Re: funny capacitor (EXCEL)

["John S. Denker" <jsd@MONMOUTH.COM>]

At 01:43 PM 3/25/01 -0500, Ludwik Kowalski wrote:
> I learned how to use Excel 97 to solve the Q(V) problems
> (see below). .... IT AVAILABLE EVERYWHERE AND
> FOR THAT REASON IT IS A GOOD IDEA TO USE IT IN TEACHING.

That's one very practical reason.

Another reason is that there are huuuuge numbers of people who are
intimidated by standard computer languages but not so intimidated by
spreadsheets.  Therefore one could argue that there are
pedagogical/psychological advantages to using spreadsheets for some
purposes, especially in a course that is nominally devoted to physics not
computer technology.

These pedagogical/psychological factors would be an interesting topic for
discussion some day.

> At the end of iterations I have potentials
> displayed in all cells (with respect to my rectangular enclosure).
> What I would like to do, but do not know how, is use colors
> to identify potentials

Look under Formatting :: Conditional Formatting.


> 1) My "universe" consists of cells from A1 to Z26. Why not?

Well, actually there are some pretty good reasons why not.  One
manifestation of the problem can be seen when trying to evaluate the charge
density using the Laplacian operator (or the equivalent).  The value of the
Laplacian at a location at the edge of the universe depends on the value of
that cell and neighboring cells in all four directions.  Therefore it
appears we have three choices:
  -- don't evaluate the Laplacian for cells near the edge
     (which means there could well be charge hiding in places
      that we can't evaluate)
  -- use the "plain" Laplacian formula in most of the universe,
     but use a "special-case" Laplacian formula near the edges
     (which is error-prone, hard to explain, and inelegant)
  -- move the universe away from the edge of the spreadsheet and
     put some sensible boundary conditions in the cells "just outside
     the universe" along each of the four edges.

I also like to leave at least one row and one column available for labels
et cetera.

It is easy to move the universe so that it starts at C3 rather than A1.  So
let's do that.  Cut and paste.

Now have a 26x26 universe that extends from C3 to AB28 inclusive.  I
recommend the following boundary condition:
  B3  := AB3 (and fill down)
  AC3 := C3  (and fill down)
  C2  := C28 (and fill across)
  C29 := C3  (and fill across)
I suggest formatting these cells in a distinctive style to help people
understand that they are not part of the main universe.

This implements Born / von Karman periodic boundary conditions.  Can you
prove that this scheme upholds the following properties?
 -- Gauge invariance
 -- Global charge neutrality
 -- Local charge conservation

These boundary conditions permit the Laplacian to be expressed everywhere
by a regular formula that is easy to understand and easy to prove correct.


> 2) I begin by populating all cells with zeros.
>
> 3) The formula =(B1+B3+A1+A3)/4 is entered into cell B2

This implicitly creates a grounded enclosure in the cells at the edge (i.e.
just inside the edge) of the universe.

Students should be _permitted_ to use a grounded enclosure but not
_required_ to do so.

In particular it is instructive to see what happens using an enclosure that
consists of a few narrow widely-spaced bars.

Using proper boundary conditions, the case of a perforated enclosure (or no
enclosure at all) can be handled with ease.  It just works, automatically,
naturally.  Without proper boundary conditions, it's a nightmare.


> Suppose I am interested in distribution of potentials
> between two plates of unequal size, one at V=0 (left)
> and one at  V=99 (right).

This asymmetry guarantees there will be nontrivial charge on the
enclosure.  This is why it is good to design the layout to permit
calculation of the charge on the enclosure.