A sphere of radius R, centered at the origin, carries charge density

where k is a constant, and r, are the usual spherical coordinates. Find the approximate potential for points on the z axis, far from the sphere.

You can use the superposition principle to combine solutions obtained by separation of variables. For example, in Prob. 3.16 you found the potential inside a cubical box, if five faces are grounded and the sixth is at a constant potential ; by a six-fold superposition of the result, you could obtain the potential inside a cube with the faces maintained at specified constant voltages . In this way, using Ex. 3.4 and Prob. 3.15, find the potential inside a rectangular pipe with two facing sides at potential , a third at . and the last at grounded.

For the dipole in Ex. 3.10, expand to order , and use this

to determine the quadrupole and octo-pole terms in the potential.

Find the general solution to Laplace's equation in spherical coordinates, for the case where V depends only on r. Do the same for cylindrical coordinates, assuming v depends only on s.

a) Using the law of cosines, show that Eq. 3.17 can be written as follows:

Where and are the usual spherical polar coordinates, with the z axis along the

line through . In this form, it is obvious that on the sphere, .

b) Find the induced surface charge on the sphere, as a function of . Integrate this to get the total induced charge. (What should it be?)

c) Calculate the energy of this configuration.

Two semi-infinite grounded conducting planes meet at right angles. In the region between them, there is a point charge q, situated as shown in Fig. 3.15. Set up the image configuration, and calculate the potential in this region. What charges do you need, and where should they be located? What is the force on q? How much Work did it take to bring q in from infinity? Suppose the planes met at some angle other than ; would you still be able to solve the problem by the method of images? If not, for what particular angles does the method work?