Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

Re: Confused by a derivation.



I said:
But Gauss' law, by its lonesome,
does not require that fields inside conductors be zero, it is only
concerned with balancing the net flux output from a region with the
charge
inside that region.

More to the point, I think the crux is that Gauss' law does not know the
superposition principle, so that it can be ineffectual to tell it that the
net field inside a conductor must be zero. It cannot make full use of
that knowledge without explicit help from the superposition principle.

Bob Sciamanda (W3NLV)
Physics, Edinboro Univ of PA (em)
trebor@velocity.net
http://www.velocity.net/~trebor
----- Original Message -----
From: "Bob Sciamanda" <trebor@VELOCITY.NET>
To: <PHYS-L@lists.nau.edu>
Sent: Wednesday, February 06, 2002 1:35 PM
Subject: Re: Confused by a derivation.


Carl,
This is a good synopsis of where we stand. Forcing the superposition of
the fields of the four charge layers to cancel to zero inside both
conductors forces the unique solution. But Gauss' law, by its lonesome,
does not require that fields inside conductors be zero, it is only
concerned with balancing the net flux output from a region with the
charge
inside that region. It is not as complete a rendering of electrostatics
as is the superposition principle. Coulomb's law plus the superposition
principle is the complete basis of all of electrostatics. Gauss' law is
only a subset of this information.
Bob Sciamanda (W3NLV)
Physics, Edinboro Univ of PA (em)
trebor@velocity.net
http://www.velocity.net/~trebor
----- Original Message -----
From: "Carl E. Mungan" <mungan@USNA.EDU>
To: <PHYS-L@lists.nau.edu>
Sent: Wednesday, February 06, 2002 12:21 PM
Subject: Re: Confused by a derivation.


Let me take a crack at this. Do I understand that the problem is the
following:

Put +Q on one plate, -Q on the other plate of a standard
parallel-plate capacitor. Consider the following two possible charge
and field distributions.

1. The charges on each plate are split between the two sides of the
plates:

^ ^ ^ ^ ^ ^ ^ ^ ^ ^
| | | | | | | | | |
+ + + + + + + + + +
====================
+ + + + + + + + + +
| | | | | | | | | |
v v v v v v v v v v
- - - - - - - - - -
====================
- - - - - - - - - -
| | | | | | | | | |
v v v v v v v v v v

2. The charges on each plate go entirely to the inner facing surfaces:

====================
++++++++++++++++++++
||||||||||||||||||||
vvvvvvvvvvvvvvvvvvvv
--------------------
====================

Which of these two configurations will actually occur? (Or perhaps
are both possible?)

We know the field due to a single infinite sheet of charge. So we can
superpose it for 4 sheets of charge. Doing this proves configuration
1 *cannot* occur because it leads to nonzero fields inside the metal
of the plates. Please check this for yourself.

Fine. But now the question is: Does configuration 1 violate Gauss'
law? Answer: No. No matter how you position the endcaps of your
Gaussian surfaces (in the metal or in the vacuum), Gauss' law is
satisfied. Again, please check this for yourself.

So if we only knew Gauss' law, we could not disprove configuration 1.
Nevertheless we know it cannot occur. So what is wrong with it then?
Answer: it is energetically too costly. The field energy in each
region is proportional to E^2. Configuration 1 would require
infinitely more energy than configuration 2.

If we think about each side as a separate plate, we have 4 plates in
all. This is a set of series capacitors and there is a unique
solution to where the charges go, imposed by requiring the energy to
be a minimum. Gauss' law alone will not tell you this. You guys
taught me about this earlier:



http://physics.usna.edu/physics/faculty/mungan/Scholarship/Capacitors.html
--
Carl E. Mungan, Asst. Prof. of Physics 410-293-6680 (O) -3729 (F)
U.S. Naval Academy, Stop 9C, Annapolis, MD 21402-5026
mungan@usna.edu http://physics.usna.edu/physics/faculty/mungan/