Two oppositely charged infinite conducting plates

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SUMMARY

The discussion centers on the behavior of electric fields between two oppositely charged infinite conducting plates, separated by distance d and with thickness D. It establishes that the assumption of constant surface charge densities arises from two key conditions: translational invariance parallel to the plates and the uniqueness of the electric field solution. This leads to the conclusion that the solution set is a singleton, ensuring that the electric field itself is also translation-invariant.

PREREQUISITES
  • Understanding of electrostatics and electric fields
  • Familiarity with concepts of charge density and surface charge
  • Knowledge of translational invariance in physics
  • Basic principles of uniqueness in mathematical solutions
NEXT STEPS
  • Study the implications of translational invariance in electrostatics
  • Explore the uniqueness theorem for electric fields in electrostatic configurations
  • Investigate the mathematical formulation of electric fields between conducting plates
  • Learn about boundary conditions in electrostatics problems
USEFUL FOR

This discussion is beneficial for physicists, electrical engineers, and students studying electrostatics, particularly those interested in the behavior of electric fields in conducting materials.

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Suppose they're separated by a distance [itex]d[/itex] and have thickness [itex]D[/itex]. One has charge [itex]Q[/itex], the other has charge [itex]-Q[/itex]. 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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