Re: Electrostatics problem

[Bob Sciamanda <trebor@VELOCITY.NET>]

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 =
0
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 - s3 -s4 )/(2*epsilon) = 0

The E field inside the right plate (due to the 4 sheets of charge) = 0:
(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

Bob Sciamanda
Physics, Edinboro Univ of PA (ret)
trebor@velocity.net
http://www.velocity.net/~trebor

> -----Original Message-----
> From: Ken Fox <kfox@STEGA.SMOKY.ORG>
> To: PHYS-L@LISTS.NAU.EDU <PHYS-L@LISTS.NAU.EDU>
> Date: Wednesday, February 24, 1999 5:59 PM
> Subject: Electrostatics problem
>
>
> > I just spent a short time wrestling with this question with one of my more
> > intuitive AP Physics student. I am stuck.
> >
> > Given two conducting metal parallel plates, as an air capacitor. If we
> > place equal
> > charges of opposite sign on each plate, we would expect to find the charge
> > on the inner facing surfaces. If the signs are alike, the charges would
> > migrate to the outer surfaces. So far so good?
> >
> > Is there a clear way ( I cannot think of it) to determine the charge
> > distribution if the two plates are charged unequally. We tried looking at
> > potentials and thinking that the final potentials will tend to a minimum
> > but found it hard to write any meaningful expressions for that.
> >
> >
> > Any help would be appreciated.
> >
> > Ken Fox
> > Smoky Hill High School
> > 16100 Smoky Hill Rd
> > Aurora,CO 80015
> > 303-693-1700(w)
> > 303-850-7537(H)
> > kfox@stega.smoky.org