Setup 1: the grounded case
When a positive point charge is near a conducting grounded infinite plate of finite thickness, the area near the charge becomes negative.
The net charge on the plate should be negative:
One way to find the total charge on the plate is to use the method of images (replace the plate with an image, negative charge on the other side of the plate). The potential in space is just the dipole potential (it vanished on the plane of the right side of the plate). From the potential we can derive the electric field, and from it the charge density on the plate.
The potential inside the plate vanishes. The area left to the plate has no charges, and the boundary conditions are those of vanishing potential, so the solution is trivial: the potential vanishes left of the plate. This means that there is no electric field there, and so no charge density on the left side of the plate.
This means that in total we have negative net charge (which can be found by integration). The result is the same if the thickness is zero.
So far so good. But could this be more subtle?
Setup 2: the neutral case
A claim that the plate could be uncharged:
This article claims that given an uncharged, non-grouded, conducting infinite plate gives the same electric field as a dipole, using the method of images. If the fields are the same as in the grounded case discussed above, then the potentials in both setups must be the same (let the potential at infinity vanish). So the plate is de-facto grounded, even if not physically.
But that must mean that the charge densities on the plate are the same in both setups!
The article does agree that the charge density on the left side of the plate vanished, but it claims that the net charge might still be finite on it (and positive).
The claim is that as we take the limit of larger plate, the charge is constant (and positive), and the area grows, the charge density is uniform, and goes to zero.
How do we resolve this contradiction?
Which of these claims is false?
- Setup 1, the grounded case, has a negative net charge.
- The article claims that the uncharged, non-grounded setup 2 has a dipole field, and this means that the fields of the two setups are equivalent.
- This means that the charge densities are the same in both cases.
- This means that the net charge must be the same, which means that the article is wrong in assuming that an uncharged non-grounded plate near a point charge would have a dipole field.
I have some thoughts on how to solve this contradiction, but I'm not sure:
- The article is mistaken in the assumption that the plate has zero potential. A large, uncharged plate near a point charge must have some nonzero potential. Looking at an infinite plate is problematic if its potential is nonzero, because the boundary conditions are contradictory: the potential at infinity must equal that of the plate, as it extends to infinity.
- For an uncharged plate, the charge density on the left side is not uniform (unlike the article's claim). For a thin plate, the positive charges gather on the outskirts of the plate, far away from the positive point charge. the negative charge stays in the center. This means that we push finite charge off to infinity and that could create an ill-defined setup, which undermines some of the assumptions used to calculate the potential.
- The question of net charge is undefined in the infinite-plate setup, and more subtle than just integrating over the charge density. This means that either the first setup of a grounded plate actually has zero net charge, or that we just cannot find out the net charge (what a strange option!).