Two oppositely charged infinite conducting plates

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In the scenario of two oppositely charged infinite conducting plates, it is assumed that the surface charge densities are constant due to translational invariance parallel to the plates. However, this assumption is nuanced; it requires both translational invariance and the existence of a unique solution for the electric field. The uniqueness of the solution implies that the solution set is a singleton, reinforcing that the solution itself must also exhibit translational invariance. This deeper understanding clarifies the conditions under which the charge densities remain constant. Ultimately, both conditions are essential for the validity of the assumptions made in the problem.
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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?
 
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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).
 
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