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Re: Confused by a derivation.



A boo-boo?
Shouldn't eq 6 read E1 = E3
They must both point in the same direction to give zero flux out of your
Gaussian cylinder.

Bob Sciamanda (W3NLV)
Physics, Edinboro Univ of PA (em)
trebor@velocity.net
http://www.velocity.net/~trebor
----- Original Message -----
From: "John Mallinckrodt" <ajmallinckro@CSUPOMONA.EDU>
To: <PHYS-L@lists.nau.edu>
Sent: Tuesday, February 05, 2002 1:52 PM
Subject: Re: Confused by a derivation.


I think Bob Sciamanda's argument involving the deformation of
concentric spheres has the germ of the final piece of the puzzle.
In other words, all that one needs to assume, beyond Gauss' law,
is

1 that we genuinely do have planar symmetry, i.e., there is no
fringing field (this is the key), and

2 that the charge on conductor 1 is equal and opposite to that on
conductor 2, and

3 that static equilibrium requires that there be zero electric
field inside a conductor.

Assumption 1 requires that there be a uniform surface charge
density on each surface. Label them s1, s2, s3, and s4 as shown
below:
region 1
--------------- s1
conductor A
--------------- s2 ^
|
region 2 | + direction
|
--------------- s3
conductor B
--------------- s4

region 3

Assumption 2 requires that

s1 + s2 = -(s3 + s4) [Eq. 1]

Finally using assumption 3 we can use a Gaussian cylinder
with its endcaps inside conductor A and region 1 to show that

E1 = +s1/eps_o [Eq. 2]

with its endcaps inside conductor B and region 3 to show that

E3 = -s4/eps_o [Eq. 3]
(note the choice of "up" as the positive direction)

with its endcaps inside conductor A and region 2 to show that

E2 = -s2/eps_o [Eq. 4]

with its endcaps inside conductor B and region 2 to show that

E2 = +s3/eps_o [Eq. 5]

and with its endcaps inside region 1 and region 3 to show that

E1 = -E3 [Eq. 6]

Now, equations 4 and 5 show that

s3 = -s2 [Eq. 7]

Equations 1 and 7 show that

s1 = -s4 [Eq. 8]

Equations 2, 3, and 8 show that

E1 = E3 [Eq. 9]

Equations 6 and 9 show that

E1 = E3 = 0 [Eq. 10]

and equations 2, 3, and 10 show that

s1 = s4 = 0

That is, all of the charge resides on the inner surfaces.

For real capacitors assumption 1 is violated and the result does
not hold although the violation is typically quite minor.

John Mallinckrodt mailto:ajm@csupomona.edu
Cal Poly Pomona http://www.csupomona.edu/~ajm