FIGURE EX24.1 shows two cross sections of two infinitely long coaxial cylinders. The inner cylinder has a positive charge, the outer cylinder has an equal negative charge. Draw this figure on your paper, then draw electric field vectors showing the shape of the electric field.

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Hi, everyone. Let's take a look at this practice problem dealing with gass's law and electric fields. So this question says, imagine a spherical shell where the inner surface has a negative charge. While the outer surface has an equivalent positive charge that's shown in a diagram below the question, you'll need to draw this figure on your paper and then sketch out the direction of the electric fields. So below the question, we're given a diagram that shows our spherical shell and the inner surface of the spherical shell has a negative charge and the outer surface of the spherical shell has a positive charge. We're given four possible choices as our answers. And each one of these answer choices is a diagram. So for choice A, we can have our spherical shell but the we do have electric field lines inside the shell itself. And those electric field lines are pointing away from the inner shell, inner side of the shell where the negative charge is towards the outer side of the shell, where the positive charges are, this is going all the way around the shell. For choice B, we again have electrofield lines being drawn inside the shell itself. And here we have electric field lines pointing towards both the inner and outer um edges of the shell. So pointing towards the negative charge and pointing towards the positive charge. For choice C, we again have the electric field inside the shell. And here the electric field lines are going from the outer edge of the shell towards the inner edge of the shell. And that means that it's pointing away from the positive charge and towards the negative charge. And for choice D, we have electric field actually inside the cavity inside the shell. And here we have electric field lines radiating outward from the center of the sphere. Now, here we need to um actually draw our diagram and I'm actually going to use the diagram that's already given in the problem. But you would need to redraw this on your paper. And we're asked to sketch the direction of the electric field lines around all the different regions. So before we do this, we're going to recall our formula for Gauss's Law. And so for Gauss's law that says that the electric flux is equal to the charge that's enclosed by the galaxy surface divided by epsilon knot. And we also need to recall our definition for electric flux and that is electric flux, it's equal to the integral of E dot D A where E here is our electric field and D A is some small element of our G Gaussian serve some small area element of our Gaussian surface. So what we're going to do is draw three different Gaussian surfaces on our diagram. The first one is going to be inside the cavity. And so we're gonna draw a Gaussian surface, it would be a sphere in three dimensions. We'll just draw it as a circle on the diagram. And if I look at how much charge is being enclosed by that Gaussian surface, it's actually zero, none of the charge is being enclosed. And so, since I have no charges being enclosed, that means I have no electric flux. And if I have no electric flux, and since I do have a surface area for that Gaussian surface, that means that the electric field has to be zero. So inside the cavity of this sphere, there is no electric field, we're gonna do something similar except this time, we're gonna be outside, the spherical sh shell will draw a Gaussian surface. And again, it'll be a circle. And again, if I do this, we have zero net charge enclosed. So we have an equal amount of positive and negative charges being enclosed. And so when I add those up, we get, we get zero net charge since I have zero net charge, I have zero flux. And again, I have no electric field. So they would have no electric field lines outside the outer edge of this spherical shell. So the third Gaussian surface that I'm going to draw is actually inside the shell itself between the inner and outer surfaces of the shell. And when I do this, I do notice that I now have charge being enclosed, I have enclosed all the negative charge. So since I have an enclosed charge, I'm going to have an electric flux and that means I'm going to have an electric field. So now I can use the fact that my electric field lines need to point away from positive charges and towards negative charges. That means my electric field is going to point from the um start at the outer surface and go towards the inner surface of the shell. And this is gonna happen all the way around. And the electro field lines start at the outer end out at the outer edge of the shell and they stop at the inner edge of the shell. They do not continue on um either outside the shell or inside the cavity. So now we have our electric field lines drawn, we can now compare these to our answer choices. And if I look at my answer choices, this corresponds to answer choice C because answer choice C has the electric field lines only inside the shell and it's going from the outer edge of the shell to the inner edge. And we have no electric field lines in the cavity or outside of the shell. So in this case, we basically had to apply Gauss's law to see if we had an electric field in a region. And if we had no electric field, then we didn't need to do anything. But once we determined that we had some uh enclosed charge, that meant that we had an electric field in the region, we just use the fact that electric field lines point away from positive charges and towards negative charges to get the direction of our electric field. So I hope that this explanation has been helpful and I'll see you in the next video.

FIGURE EX24.1 shows two cross sections of two infinitely long coaxial cylinders. The inner cylinder has a positive charge, the outer cylinder has an equal negative charge. Draw this figure on your paper, then draw electric field vectors showing the shape of the electric field. <IMAGE>

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Key Concepts

Electric Field

Gauss's Law

Superposition Principle