 I'm sure you've seen these electric field lines and it's always tempting to think that you know something is actually flowing out of this positive charge or something is flowing in to this negative charge even though there isn't any. But in this video we're going to take that temptation to the next level. We're going to figure out how to calculate how much the electric field flows through the through a given area. And that's the idea behind electric flux. Flux is a measure of how much electric field is flowing through something even though it's not really doing that but we imagine it to be a flow and it's a number that tells you how much the flux field is flowing. And just to give you a little bit of spoiler, this is what the equation of flux looks like. Electric flux looks like. Now I know this looks super complicated and everything but the goal of this video by the end of this video we would have derived this from scratch. But before we begin you might be wondering why? Why do we even study this new concept? You should always keep asking that question. So why are we studying this new concept of electric flux? Why are we imagining electric field to be a flow? Well we will see later on that this will help us look at Coulomb's inverse square law in a completely different way. It's going to blow your mind away. That's what we call Gauss's law. It's going to come later on. That is going to help us solve certain problems which is very tedious to do just by using Coulomb's law. So that's where we're going towards. So to eventually go towards Gauss's law we need to learn this vocabulary of electric flux. Imagining electric field as a flow. Alright so where do we start? How do we start building this equation? I thought maybe we can start with something that actually flows. Let's say water. So imagine we have some water and let's say this water is flowing. Let's assume that the water is moving to the right with a velocity of let's take some number 10 meters per second. And imagine we keep some kind of a window inside that water and let's say that window also has some area. So let's say that the area of this window is, I don't know, maybe something like 2 meters per, 2 meters square, sorry. Now the question is how much is water flowing through this window per second? Alright, that's what I want to calculate. How much water flows per second? That's what we'll call as flux of water and the symbol for flux would be phi. So that's what I want to calculate. It'll be a great idea for you to pause and think about this. This is not electricity. This is nothing to do with electric fields. Just water and area. I want you to think about it and I'll give you a small clue. Think about the units in which you're going to calculate this answer. When I say how much water, what units are you going to answer it in? Well how much water would be something like in liters, right? So we're looking at volume. So we're looking at something like so many meter cubes per second. So great idea to pause the video and see if you can first try and answer this yourself. Alright, here's how I like to do it. I will imagine a cube of water right behind my window and you will see in a second why I like to do that over here. Let's say I have a timer with me and right now the timer is zero. I will start my timer and then I'll stop it after one second because I want to know how much water flows through the window in one second. So let's say I start my timer and I see this cube of water flowing through the window and after one second I stop. Now this is the amount of water that flew through this window in one second. So whatever is the volume of this cube, that must be the flux. Does that make sense? So all we have to do now is calculate the volume of this cube. And again if you couldn't find the answer before, again great idea to pause and see if you can do it now. Alright, so how do we calculate volume of this cube? We do length into breadth into height. So let's see if we know these things. Do we know what this length is? Do we know, let me use white. Do we know what this length is? Yes we do. Since this is the distance traveled by water in one second, remember we timed it for one second, and this must be 10 meters. If this was 20 meters per second, this would have been 20 meters. So this is 10 meters. So we know the length. Do we know the breadth and the height? Well we know the entire area, so we can just multiply it by the area, right? So the area into length would give me the cube. This area would be just the area of the window. And that we know is 2 meters square. So the flux of water is going to be the length 10 meters into 2 meters square. And remember this is in one second, so strictly speaking I should say this is 10 meters per second into 2 meters square. And that would be my flux. And so what's that value? That's going to be 20 meter cubed per second. Alright? So what's that value? If this window was a little smaller, and this cube would have been smaller, so I would have smaller flux. So the area matters. And if the speed was less, then this distance trial would have been smaller, making this volume smaller. And again the flux would have been smaller. So can you see that the flux is the product of the velocity, this is the velocity, and the area of that window. So we can go ahead and write this in terms of an equation now. We could say the flux of water is going to be the velocity of the water times the area of that window. And if this was air, this would be the flux of air, and this would be the velocity of air. And there we have it. We have calculated the flux of water. But of course you say hey, we don't want the flux of water, we want the flux of electric field, electric flux, right? Well now what we can do is we can replace this water with electric field. And so the way you think about it is you imagine the electric field is kind of the strength of the electric field is kind of like the velocity of the water. So then what would be the formula for electric flux? Again, I want you to think about this. I'm pretty sure you can do this yourself. I'll give you two seconds. All right. So the flux of the electric field, right over here, phi of e, what's that going to be? Instead of velocity, we'll use electric field strength. We'll kind of think of this as similar to velocity, but it's not. Nothing is moving over here, but it's kind of like that. So electric field strength e times the area. This is how you calculate the electric flux. So if you want to calculate the flux of any field for that matter, you just multiply it by the field strength and the area. Of course, over here, the units would be different. I am pretty sure you can do that. What would be the units of the flux over here? The electric flux units would be the units of the electric field. What's the unit of the electric field? It's force per charge. Remember, electric field is force per unit charge, right? So it's going to be Newton per Coulomb times the area, which is meter square. Now the units don't make much sense to me, Newton meter square per Coulomb. I don't know what that is supposed to be, but you can kind of now understand this number is kind of like how much the electric field is flowing through that area, even though it's not really doing that, but that's basically what it is. So are we done? Well, most of the hard work is done, but this is not the general equation because we took a very specific case where the electric field is very uniform and the window was perpendicular to it. We now have to generalize it, which is just going to be a little bit more steps. The hard work is done, okay? So let's assume that the window was not perpendicular to the electric field. Let's assume that the window was tilted at some angle. Now what would be the electric flux? Same window, same area. I want you to first think about what would happen to the value of this flux. Would it remain the same or would it be different? Would it be more or less? And again, you can imagine this to be water if it helps you. And by the way, I can show you a side view, which is going to be a little easier to imagine. All right, there you go. All right, so pause the video and think a little bit about this. What do you think will happen to the flux now through this window? Same area, same window but tilted. All right, hopefully you answered. You can kind of see in the drawing itself that in this particular window, there are three electric field lines passing through. But because the window has now been tilted, you can kind of see that these field lines are no longer passing through it, so they no longer contribute to the flux. So you can kind of see that the flux has reduced because of the tilting of the window, right? So now the question is, because the flux has changed, now comes the question, how do we calculate the electric flux now? You can't just multiply electric field into the area. This would be true if it was perpendicular, but what now, how do we calculate it? Well, here's how I like to look at it. Let's imagine, let's get rid of that window. And let's imagine that you're looking from this side in the direction in which the electric field is sort of flowing. That's what we're imagining. Now, as far as you are concerned, the top of the window is somewhere over here at this level. And the bottom of the window from your angle is somewhere over here. So as far as you are concerned for you, the window through which the electric field is flowing is just this. This is the effective area we could say through which the electric field is flowing. And so now to calculate the electric flux, you have to multiply with not this area, but this area, the area that the component of the area that's perpendicular to the electric field. And we can go ahead and call that area as a perpendicular. If we call that, then all you have to do is, you know, calculate what the a perpendicular is and multiply it by that. So our simple modification to this equation is you calculate, you multiply it with the a perpendicular. And if you're wondering, how do I calculate this value, this effective area? It can be done with trigonometry. If you knew this angle, then you can use trigonometry and figure this out. But don't worry, we'll do that some in some other video, not over here. And now hopefully you'll agree as you tilt it more and more, as the angle becomes more and more, the effective area reduces, the a perpendicular becomes smaller and smaller. And eventually, when the window becomes completely parallel to the electric field, the effective area goes to zero. And that's when we say the flux is zero. And hopefully that makes sense. Now nothing is flowing through this window because it is parallel to the flow, parallel to the electric field. All right, so the angle matters as well. So is this the most general equation? Not really. There comes the last generalization because over here we assume the electric field was constant or uniform over the entire window. But what if that's not the case? What if we have a non-uniform electric field and let's say the area is also all, you know, curvy and everything. Now how do you calculate the flux? The electric field is changing everywhere. So what value will you substitute? And how do you calculate a perpendicular? The angle is changing everywhere. Now what do you do? Again, one last time I want you to pause the video and think a little bit about how would you calculate it? Well, now what we could do is we can take that area and we can divide it into tiny, tiny little pieces. Very, very, very, very tiny pieces. And the idea is if the piece is small enough, you can assume each of these piece to be very flat. And you can assume the electric field over that piece is pretty much uniform. You make it small enough and you can assume that the electric field over there is uniform. And then you can calculate what the flux is over there and do the same thing and add up. And so if you were to write it mathematically, the way we'll do that is let's assume a tiny piece over here. Let's zoom into that section over there. So let's take the tiny piece and its area. We're going to call it as DA in the speed of calculus because this is an infinitesimally small piece. That's how we love to think about it. It has an infinitesimally small area, we would say DA. And let's say the electric field at that point, at that point is somewhat this way, which has an electric field, strength is E. Then how do we calculate flux over here? The same formula, the flux would be E times the perpendicular component of that area. So the tiny, tiny flux over there, we'll call d phi. The tiny flux over here is going to be the electric field at that point times the perpendicular tiny area over there. And then you calculate that same thing everywhere and you add up. And when you add up, you're doing an infinite summation because you're taking these very tiny pieces in a limit that the area is about to go to zero and it's the calculus now. So when you add up, we use now the term integral. So the total flux, the total flux, which we're going to call this phi now, the total flux now becomes the integral of E times DA. And so that's how you calculate flux in general. This is the most general equation now to calculate flux. Of course, we can also write this in the vector form, which we'll see in another video. But yes, this is where we stop. So to sum it up, electric flux is a number which tells you how much electric field is quote unquote flowing through an area. And the way to calculate that is we multiply the field strength with the perpendicular component of the area. And of course, if the field is changing everywhere and the area is all crooked and everything, then you calculate the flux over a tiny surface and then you integrate it over the entire surface. And remember, this is not just the flux for electric field. Tomorrow, if you're learning about magnetic flux, just replace E with B. This can be a flux of any field that you want.