A positive charge kept inside a conducting shell

[Titan97]

Homework Statement

A positive point charge q is located off-center inside a conducting spherical shell.We know from
Gauss’s law that the total charge on the inner surface of the shell is −q. Is the surface charge density negative over the entire inner surface?
Or can it be positive on the far side of the inner surface if the point charge q is close enough to the shell so that it attracts enough negative charge to the near side? Justify your answer.

Homework Equations

None

The Attempt at a Solution

The author asks the reader to think about the electric field lines.
Why isn't it possible? I can have electric field lines like these:

 

[gneill]

The field lines from +q must terminate at negative charges on the inner surface of the shell. Because it is a conducting shell it must have a single uniform potential (no static potential difference can be maintained in a conductor). So what would happen to your positive and negative charges on the conductor?

 

[Titan97]

If there is positive charge on the shell's inner surface, there will be electric field lines from the negative charge of inner shell to the positively charged part. But if that's the case, the potential difference between any two points will not be zero.

 

[gneill]

If there is positive charge on the shell's inner surface, there will be electric field lines from the negative charge of inner shell to the positively charged part. But if that's the case, the potential difference between any two points will not be zero.

Right, and that is an unstable situation for a conductor. The charges MUST move to combine and cancel unless some field working tangentially to the surface opposes their movement. It's akin to finding a hill of water on a lake