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Illustration 24.3: A Cylinder of Charge
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In this animation each charge is a line or rod of charge into and out of the screen. You can create charge distributions and see the electric field that results (position is given in meters and electric field strength is given in newtons/coulomb). Restart.
Begin by adding one line charge. Notice the field lines. Imagine that this charge extends both into and out of the computer screen. Clearly, there is cylindrical symmetry. You could imagine putting a tube centered on this charge. How could you convince another student that the magnitude of the electric field at any point on the tube would be the same as any other point on the tube? Well, there is nothing special about the placement of your tube in the direction out of the screen. There is nothing special about how long that tube is.
Now, add another line charge. What happens to the symmetry? Far away from the two charges, there would still be cylindrical symmetry. Why? If you are far enough away it looks (approximately) like a single rod. Look at what happens to the field as you add 10 more line charges and then half a cylinder. When you create a full cylinder, what is the symmetry? The field everywhere inside the cylinder is zero. What Gaussian surface could you use to find the electric field inside the cylinder or outside the cylinder? The full cylinder has cylindrical symmetry about the middle of the cylindrical shell of line charge. Since there is a symmetry, we can use Gauss's law to calculate the electric field. Since there is no charge enclosed by a Gaussian surface of radius 1.65 m, the flux is zero, and because of the symmetry we can say that E is zero inside the cylindrical shell. We could also use a cylindrical Gaussian surface to calculate the electric field outside of the cylindrical shell of line charge.
As you work problems using Gauss's law, you will need to be able to identify symmetry (look at the symmetry of the charge configuration), as well as recognize the direction of the field and where it cancels out. Be sure that you can do that for this configuration.
Illustration authored by Anne J. Cox.
Script authored by Wolfgang Christian.