Two oppositely charged infinite conducting plates

[AxiomOfChoice]

Suppose they're separated by a distance $d$ and have thickness $D$. One has charge $Q$, the other has charge $-Q$. Why can we assume that each of the four surface charge densities are constant?

 

[vanesch]

The simple answer would be that there is translational invariance parallel to the plates in this problem.

However, after some thinking, you realize that this is somewhat too easy.

Because this only assumes that the set of solutions is invariant under translation, but not each individual solution.

So there are actually TWO conditions: the fact that there is translation invariance, AND the fact that there is going to be a unique solution for the electric field.

In that case, your solution set is a singleton, and in that case, the solution itself must also be translation-invariant (and not just the set).

 

[astrokreng]

The assumption of constant surface charge densities in this scenario is based on the fundamental principles of electrostatics. In an infinite conducting plate, the charges are free to move and distribute themselves evenly across the surface. This results in a constant surface charge density, as any excess charge will quickly redistribute itself to achieve equilibrium.

Moreover, in the case of two oppositely charged plates, the charges on each plate will create an electric field that is perpendicular to the surface. This electric field will act as a repulsive force, preventing any accumulation of charge on the surface. As a result, the surface charge density remains constant.

Additionally, the charges on the plates are assumed to be spread out uniformly over the entire surface, resulting in a constant distribution of charge. This assumption is valid as long as the thickness of the plates is much smaller than the distance between them (D << d). In this case, the charges can be considered to be distributed evenly across the surface, leading to a constant surface charge density.

In summary, the assumption of constant surface charge densities is based on the fundamental principles of electrostatics, the behavior of electric fields, and the uniform distribution of charges on the infinite conducting plates.