Hints for Assignment 3

12. As in problem 10, we are talking about getting the field at a given point given a continuous charge distribution. Make each little piece of the charge small enough that 1 / r2, with r being the distance to the point P, does not vary significantly over the extent of the charge. Try to visualize the diagram in 3 dimensions. If the sphere gets bigger but the (solid) angle subtended by the piece of charge does not, what happens to the size of the charge? What happens to the size of a constant amount of charge?

You will not understand part (d) until you have read the section on Gauss's law or heard the Gauss's Law lecture. Come back to (d) if necessary, but do not skip it entirely.

13. Remember what we did in class for charged disks.

14. First find the field due to either one of the line charges everywhere along the x axis.


In general, if you can use Gauss's Law to calculate an electric field, do so, as Gauss's Law when applicable is always easier than direct integration. Only if you cannot specify the direction of the field and the surfaces over which it is constant should you do a direct integration of Coulomb's Law over the charge density.

The surface used in Gauss's Law is a mathematical one, and need not necessarily correspond to the edges of any physical body. Choose the surface by first noting the point where you want to know the field. Then extend the surface away from that point in such a way that the field is constant everywhere on the surface. If you can get a closed surface this way, or make a closed surface by adding on areas where the field is parallel to the surface, you can very probably use Gauss's Law.


15. Use spherical coordinates, of course. Part (c) reminds you that even though the shell has a net charge of zero, its charge density doesn't have to be zero everywhere. What does "conducting" tell you that is relevant to the solution?

16. You have to do the volume integral explicitly for this problem. There are no shortcuts. Which coordinate system should you use?

In some ways problem 17 is easier than this one, and in some ways it is harder. If you have difficulties with problem 16, trying doing 17 and coming back to this one afterwards.

17. You need to calculate the field inside separately from the field outside.

All integrals reduce to known volumes and areas, so you can avoid explicit integrations. You should set up in cylindrical coordinates to make it obvious that the integrals are that simple.

18. You have to do the volume integral in this problem explicitly. How many regions have to be done separately? What system of coordinates should you use? What is the proper choice of the Gaussian surface in each region? Problems 16 and 18 are logically similar, but require different coordinate systems.

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Last revised 1998/02/19