Force on Point Charge (B) in Conductor Cavity

[sspitz]

Suppose there is a cavity inside a conductor. Outside the conductor there is a point charge (A). E inside the cavity is zero because the field from the conductor and point charge cancel. That I believe.

Suppose I add a point charge (B) inside the cavity. Obviously, there is a radial field inside the cavity from the point charge (B). But won't the point charge (B) also mess up the distribution of charge on the surfaces of the conductor? Couldn't this new distribution produce a force on the point charge (B)?

My book treats as trivial that the force on B must be zero. I don't see it. Maybe an argument about the uniqueness of the potential function...

 

[sunjin09]

Your book probably assumes the charge B is a "test" charge, small enough to not perturb the system. The idea is probably that the metal shell "shields" any outside fields.

 

[sspitz]

Alas, no. It is definitely a charge of arbitrary Q for both A and B. I think in the specific example, A is at the center of a spherical cavity, but not centered in the conductor. Not sure if that matters and why or why not.

 

[sunjin09]

Have you figured it out? I'm curious. I think you can ignore the charge A, so that an arbitrary charge inside a metal cavity will not feel any electric field due to induced charge. Can you prove this is right?

 

[sspitz]

Well, you definitely can't ignore B because it must affect the distribution of charge on the conductor. Maybe you can prove the new distribution still exerts no force on B.

I find it bizarre that ch. 3 of an intro EM book would have this problem and no explanation.

I have no rigorous proof that F on B is zero, but here is my best guess for B at the center of a spherical cavity.

Replace B with a very small conductor. The big conductor and B are equipotentials. One possible solution for the potential is the potential of a simple radial field from B to the surface of the cavity. Since the potential must be unique, this is the potential, and the field is radial in the limit as B becomes a point charge.

I don't even believe this argument, and I wrote it. I'm sure there is some very simple solution. I would appreciate someone pointing it out. Thanks.