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Anderson Video - Gauss's Law Intro

Professor Anderson
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 Let's talk about Gauss's law. This is one of the biggies and one of the more difficult concepts to understand. Gauss's law is an approach to calculating electric fields for different charge distributions and if you keep going on in physics, you're gonna get to a very complicated form of Gauss's law. Okay, and just for fun let me show you what it looks like. That's what it looks like. Del dot E equals Rho over epsilon naught. Okay, this says the divergence of the E field is equal to the charge density Rho divided by epsilon naught. You can also write this in integral form. Integral of E dot DA equals Q enclosed over epsilon naught. That is a closed surface integral of E. It's equal to how much charge is enclosed divided by epsilon naught. So some of that stuff looks completely foreign to you but some of it looks familiar. Right? There's an E. Oh, we know what that is. There's a Q, I know what that is. Epsilon naught, we heard about that. That's permittivity. This integral stuff, maybe it's a little complicated but it's really not that bad. This simplifies greatly in a lot of cases to the following: E A equals Q over epsilon naught. And that's what we talked about today in class. Know that there is a lot more to it, but it simplifies in a great number of problems to just this: E times a equals Q over epsilon naught. So let's attack it from that point of view. You can immediately forget what I just said. Let's say we have the following, let's say I have a positive charge Q and we're gonna draw some field lines. All right, we know how to do that. Field lines leave positive charges. This is the electric field E, but now let's double the Q. Plus 2Q. And so we've got this, this, and so forth. We're gonna get 8 lines instead of 4 lines. Okay, and we had 4 lines here, now we have 8 lines. And you can see what's happening, right? When I double the charge, I double the field lines. And this is really the whole point of Gauss's law, if we do one more step and that's– let's draw some spheres around these things. So here's my plus 2Q. I'm gonna put a spherical shell all the way around. Hopefully you can see that okay. And that shell is called a Gaussian surface. In this case it's a spherical shell. So, think like a basketball, right? Just the outside edge of a basketball well let's Well let's see if we can figure out what's happening inside. What we said was 2Q is gonna give us 8 field lines. So I'm drawing them with dashes because they're inside the sphere, but eventually they're going to poke right on out. And here they come out of the sphere. Okay, eight field lines are coming out of this sphere. If I drew it with one Q in there, then four field lines would be popping out. And so this is Gauss' Law. The field lines through the surface is proportional to the charge inside. That is it in words. We just wrote it down mathematically, this is what it says in words. I've got 2Q so 8 lines lines came out. If I had 1Q than four lines would come out. It's proportional to the charge inside. This right here is called the flux. What's the flux? How many field lines are coming out of the surface? This is proportional, right? That's what that little alpha means, it means proportional. That's of course our charge Q. Alright, but electric field lines is not flux if I multiply it by the area of the surface. There's some area A of that surface. That is flux, okay? That is flux and that is equal to Q over epsilon naught where epsilon naught is our proportionality constant. It's proportional to Q over epsilon naught. Epsilon naught is called the permittivity of free space. It has a value in SI units of 8.85 times times 10 to the minus 12. And you can put the units on there, it's some funky units. So, when you are asked the question, "What is the flux?" This thing is what you want. The flux is E times A. Alright so let's take a look at one of your homework problems that in fact deals with that question.