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Sorry, I am repeating my last post with some typos corrected:
( I also add the solutions to the 4 simultaneous equations):
Assumptions, previously proved theorems, and definitions:
The two parallel conducting plates are infinite in length and width
(neglect end effects). Once electrostatic equilibrium is achieved, the
E field inside each plate =00:
Once electrostatic equilibrium is achieved, non zero net charge
resides only as sheets of surface charge on the 4 plate surfaces.
A sheet of surface charge of density s(Coulombs/m^2) produces
everywhere an E field of s/(2*epsilon) - directed away from the
sheet if s is positive and toward the sheet if s is negative.
S1 = the net charge density on the left plate (the net plate charge
divided by the area of one side);
S2 = the net charge density on the right plate (the net plate charge
divided by the area of one side);
s1 = the net charge density on the left surface of the left plate;
s2 = the net charge density on the right surface of the left plate;
s3 = the net charge density on the left surface of the right plate;
s4 = the net charge density on the right surface of the right plate.
The problem:
Given S1 and S2, calculate s1, s2, s3 and s4
We need 4 equations to determine the 4 unknowns:
By definition:
(1): s1 + s2 = S1
(2): s3 + s4 = S2
The E field inside the left plate (due to the 4 sheets of charge) =0:
(3): (s1 - s2 -s3 -s4 )/(2*epsilon) = 0
The E field inside the right plate (due to the 4 sheets of charge) =
(4): (s1 +s2 +s3 - s4 )/(2*epsilon) =0
The above 4 numbered equations determine s1, s2, s3 and s4,
given S1 and S2. The solutions are:
s1 = s4 = (S1 + S2)/2
s2 = - s3 = (S1 - S2 )/2