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: Calculating Fields etc - Another Approach

Good morning Leigh:

Your pedagogical approach (algorithm) is fine with me.
Why are doing this?

1) Something drives us "to finish what was started".

2) We want to convince ourselves that the numerical and
analytical solutions are in reasonable agreement.

3) We know that our object is a cylinder, rather than an
ellipsoid of revolution. So we are not solving the same
problem as David, except, in the limit of the "flat disk"

4) After satisfying ourselves we want to share what we
learned. Phys-L is one place; we can also write an article
for The Physics Teacher, for example. But it is too early
to discuss this aspect; we still did not compare our C,
and our S(r) with those of David.

Do you agree that we have a common purpose, at
least in general terms? If not then try to reformulate
the goals; I am improvising now.

Two methods of averaging (your and John's) give
different gradients under identical conditions. Therefore,
at least one of them must be wrong. Unless you object,
I would like to make this statement on phys-L today.
On the other hand, it may be better to wait with this till
the step 4 of our project is reached. What do you think?

Leigh Palmer wrote:

I had a different objective in mind when I took up the task of
calculating a potential field on a spreadsheet. I wanted a
pedagogically useful algorithm which exemplifies the geometrical
meaning of Laplace's equation in the most transparent way
possible. I did not intend to discover the most rapidly
convergent algorithm for calculating the field.

The geometrical meaning of Laplace's equation (div grad V = 0) is
that in free space the potential V at any point is equal to the
average potential on the surface of any sphere centered on that
point. When we partition space into cells we can take crude
averages in the manner described earlier. The problem is simple
in the case of Cartesian coordinates, but symmetrical problems
can be approached more easily by exploiting the symmetry. (No one
has yet suggested the one-dimensional case, which corresponds to
spherical symmetry, but that, too is worthy of consideration.) I
have been trying to construct my algorithm with the pedagogical
goal in mind. In doing a practical calculation in which the sole
object is to find the answer, of course, other methods are to be