How to determine the surface charge density for a parallel plate capacitor

[ppoonamk]

Homework Statement

The problem statement is given in the figure below

Homework Equations

E=σ/(2* ε)

The Attempt at a Solution

The top plate is assumed to be negative charge. The bottom plate is assumed to be positive charge. Since negative charge pulls and positive charge pushes.

surface charge density for upper surface of the top plate = σ1
surface charge density for lower surface of the top plate = σ3

surface charge density for upper surface of the bottom plate = σ2
surface charge density for lower surface of the bottom plate = σ4

So I tried to derive σ3 and σ4 in terms of σ1 and σ2

I know that the electric field in the metal plates is 0.
So the electric field in the top plate = σ1/(2* ε)+σ2/(2* ε)+σ4/(2* ε)-σ3/(2* ε)=0

Electric field in the bottom plate = 0= σ1/(2* ε)+σ2/(2* ε)+σ3/(2* ε)-σ4/(2* ε)

I seem to be stuck after this. Please help :)

How do I get σ3 and σ4 in terms of σ2 and σ1

 

[anathema]

I would approach this problem by using the given equations and principles of electrostatics to solve for the unknown surface charge densities.

First, I would recognize that the electric field inside a conductor is always zero, so the equations provided in the problem statement should be used to solve for the surface charge densities on the top and bottom plates.

To start, I would set up the equations for the electric field in the top and bottom plates, using the given surface charge densities for each plate. For the top plate, the electric field would be equal to σ1/(2* ε)+σ2/(2* ε)+σ4/(2* ε)-σ3/(2* ε)=0, and for the bottom plate, the electric field would be equal to σ1/(2* ε)+σ2/(2* ε)+σ3/(2* ε)-σ4/(2* ε)=0.

Next, I would use the fact that the electric field is zero to solve for σ3 and σ4 in terms of σ1 and σ2. This can be done by setting the equations for the electric field in both plates equal to each other and rearranging to solve for σ3 and σ4. This would result in the following equations:

σ3 = σ1 + σ2 - σ4
σ4 = σ1 + σ2 - σ3

These equations show that σ3 and σ4 are dependent on σ1 and σ2, and can be solved for using the known values of σ1 and σ2.

In summary, as a scientist, I would use the given equations and principles of electrostatics to solve for the surface charge densities on the top and bottom plates. By setting the equations for the electric field in both plates equal to each other and rearranging, I would be able to solve for σ3 and σ4 in terms of σ1 and σ2.