edgard (02/03/83)
We are in search for the solution of the following problem:
What is the potential distribution on the surface of a homogeneous
and isotropic conducting cylinder containing one point current
source and one point current sink (both of the same time in-
variant magnitude)?
The cylinder is surrounded by an insulating medium.
We have formulated this problem in cylindrical coordinates (r,fi,z).
In its simplest form (sink at the origin/source on the symmetry axis
/infinite cylinder), the mathematical expression becomes:
-Basic differential equation:
LAPLACIAN(p)=constant*(delta(x'-1'z*a)-delta(x')) (Poisson problem)
(x' and 1'z are vector quantities; 1'z=unit vector in z-direction;
delta is a Dirac function)
delta(x') can be written as (delta(r)*delta(z))/(pi*r) and
delta(x'-1'z*a) = (delta(r)*delta(z-a))/(pi*r).
(no delta(fi)-factor because of axial symmetry)
-Boundary conditions:
p=0 at infinity and dp |
-- | = 0 (Neumann type boundary condition)
dr |
|r=R
d
(R=radius of cylinder; -- = partial derivative )
dr
All attempts to compute a plausible solution for our problem failed.
(Our method was based on "Classical Electrodynamics" of J.D.Jackson
-the most complete work we know for do-it-yourself potential problem
solving.
We think that the weak point in our strategy was the decomposition
of delta(r)/r into a Fourier-Bessel series since this function is
very badly behaved for r=0)
In the literature one can almost only find solutions to problems
in spherical coordinates or with Dirichlet boundary conditions
(p=p(r,fi,z) implied at the boundaries).
Did someone resolve a similar problem yet?
Does anyone know any literature, describing the solution of this
problem or describing a solution strategy for this class of problems?
Please mail solutions and/or references to:
..!philabs!mcvax!vub!edgard or:
Edgard NYSSEN
Brussels Free University (VUB)
Fac. of Medicine - unit HART
Laarbeeklaan 103
1090 Brussel
BELGIUM