Charge distributions of two infinite parallel plates

[MaximumTurtles]

Homework Statement

Two infinitely large conducting plates with excess charge 2Q and 3Q are placed parallel to one another, and at a small distance from one another. How are the charges 2Q and 3Q distributed? You may assume that infinitely large sheets of charge produce electric fields that are distance-independent. Make sure not to just guess or list the charge distribution, but to prove it! Based on your result, can you state the formula for the magnitude of the field between these two parallel very large conducting plates, as well as the formula for the field between any two parallel very large conducting plates?

Homework Equations

This is in a practice test for a chapter on Gauss's law.

The Attempt at a Solution

I have tried setting up two Gaussian surfaces, the first is a cylinder with one end inside each plate, the other with each end on the outside of each plate. I can relate the surface charge densities to the electric fields outside, and I can get the relationship between the inner surfaces, but I can't figure out how to get a relationship between the charges 2Q and 3Q (written as Q and Q' in my diagram) and the surface charge densities.

Here is my attempt:

https://drive.google.com/open?id=16OxjQOXjbZ0hBXws6ZkEGEu0gQ0aWe9t

This is my first time posting, so I apologize for being a noob, let me know if there's anything I can do to help out.

 

[BvU]

Hello MaxT,

A good starting post, but my neck hurts from craning sideways. You place Q and Q' in ambuiguous locations: it is known that positive charge repels positive charge, so for a single plate the charges would not stay in the bulk volume, but move (conductor!) to a suitable surface.
The exercise is a bit complicated by the repeated use of the term 'infinitely', but never mind (replace by 'very'). And: you do the right thing by switching to surface charge density.
I agree with σ₁ =-σ₂ [edit - now that I can see the subscripts] $\ \sigma_2 = -\sigma_3\$ but that's about all I can decipher.

Any other useful Gauss surfaces ? How should the total charge distribution behave when seen from 'infinitely' far away ? All field lines have which direction ?

 

[MaximumTurtles]

A good starting post, but my neck hurts from craning sideways.

Haha, woops, I fixed that! Sorry about the wording of the question, the professor's English isn't too good, but I copied down what he wrote exactly. Practically speaking, he just want the plates to be large enough that we can consider the electric field they create to be distance-independent (E = sigma/(2*epsilon naught)).

I can see that the areas outside the plates will have a field strength equal to the sum of the fields created by all 4 surfaces, the inside of the plates will have a zero field, and the area in between will be the difference between the fields created by the top plate and the bottom plate. I still can't see how to get the surface charge densities in terms of the charges of the plates though.

 

[BvU]

Fixed my sigma subscript numbers. What do you know about $\ \sigma_1+\sigma_2\$ in relation to $\ \sigma_3+\sigma_4\$ ?

 

[haruspex]

the inside of the plates will have a zero field

what equations does that give you in terms of the four surface charge densities?