# Concentric conducting spherical shells cut by a horizontal plane

**1. The problem statement, all variables and given/known data**

A conducting spherical shell of outer radius a and inner radius 3a/4 is cut in two pieces via a horizontal plane a distance a/2 above the center of the spherical shell, as shown in Figure 1. Let us label "A" the upper part of the shell and "B" the lower part of the shell. The shell is initially uncharged and the two pieces that result from the cutting procedure remain in perfect electrical contact. A new conducting sphere of radius a/16 and total charge +Q is inserted in the shell and it is centered on the shell’s center as shown in the same figure.

(a) Are there any charge densities on the inner (r=3a/4) and outer (r=a) surfaces of the shell as well as within it? If yes, derive them.

(b) What (if any) is the force per unit area on the inner and outer surfaces of the shell?

(c) From now on we focus only on the "A" part of the shell: set up an integral that will yield the net force acting on the "A" part. What is its direction? Identify which variables you are integrating and what are the limits of integration. (You are not asked to perform the integration!)

(d) Do the same as (c) for the inner shell of the "A" part of the shell.

(e) Do the same as (c) for the "A" part as a whole.

**2. Relevant equations**

Gauss’ Law (cgs): [tex]\nabla \cdot \mathbf{E} = 4\pi \rho[/tex]

Poisson’s equation (cgs): [tex]\nabla^2 \varphi = -4\pi \rho[/tex]

**3. The attempt at a solution**

I just don’t even know where to start with this problem. I know without the horizontal plane I could apply Gauss’ Law within the inner shell to see that a charge −Q must be distributed on the inside surface of this inner shell (the one at radius r=3a/4), and that it would be uniform by symmetry since the +Q charge is right in the center. But how would I even approach this with the horizontal surface there? Gauss’ Law shows there should still be charge −Q along the inside of the inside shell and on the horizontal plane inside it, but no idea how it’s distributed. Since the plane is not assumed to be a conductor the method of images wouldn’t seem useful. I have no idea what the field should look like since the plane isn’t necessarily an equipotential, and thus I can’t assume the field is perpendicular to the plane. Anyone have some ideas as to how to start this problem that is laid out below? It’s from problem set 3 of the 8.022 course from Fall 2004 on MIT OCW. Thanks!

http://ift.tt/1lLb6Fw

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