# Can the surface charge density be negative somewhere on the inner surface of a spherical conductor shell?

A positive charge $q$ is located off-centre inside a conducting spherical shell. We know 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 the shell so that it attracts enough negative charge to the near side?

1. Is it possible?

2. Would there be a force on the point charge, why?

3. Why the uniqueness theorem cannot be applied to suggest that the field anywhere inside the shell is $0$?

## 1 Answer

When the point charge is not at the center of the sphere, the electric field lines will not intersect the sphere at right angles. Consequently, there is an initial component of electric field along the surface of a conductor. We know this results in a force on the charge carriers inside the conductor, and these charge carriers will re-arrange until the electric field is once again perpendicular to the conductor.

At that point you have a charge concentration (positive or negative, depending on the polarity of $$q$$ and whether you are looking at the point closest to $$q$$, or furthest from it). The polarization would result in a force on $$q$$ - attracted towards the point of the sphere that it's closest to, since the charge concentration there is greatest.

There is no conflict with the uniqueness theorem. The UT only states that "if the solution meets the boundary conditions, it is the solution". And the above describes such a solution - the boundary conditions (E perpendicular to conductor surface) are met because there is a redistribution of charge.

The following diagram illustrates this:

I drew the field lines inside the sphere as straight lines initially, then attempted to show how they have to bend in order to meet the conductor surface at right angles. The only way I can think of bending the field lines at B is to have a charge of the same polarity as q on the surface. There can be no net field on the dotted surface (inside the conducting shell).

So getting down to brass tacks: "can there be a region where the surface charge is positive" (assuming that $$q$$ is positive)?

This is a question being asked on this site - and I reproduce the diagram given (for the correct answer):

The reasoning follows from the two hints supplied:

The field inside the conductor must be zero. The charge inside the shell is off-center, and hence the charge on the inner surface of the shell will arrange itself asymmetrically to cancel the field of the large positive charge.

The field inside the conductor must be zero. To the charges on the outer surface, it is as if the inside of the conductor were completely neutral. Thus, the charges on the outer surface will feel no force other than their own mutual repulsion, and will therefore have no preferred direction.

Now if you think about this, you can see that at every point where the electric field lines from $$q$$ hit the surface, you need an opposite charge to cancel the electric field - otherwise you end up with an electric field inside the conductor. And we know that can't happen. In the diagram below, $$E_1$$ and $$E_2$$ must point in opposite directions to cancel inside the conductor.

• So is it possible for the surface charge to be positive somewhere? I've changed my questions slightly. – Rescy_ Nov 9 '15 at 22:15
• Your edit changes the question in a very significant way. You didn't state that the sphere was grounded before: but that must be the case if the sphere has a total charge of $-q$ (new information). I am not sure my argument is valid for a grounded sphere - while there will be a non-uniform charge distribution, I'm not sure there will be a point where the surface charge on the surface will have the same sign as $q$. – Floris Nov 9 '15 at 22:19
• No the sphere is not grounded. It has total charge $-q$ on its inner surface and there is a total charge of $+q$ on its outer surface (it is a shell). – Rescy_ Nov 10 '15 at 5:14
• I need to improve my reading skills... – Floris Nov 10 '15 at 11:32
• Thanks for the answer...but...can there be some region of the inner surface that has positive charge density? – Rescy_ Nov 10 '15 at 22:43