# Confusion with Grounded Conductor: bringing in a point from infinity

Suppose, for sake of argument, we have a spherical grounded conductor at the origin. Additionally, let our reference voltage be at infinity. Now, I view the potential of a point in space as being the work that I personally would have to do to move a unit charge from the reference to the point in question. Thus, I interpret this grounded conductor as points in space at which I could bring in a unit charge from infinity and end up doing no work. However, that interpretation certainly cannot be true in this case. Suppose the charge that I am bringing in is unit positive. The entire time I bring that charge towards the conductor there will be a larger and larger negative charge build-up, so I will have to exert a larger and larger force in the opposite direction the entire time I bring it in. The net work that I do in moving this charge towards the conductor will clearly be negative, not zero as the potential value would seem to suggest.

Can someone resolve these conflicting interpretations? Much appreciated.

First, one of the implications of the electrostatic potential satisfying Laplace's equation is that the extremes are at the boundaries.

If the potential of the surface of the sphere is zero and the potential at infinity is zero, the only solution for the potential outside the sphere is the trivial solution, i.e., the potential is zero everywhere outside the sphere.

Second, as you may know, if there is a point charge outside of the sphere, the potential outside of the sphere is identical to that of two point charges, the original and an image, and no grounded sphere. The image charge is located inside the radius $R$ of the sphere and its location depends on the location of the outside charge.

Since the location of the image charge changes with the location of the outside charge, moving the point charge changes the potential due to the image charge.

The point is that, interpreting the potential of a system of charges as the work done to bring a charge in from infinity assumes that the system of charges is fixed, i.e., undisturbed by our bringing in the charge from infinity. Thus, the notion of a test charge.

• Great answer, thanks! Before I accept it I just have a clarifying question about your Laplace comment. So does it not make any sense to have inf. be our reference? In which case, as you mentioned, the potential everywhere will be zero.
– dsm
Commented Oct 9, 2014 at 3:50
• It often makes physical sense to have the zero reference at infinity. In some cases that's impossible, e.g., uniform electric field over all space. In this case it's OK but, since the sphere is also at zero volts, the potential on the boundaries of the outside volume are equal so the problem is uninteresting (trivial) unless there is charge present somewhere. Commented Oct 9, 2014 at 11:31

In electric potential, you must have encountered the notion of test charge.

When you want to determine the electric potential at any point in space resulting due to a certain configuration of charged particles, you bring a test charge from infinity to the said point and calculate how much work you had to do during that process. That work is the potential at that point. The test charge is a unit positive charge that itself does not exert any electrostatic force, thus does not alter our configuration of charged particles. i.e, it can only experience electrostatic force but not exert it.

Back to your question; yes its true that bringing a unit positive charge near to a metal sphere would induce negative charge on the side of sphere facing the charge, virtually attracting our unit positive charge. Thus, we practically do a net negative work, which can not be estimated by the formula W = V x Q. It is because V is the work done to bring a test charge (that exerts no force, resulting in no induction in our sphere) from infinity, not a real charge (that causes induction)

On the one hand, we let our reference voltage be at infinity.

On the other, we say the sphere is grounded. "Grounded" could mean two things:

1. The potential of the sphere is defined to be the reference. This contradicts the previous definition of the reference to be at infinity.

2. The sphere is connected by an ideal conductor to an infinite source of charge at the reference potential. This is not physically possible in cases where the reference is taken to be at infinity.