'll do with this equation is we'll try to somehow bring electric field into this equation and get rid of C and get rid of V. So I want you to think a little bit about, how can we do that? How can we bring electric field in this equation? Well, I'm looking at potential difference over here. We've seen the connection between electric field and potential difference again in previous videos. We've seen that electric field can be thought of as the negative potential gradient. What does that mean? It basically means you can think of electric field as how much potential difference exists per meter, right? That's a way to think about electric field. And over here, D is the distance over which that electric field exists. That basically is the distance between the plates of the capacitor because this is where the electric field will exist. So that's how I can bring electric field into the picture by substituting for V. So V happens to be from this equation minus E times D. Okay, that's great. I also want to get rid of that capacitance. I don't want that, okay? How can I do that? How can I get rid of the capacitance? Can I substitute something for this? There's a parallel plate capacitor over here. So probably we can substitute the expression for parallel plate capacitor which you probably studied earlier. And we can see what we get. So why don't you pause the video, substitute that expression and see what we get. All right, so the equation for the capacitance or parallel plate capacitance is, we've seen before, it's epsilon naught into A divided by D where A is the area of the plates and D is the distance between the plates. So if I substitute that over here, let's see what I get. What I'll end up with is half C which is epsilon naught into A divided by D multiplied by V, which is a potential difference which I can write as negative E times D. When I square that, the negative sign goes, that's great. So I'll end up with E square D square because I'm squaring it. All right, so we can simplify. The D goes away. And so what I get now is U that is the energy stored in the capacitor equals half epsilon naught. Let me write the E square first. I got what I wanted. Multiplied by area times the distance. What is that? Well, in general, that may not make any sense, but over here, this is the area of the plate multiplied by the distance between the plates. When you multiply that, you're actually calculating the volume, the volume of the space in between the two plates, right? So that's the volume. So I'm just gonna write that as the volume. And because I only want electric field on this side, I'm gonna divide by volume on both sides. And when I do that, what I'll end up with is let me just cancel that volume over here, bring that volume over here. This is the expression that I end up with. Now, why is this mind blowing? Because all I did is took an equation that we were familiar with, did some substitution and yeah, I got another equation. But let's see what this equation is trying to tell us. So on the left-hand side, I have energy stored in the capacitor divided by the volume of the capacitor. What is this? What is this? Well, when you take mass and divide by volume, you get density. It's a measure of how crowded masses or how compact masses. Charge divided by volume is charge density, measure of how compact or how crowded charges. So what do you think is energy divided by volume? Well, it has to be energy density. So let's write that. This is energy density. It's a measure of how compact or how much energy is crowded in this space. So why is this concept so special? Well, if I asked you, hey, where is this energy stored? What would your answer be? Well, you would say, hey, this is basically electrostatic potential energy. It is the energy stored in the charges, the configuration of charges, just like how gravitational potential energy is stored in the masses, configuration of masses, right? Yes, now a different way to look at it is we are saying, hey, this is the energy stored in space, in vacuum. And how can vacuum have energy? Well, because there's electric field. So what this is saying is, in separating the charges, we have created an electric field. And that energy is not stored in charges, but it's stored in electric field. Field itself can have energy. And this is so incredible because this means vacuum. There is no stuff over here, even though vacuum has nothing, due to fields, it can store energy. Have you ever imagined vacuum having energy? That's a pretty radical concept. Now, don't think that this is a new kind of energy, not at all. It's just that we earlier thought of this energy as energy stored in charges. Now we're looking at that same electrostatic potential and looking at it in a different perspective. Looking at this new radical perspective, we're saying it's energy stored in electric fields, energy stored in space. And this can, you know, motivate all sorts of deep philosophical questions like, hey, is energy real? Is the field real? And you can debate about that for hours together. And that's why this concept is so incredible and deep. But besides all these thought-provoking motivations, there's another practical reason why this equation is so awesome. You see, although we derived this expression for a very special case where we had a parallel plate capacitors and we have this nice and uniform electric field, turns out that this expression holds true in general. Doesn't matter whether you have capacitors or not, in general, if you have a field, electric field, uniform or not, the energy that the electric field carries in space, the energy density in space will always be this number, half epsilon naught E square. And the units, of course, would be joules per meter cube. So this is a general expression for how much electric field carries energy density. And why that's important is because you might know that light is a form of energy, right? However, about 300 years back, if I asked anyone in the world, hey, how do you calculate the energy that the light carries? Nobody would have a clue because nobody knew what was light made of. Of course, you would have some clue that brighter light has more energy, but you wouldn't know how to mathematically calculate it. However, you might now know that light, today we know that light is an electromagnetic wave, which basically says that light is made of electric and magnetic fields. So this means if I have to calculate how much energy light carries, all I have to do is figure out how much energy is present in the electric fields and the magnetic field. And since this is a general expression, I can use the same expression to calculate how much energy is present, at least for the electric field part of the light. And a similar expression can be derived. And so I can also use that to calculate how much energy the magnetic field part contains and that total energy will now represent the energy of light. And so what really blows my mind away is that never in my wildest dreams that I think that by calculating how much energy a capacitor has, I would be equipped to at least in part calculate how much energy light carries. That's insane if you ask me.