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GL
University of California, Berkeley

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Gauss's Law Example

Gauss's law is a law of magnetostatics, which states that the magnetic flux through a closed surface is proportional to the total magnetic charge enclosed by the surface. The constant of proportionality is the magnetic permeability of the material. The law was formulated by Carl Friedrich Gauss in 1833. The SI unit of flux density is the tesla (T).

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So in this video we'll look at some examples of KAOS is law, which I've rewritten here. Let's phi equals the integral of the electric vector the electric field through a small surface integrated over on entire surface. And so perhaps we have two charges now, plus Q and some minus que we one of the of the flux through several services. The first one, it's simply the positive charge. Well, we know the flux through this. What's equal, que enclosed over Epsilon not. And so here we simply get Q over, you know, now say instead, we wanna know a surface over here. Well, once again, Fi is equal to Q and closer, perhaps or not. So we get minus que over. Absolutely not. Now finally say, we want to know some large surface that encapsulates both of these charges. Well, are charging closes now the plus Q plus the minus que so it's simply zero over. Absalon not or total flux is zero. Now, before we get into more concrete examples of KAOS is law. Let's consider a few things about conductors and there are two facts that we're gonna have to remember as we work one. The electric field inside a conductor is zero everywhere beneath the surface of the conductor. To see this, let's consider a conductor. So we have some sphere made of conducting material and as we know, any materials made out of atoms, and these atoms have particles such as protons and electrons that have charge on them, you know, draw some of these particles here in red. Now let's assume the electric field inside is not zero. It's some finite value greater than zero in magnitude. Well, what's gonna happen to these charged particles? They're going to begin to move because this electric field applies a force on all of them, and as long as a fill of force, they'll keep moving and they'll keep moving until they find a place where there is no force. The only way to not have a force on electric particle or in a charged particle, it's Ah is toe have the electric field B zero. So basically, we can consider this and say, Well, the electrons inside a conductor will move around until they kill any internal electric field. And for a better example of that, let's consider are neutral material again, and we know the field is zero. We've just shown why that ISS inside this conductor and now we're going to add on external electric field of some constant magnitude E this is gonna pass through the conductor. And what happens is it does that Well, assuming this is a positively directed field, the electrons inside this conductor are gonna be feeling of force and want to move to the left. And as they gather this way, we get a net positive charge which I'll draw in blue on the surface, because the electrons have left there and we have more Bear Adams more bear nuclei over there with positive charge. And so we get a net electric field. Well, the positive on one end, the negative on the other, we can see this will create a field in this direction in this field will continue to be created as electrons move until it matches the external field we've created over here or we've administered over here. So in all cases, inside, in a conductor, anywhere beneath the surface, the electric field is zero. Now, the second important thing to remember and it falls from the first is that any excess charge that we place on a conductor must be on the surface. And the proof of this is very similar to what we've done already. So let's take our conductor again, which has zero net charge and we're gonna place just one point charge Q inside of it. Well, this is going to create an electric field outward radial e outward. And that field will begin to affect the other particles there until those particles move to the surface where they can't move any longer and again re balance themselves so that the field inside goes back to zero. And we know it has to be zero in there so that the charges stop moving. So any charge excess charge that we add to this conductor positive or negative has to reside out here on the surface. And these two facts will be very important as we look at further examples of gas is law. Oh, and there is one more important note to add. Yeah, surface does not necessarily mean outer surface. We can have ah, conductor that has a cavity inside and it can be possible for charge that reside on this internal surface as well. We'll see when this arises and why? Further examples. So first, let's consider a simple example of, ah, conducting sphere that's neutral. And then we had some charge. Q. On it. We want to find the electric field everywhere in space. Well, we know Q will consist of many little charged particles. How will they distribute if I put them all in one spaces I have here? Well, we know there's a repulsive force between these charged particles, and they'll start to spread out and they'll spread out evenly over the surface of the sphere and evenly because this is a symmetrical surface, so there's really no prefer place to go. They'll just spread out as far as possible. So we get a nice uniformed distribution over the surface of the sphere and over the surface, because we've said there can be no electric field inside of a conductor, and any excess charge will go to the surface. So now we consider the center and what happens inside where we draw the ocean surface and we say there is no charging clothes instance. Why equals E A, which is equal to que enclosed over epsilon, not which is zero, because there is no charge then e inside is zero or e zero when our radius from the center of the sphere is less than the radius of the spirit itself. Capital are now. Let's consider what happens if we draw some surface outside centered on the center of our sphere. Well, now we can treat all of this as if all the charges collapsed to the center of the surface. This does not mean that the charges in the middle of this conducting spirit, obviously it's not. But we can treat it that way. And in the same way we can treat a satellite orbiting around the Earth as reacting to all of the mass of the Earth centered at the center or at the Earth's core. Obviously the massive spread everywhere throughout the Earth. But we can make the exception to treat it as centered there in the middle on we could do the same with charge. So now our fi, it's the integral of e dot ds over the surface. Of course, Now we know there's no preferred orientation. Everything has to radiate outward. And so it s a T s or the electric field this perpendicular everywhere to the surface and it's constant on a surface of radius R, but we can bring that out. And now we're just integrating over the surface itself, which is spherical. And we know the surface area is here is four pi r squared. This is equal to Q and closed over epsilon not or capital Q over Epsilon Not and so we find that e is equal to queue over four pi r squared epsilon. Not what You'll also recognize this cake over r squared what we've come to expect from what point charge So we can grab this. We have radius on our X axis and our electric field as a function of radius on the other axis. And we know up until this radius of capital are, our electric field is simply zero, and from here it falls off in the one over R Square fashion. What is this value? Here is Q over four four pi capital R squared. Absolutely not

GL
University of California, Berkeley
Top Physics 102 Electricity and Magnetism Educators
Andy C.

University of Michigan - Ann Arbor

Marshall S.

University of Washington

Zachary M.

Hope College

Aspen F.

University of Sheffield