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Re: Calculating resistance



Regarding Bernard Cleyet's comment:

I'm not sure it's conformal transformation. He uses various complex functions
as solns. for Laplace's eq. log for cylinders linear for planes etc.

Then they most likely *are* conformal transformations that he uses.
Consider a complex analytic function w = f(z) where w = u + i*v and
z = x + i*y are complex variables (here we assume the variables x, y,
u, & v are all real). If we look at the real part of the function f,
i.e. g(x,y) == Re(f(z)) we see that it is a real-valued function of
the two real variables x & y. Similarly, if we look at the imaginary
part of g, i.e. h(x,y) == Im(f(z)) we see that it, too, is another
real-valued function of the real variables x & y. The complex map
w = f(z) when transcribed in terms of the real components describes a
coordinate transformation for the x-y plane. In particular we have
u = g(x,y) and v = h(x,y) as the new coordinates transformed to.
This coordinate transformation *is* conformal whenever the complex
function f(z) is analytic. By definition, a conformal transformation
is one for which the metric form ds^2 = dx^2 + dy^2 *locally*
preserves the geometry, i.e. when written in the new coordinates the
metric form has the structure ds^2 = A(u,v)*(du^2 + dv^2) where
A(u,v,) is some locally varying (but isotropic) dilatation scale
function. As long as f(z) is complex analytic then the coordinate
transformation it defines *is* conformal.

The reason that a conformal transformation is useful in solving
potential field problems is that if some function V(u,v) does satisfy
Laplace's equation in terms of the variables u & v, then the new
function V' defined by the substitution V'(x,y) == V(g(x,y),h(x,y))
*also* satisfies Laplace's equation in terms of other variables x & y
whenever the coordinate transformation between the sets of coordinates
is conformal.

The technique of conformal transformations implemented by an analytic
complex function relies on our ability to take a 2-d potential field
problem whose geometry is 'hard' in some sense (i.e. in the sense of
having difficult boundary conditions) and transforming it to another
potential field problem whose transformed boundary conditions are
easily dealt with. We solve the easy problem and then transform back
to our original problem by just substituting the original coordinates
into our solution for the easy problem. This substituted solution
then solves our original 'hard' problem.

Maybe my understanding of the math. is rather weak, but it does seem more
direct than knowing a little known plane geometry fact, and appeal is also made
(note passive voice) to the log character of the potential.

Well, the logarithmic nature of the potential due to a point charge
in 2-dimensions is pretty commonly known, even though the fact that a
circle is the locus of points in a plane that have a constant
quotient of distances from 2 fixed points to the locus is not so
commonly known. For some reason (of course I'm biased) it seems to
me that it is easier to construct the solution following the method I
proposed (assuming one already knew about the fact that of circle
being defined by a quotient of distances) than it is to try to divine
the correct magic analytic conformal transformation that solves the
problem.

BTW, does anyone out there wish to try to characterize the locus in a
plane defined by a fixed *product* of distances to a pair of fixed
points?

They both appear
from ???, in one case a soln. try of an eq. known basically to describe
electrostatics and the other from geometry.

Everybody has their own preferences. There is not much accounting
for taste.

I agree it's difficult method, but not necessarily of limited applicability, as
I suspect a computer could iteratively solve for irregular shapes beginning
with one of high symmetry. (a la JD)

It's not so much that the method (of complex functions/analytic
functions) is so difficult as that it relies deeply on a lot of
seemingly unmotivated prior insight to be able to pick the right
magic transformation function that miraculously solves the
problem. But that sort of thing kind of goes with the territory
when doing exact solutions by other methods too (as my method
sort of illustrates).

I have never heard of anyone using complex functions/conformal
transformations as part of an iterative numerical approximation
scheme for realistic irregular boundary condition problems, & don't
know just how it would/could be implemented productively. That
doesn't mean the method can't be used in some way for such a scheme
though. Usually such 'real life' problems are just solved 'brute
force' via some sort of direct numerical solution of the discrete
finite difference equations generated from the actual PDE when a
discrete grid/tessellation is imposed on the problem, and one of
the methods of either finite differences or finite elements is
used.

bc

P.s. DB left to the reader the math. manipulation at the end. I wonder, if as
in so many texts, this is the hardest part.

That sort of depends on what your definition of what 'hard' is (sorry
to sound so Clintonesque). The part left to the reader is not
conceptually or intellectually all that hard. It is just some
relatively straightforward variable matching. The conceptually hard
part was to come up with a way to contract the distributed charge on
the silver dots down to points in a way that preserved the exterior
potential function and allowed the superposition principle to be used
with the concentrated charges. But the tedious algebra involved in
solving the remaining matching equations is at least as hard (& maybe
a little harder) than is required to find the properties, e.g. center
and radius, of the circle locus for a given fixed K-ratio value.

David Bowman
David_Bowman@georgetowncollege.edu