Gauss' Law

by Patrick Ford
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Hey guys. So in this video, I want to cover a concept called Gospels Law, which is a super important topic in electricity, that you need to know now in some textbooks. It's even an entire chapter and it can be kind of confusing because it ties together a lot of concepts and ideas about charge and flux. What I want to show you by the end of this video is that God's Law is really just a very straightforward relationship between flux and charge. And I'm gonna show you the three main types of problems that you'll need to know when solving problems. So let's go ahead and get started here. I'm just gonna jump straight in. So this guy goes a long time ago, he did a lot of calculations between charges and flux is, and what he came up with with this was this law. And basically what it says is that the net flux that goes through a closed surface, which is really important. They're the closed part depends only on the charge that is enclosed within that surface. So basically, here's what I mean, right? Imagine I have this little box here and in the center of this box, I have a charge. And I'm just gonna pretend that it's a positive charge. Now, this positive charge admits electric field lines that go to the surface of this box. And instead of using, you know, some equations like e a Cosine theta or whatever to calculate the flux, basically what God's law says is that the total amount of flux going through all of the surfaces. So I'm gonna call this fine net here is directly proportional to how much charge is in the box. And the equation for this is pretty straightforward. It's fine. It equals Q enclosed over epsilon. Not. So this epsilon here is just 8.85 times 10 to the minus 12 is a constant that we've seen before. But that's basically all there is to it. So this charge here that's enclosed within the box, we call this Q enclosed. Alright. So what I want to do now is show you the three most common types of problems that you'll need when using Gauss is lost. We're just gonna jump straight into the first one here. The first one is where you're given some kind of charges and you're asked to calculate the flux. Let's take a look at this first example here. What is the net flux that goes through this surface A. Over here. Alright, so we've got this little surface a. We've got these two charges. One is positive and one is negative. So, I want to kind of suffer through this a little bit with you. Show you why this is super useful. Now, pretend for a second. We didn't actually have this Gospel slaw equation. The only way we can calculate and defy is by using an old equation, which is E. A. Cosine of theta. So what happens is these point charges will admit electric fields like this? They'll have electric field lines like this and we have to calculate at each point, what is the a cosine theta? The problem is is that this E depends on our right. So it depends on the distance. And what happens is this this surface over here is going to be at a higher distance than this piece right here, and it's gonna be higher than this one are different than this one. And also what happens is that the the normal vector may sometimes point in a different direction than your electric field lines. Notice how this E. And this a don't point in the same direction. So the angle is gonna be constantly changing. What this means here is that this is basically impossible. We can't calculate ea using ea cosign theta. But now that we have Galaxies law, we can have a much more straightforward relationship between flux and charge. So this is finance and this equals Q. And closed over epsilon knots. So, what this means here is that the net flux through the surface only is directly proportional to how much charge is inside this surface here. So, how much charge is in the surface? Well, I've got a five. Cool um and a negative three Coolum charge. So, if you kind of group these together and combine them, the total amount of charge I have in the surface here is going to be five columns plus negative three columns And you'll just get too cool. Um so there's basically two columns of charge inside of the sphere. So we just pop that into this equation. So what God's law says is that fine? It is equal to two divided by 8.85 times 10 to the -12. So when you work this out, what you're gonna get here is 2.26 times 10 to the 11 Newton meters squared per Cool. Oh, so this is the power of Gaza's Law. All you have to do in a surface here is just know how much charge there is. And then you can figure out how much flux the total amount of flux that goes to the surface. And you don't have to use this E A. Cosine theta equation anymore. All right, so, let's move on to the second problem now, which is kind of reversed. In some cases you'll be given the flux either through one or multiple surfaces and you'll be asked for the charge. So, for example, we've got the flux through four sides of a closed pyramid and we've got these numbers over here. So, I'm just gonna draw this out really quickly. Imagine I have this little pyramid like this. And basically, right, so, I've got, you know, 123 and then four sides, the sort of underside like this. It doesn't matter which one is labeled, which it really doesn't matter because remember that God's Law is only concerned with the net flux, not the ones through individual surfaces. So here, what we want to do in this problem, if we want to calculate, well, what's the charge that's enclosed? Now, now we actually have the net flux and we want to figure out Q. So all we have to do here, I'm actually just gonna go ahead and move this down over here is figure out, Well, what is the net flux? Well, if you were told that the flux through each one of the surfaces, then the net flux is just gonna be adding all of them up. So I'm just gonna add 51 plus five to plus 53 plus 54. That's the net flux. So this just means that your net flux here is equal to we've got 10 plus 20 plus eight. So this is eight plus negative 15. And you're just gonna get 23 this is newton meters squared per cool. Um So what I'm gonna do here is I'm just gonna replace my fine it with 23 then I have to multiply by this, my epsilon, which is 8.85 times 10 to the minus 12. And this is gonna be my cue and closed. And when you work this out, what you're gonna get is 2.04 times 10 to the - columns. So this is kind of the opposite here, you're using the flux to work backwards and figure out. Well, if I know all the flux is and that means that there's there's basically this amount of charge that's inside of the pyramid. I don't know how it's all arranged. It could be on the surface, it could be on the center. It doesn't matter. All I know here is that this must be the total amount of charge that's enclosed within that surface. Alright, now let's move on to the last problem here, the last kind of problem, which is a little bit more tricky. So in some problems, you may be given some kind of charges or a charge and you might be asked to calculate the electric field. This one's a little less straightforward. So, let's take out our problem here. We're gonna use gasses law to write an expression for the electric field due to a point charge at some distance R. So for example, I've got my charge like this. What if I wanted to figure out the electric field here at this point, what I'm what I'm gonna call our over here. Well, we actually already know the equation for this. So remember that the equation for the electric field of a point charge is just Q. Is K. Q over R squared. So really what we're doing in this problem is we're gonna use Galaxies law to confirm this equation here. Alright, that's basically what we're gonna do. We should hopefully theoretically get this answer. So, let's start off with God's law, God's Law says that net flux is equal to Q. And closed over epsilon. Not now we want to calculate the electric fields, where does that equation pop? Or what is that variable? Pop up in our equations, remember it pops up in five net. So this is basically saying that E a cosine theta is equal to Q. And closed over epsilon knots. Now we have some kind of an area and a cosine theta, but we don't have a shape, right? We don't have like a pyramid or a circle or something like this. So the most important sort of the trickiest part of these problems is that when you're whenever you're solving for the electric field, when you don't have a surface, you're gonna have to choose one, you're gonna have to choose what's called a Gaussian surface. And this is an imaginary surface. It's not a real thing, it's not like I actually have a sphere that's enclosing this thing, it's an imaginary thing that I'm using so that I can evaluate what this electric field is. Now, here's the most important thing. You're gonna pick a Gaussian surface with symmetry where the electric field is going to be constant everywhere, there's three main types of shapes that you're going to see what you're gonna be boxes, cylinders, or spheres. So if you think about what's happening with this one, I need to choose a Gaussian surface. Do I choose a box? But what happens is if I choose a box. Some parts will be farther from the charge than others. So the electric field is not going to be constant everywhere. So it's not gonna be a box. What about a cylinder? If I use the cylinder, then I run into the same problem. Some parts in the cylinder are gonna be farther away and the electric field is still not gonna be constant. So what happens here is that the best sort of shape to use is actually going to be a sphere? So imagine I have a sphere that's sort of enclosing this charge. Again, it's imaginary, it's not a real sphere. What happens is that we know this charge is going to emit electric field lines like this, they go outwards everywhere. And what happens is my sphere right, is going to be at a constant radius like this. And what happens is my area vector, the normal, the one that's perpendicular is always going to be parallel to the electric field. So this E and this a the angle between these is going to be zero everywhere. So this actually makes my equation a lot easier because basically what happens is that the cosine term will just go away. Remember the cosine of zero is just one. So, if I want to figure out my electric field then all I have to do is just move the area to the bottom here. Now remember the area of the sphere is equal to four pi R squared. So basically what I end up with here is E equals Q enclosed over four pi R squared epsilon knots. Now this may not look exactly like this equation over here, but that's because we have one last thing to do this variable K. That we see in our K. Q over R squared is actually equal to 1/4 pi epsilon. Not. So what this means here is that E is equal to Q enclosed. Actually this is KQ enclosed over R squared. And now basically gotten back to our equation here. Alright, so using Galaxies law, we got the same equation for the electric field of a point charge. Alright, so that's it for this one. Hopefully this makes a lot of sense. Let me know if you have any questions.