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Using what we just learned, we are going to derive a few situations involving Gaussian surfaces. Why? Not because it is fun or pleasant or not all together mental bliss, but because it is interesting and profound. In addition, it will give you some more experience in solving other problems in this area.
Location of Excess Charge On a ConductorWe know and will prove that the electric field E = 0 at all points within a conductor when the charges in the conductor are at rest. We can make an imaginary surface in the interior of a conductor, such as surface A in the illustration at right. Becuase E = 0 everywhere on this surface, the net charge inside the surface has to be zero. Now compress that surface so it encloses just a point, and since E = 0 everywhere on the surface, the net charge inside that point is zero. That also means that there cannot be a net charge at any point within the conductor, because that tiny point could be put anywhere in the conductor. If that's so, that means all the charge must be on the outer surface of the conductor, as shown.
Let's cut a hole in the conductor as shown in the illustration to the left. If we look again, removing that piece doesn't do anything because everywhere in the conductor there is no net charge at any point. If you were to surround the cavity with a surface B, you will find it encloses no net charge and so there is no charge on the internal cavity. Again all the charge is deposited on the outer surface.
In the third case shown in the illustration on the right, let's say we put a charge q inside the cavity of the original conductor. We insulate it so no charges can jump from one surface to another. Now if we use Gauss's Law with surface B again, the net charge of what it encloses has to be zero because there was no charge transfer between the charge and the conductor. What that also means is that there must be a negative charge buildup on the wall of the cavity to negate the positive charge q it surrounds in order to maintain a net charge of zero. Now if our outer conductor is initially uncharged before charge q is inserted, there must be a charge on its outer surface to negate the negative charge on the inside cavity in order to maintain its uncharged status. So the other surface must have a charge equal but opposite the charge of the internal cavity, or the outer surface's charge is equal to that of charge q.
Field of a Charged Conducting SphereWe now know that any excess charge on a isolated solid conducting sphere is distributed uniformly on the outer surface. We also know that the field of the sphere is radial everywhere, because it similar to a point charge in that there isn't any reason why the field would deviate in any particular way from the radial direction. Again, the magnitude of the field is uniform over a spherical surface of radius r, as long as its concentric with the conducting sphere. This is shown on the illustration at right (click on the illustration for a VRML model of the sphere).
If r > R, we know that outside the sphere we can treat the sphere as if it were a point charge. Another way we can look at it is that the Gaussian surface encloses only that charge Q. And since the field lines are all perpendicular to the surface and uniform all over, we get:
If r < R, the Guassian surface would be inside the conductor. Thus, the field would be zero since all charge is on the outside surface of the conductor.
Field of a Line Charge (Charged Cylindrical Conductor)Notice that earlier with Coulomb's Law we derived the field of a thin line of charge, which we now will call simply a line charge, but now we will do it more easily with Gauss's Law. Now our wire is very long and we aren't near the ends of it, so we can say the field lines outside the wire are radial and lie in planes perpendicular to the wire (Field lines will not behave that way when you get close to the ends; they cease to be parallel to each other and that gets all messy). We also know that the field has the same magnitude at the same radial distance from the wire (Is this starting to sound familiar? It is because before we can use Gauss's Law we need to establish that the field is both constant and perpendicular to the Gaussian surface that we use.).
Knowing all that, we need to select a surface that we can use to find the field and naturally we would choose a cylinder of an radius r and length of l, with the ends of the cylinder perpendicular to the wire (parallel to the field lines). You can see the illustration at right to visualize this setup (click on the illustration for a VRML model of the wire).
Now, l is the charge per unit length of the wire. So that means that the charge in the Gaussian surface is l times the length. Since E is at right angles to the wire, the component of E normal to the end faces is zero. That means that the end faces don't contribute to the integral in Gauss's law. On the other hand, the side of the cylinder is perpendicular to the field lines, so they will contribute to the integral. Since the surface area of the Gaussian surface is 2prl>, we get this for the solution:
Now if you notice that the difference between a charged cylindrical conductor and a line charge is thickness, you will also realize that the relationship between the cylinder and the line is the same as the sphere and the point charge. Since excess charge on a conductor collects on the surface, there is no field inside the cylinder. Outside the cylinder you notice that you can treat the cylinder like a line, and you will arrive at the same answer.
Field of an Infinite Plane Sheet of ChargeLike last time, the infinite sheet of charge has a charge per unit area of s. However, this time we are going to use Gauss's Law. Create a cylinder whose ends have an area A and whose walls are perpendicular to the sheet as shown in the illustration to the right (click on the illustration for a VRML model of the sheet). By symmetry, and the fact the sheet is infinite, the field E has the same magnitude on both sides, is uniform, and is directed normally away from the sheet. The component of E crossing the walls of the cylinder, the curved part, is zero. What that also means is:
We used 2EA because we need to take into account both sides of the sheet. We see again that the magnitude of the field is independent of distance. That is because the field lines everywhere are straight, parallel, and uniformly spaced. Why this is true is because the sheet is infinitely large. Actually this is just an idealization of reality since nothing can really be infinitely large and that it can be applied to finite sheets as long as we stay away from the edges (where the field lines aren't parallel or uniformly spaced).
Field of an Infinite Plane Charged Conducting PlateIf we give a flat metal plate a net charge, the charge distributes itself over the entire outer surface. This is just like two infinite plane sheet of charge, one on each side of the plate. The fields from the two sheets (named 1 and 2) will thus be superimposed on one another in each of the two directions, as you can see in the illustration at right. Thus, outside the plate, the electric field will be:
But what about inside? Well, the field lines of the two sides of the plate cancel each other out. The resultant of the field lines is zero, which you realize is the same as in any other conductor. Inside a conductor, there is no charge and thus no electric field. In that derivation we used a previous derivation in order to do this one.
If you're a purist and want to do it the way Gauss would've wanted you to, with his law directly, it is:
Same diagram, same situation, different way. Inside the conductor, where the inside face lies, the field is zero; it is always zero inside a conductor. Outside the conductor, it is perpendicular to the plate, so the normal component of E is zero over the walls and equal to E on the outside face.
Field between Oppositely Charged Parallel Conducting PlatesThe illustration to the right shows what the field lines of two plane parallel conducting plates are. The field between the plates is basically uniformily spaced but there is a little fringing, at the edges. We are not concerned with the edges so we can assume it is uniform. These parallel plates are the basis of capacitors, which will be discussed in a later topic.
These two plates can be treated as two sheets of charge of opposite charge and we can use Gauss's Law. Outside the two sheets, the components of the electric fields are of the same magnitude but oppositely directed so their resultant is zero. At any point between the plates the components are in the same direction so their resultant is the sum. Basically giving you the same result as that of the charged conducting plate:
|Created by TQ Team 16600:||
Clyde, Chetan, Jim
Melanie Krieger, Chhaya Taralekar