Re: funny capacitor

["Glenn A. Carlson" <gcarlson@MAIL.WIN.ORG>]

Isn't all of this just an exercise in numerically solving Laplace's
Equation with specified boundary conditions?

It's well known that the solution of Laplace's Equation is unique
(Statement 6a), continuous (Statement 6b), satisfies the mean-value
property (Statement 6c), and can be numerically solved using the relaxation
method (Statement 7).

Griffiths describes how to do the calculation ("Introduction to
Electrodynamics," 3rd ed., at page 112); as does Purcell, "Electricity and
Magnetism," 2nd ed., problem 3.30, p. 119 (cited by Griffiths).

I thank Mr. Kowalski for taking the time to write a program which performs
the calculations and saves us from having to reinvent the wheel (or, in my
case, to dig out my sophomore-level heat transfer notes).  I'll also assign
this problem in my scientific programming class since the mean-value
property is easier to understand (and, therefore, code) than the Laplacian.

The idea that the solution to Laplace's equation represent states of
"minimal potential energy" is more intriguing.  A variational calculation
should show this.  The potential energy density u is proportional to E^2 =
(grad V)^2, and integrating u over all space gives the total potential
energy U.  To find the extremal, set the variation of U equal to
zero.  Integration by parts would give you that grad.(grad V) = 0, which is
Laplace's equation.

An interesting modification to Mr. Kowalski's program would be to calculate
the total potential energy for the space at each iteration and show that it
does in fact approach a minimum.

I confess I did not read all the posts in this thread, which at times were
tediously pedantic.

Glenn A. Carlson, P.E.
gcarlson@mail.win.org


> Date: Sat, 17 Mar 2001 11:46:17 -0500
> From: Ludwik Kowalski <KowalskiL@MAIL.MONTCLAIR.EDU>
> Subject: Re: funny capacitor (URL)
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> The recent messages of the FUNNY CAPACITOR thread
> have been added to my website at:
>
> http://alpha.montclair.edu/~kowalskiL/funny/funny.html
>
> Why did I spend time in collecting them? Because at one time
> I was under the impression that a major misconception was
> discovered by JohnD. I wanted to document the process.
> But it was not a misconception. The problem does have
> "one and only one solution" when Vs are used as potential
> differences, for example, with respect to "infinity".
>
> P.S. The next message will be long. I will post a tutorial
> on the number crunching aspect of electrostatics; essentially
> a summary what I learned in the Funny Capacitor thread.
> Feel free to use the essay in any way you wish.
> Ludwik Kowalski
>
> Ludwik Kowalski wrote:
>
> 6) Here are important statements about problems of that nature.
> A mathematician would know how to justify them but we must
> take them on faith.
>
> a) The problem has a unique solution.
>
> b) The solution is such that potentials change gradually from one
> cell to another. For example, the potential in cell (8,9) is likely
> to be close to ?5, perhaps ?4 or ?3.5 while the potential in cell
> (11,9) is likely to be close to +9, perhaps 8 or 6.85. Keep in
> mind that potentials of cells occupied by conductors remain
> constant.
>
> c) In the electrostatic equilibrium the potential of any empty space
> cell is equal to the mean value of potentials in the four surrounding
> cells. This is the most useful property of the solution. The above
> statement can be illustrated with an example. The potential of cell
> (6,13) is one quarter of potentials in cells (5,13), (7,13), (6,12)
> and (6,14). Two of these four cells are inside the T object.
> Likewise, the potential in (17,6) is one quarter of the sum of
> potentials in cells (16,16), (16,18), (17,15) and (17,17). This
> time all neighboring cells belong to the empty space.
>
> 7) Equipped with this knowledge we can begin solving the
> problem. You must make a first guess about the solution is.
> We will make a highly unrealistic guess by assuming that all
> potentials in empty space are zeros. Then we will show that
> the guess is not acceptable because the statement c is not
> satisfied by all empty cells. We will be trying to make the
> statement valid by changing potentials in all cells. The
> acceptable solution will emerge gradually. In principle, it
> is possible that the initial guess corresponds to the exact
> solution but we will not count on this.