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Re: Calculating resistance



Regarding the exchange between Bernard C. & John D.:
...
P.s. Harnwell uses complex variables to solve the two dimensional problem
(only applicable when the third is infinite in length).

Yeah. Students are often not too pleased with that
solution. And they've got a point. Learning that
method of solution in this context is IMHO just an
advanced form of rote learning. It's nearly impossible
to connect it with anything else they know. And
there are only a handful of problems that yield to
that technique, so it's not clear what the point is.

Students think the "method" here is to pull a
hypothetical solution out of thin air (or somewhere
else) and then verify it. That hardly counts as
a "method". And explaining where this solution
really comes from would take a week.

I have never been much of a fan of the method of conformal
transformations either. I have never seen a problem that it solved
that couldn't also be solved more directly by some other more
constructive method. (Also it is pretty much stuck in only 2
dimensions.)

When solving the recent problem of the 2 circular disks/parallel
cylinders I did not make use of conformal transformations at all.
In fact, I simply used a modified version of the superposition method
that John Denker had earlier advocated, but one designed to respect
the appropriate boundary conditions. I was inspired by a little known
fact of plane geometry pertaining to circles. An outline of the
solution method is given below.

It is very well known by nearly every high school student who passed
a HS analytic geometry course that an ellipse is the locus of points
in a plane such that the sum of the distances from two fixed points
in the plane to any point on the locus is a constant. Similarly, it
is just as widely known that a hyperbola is the locus of points in a
plane such that the difference in the distances from two fixed points
in the plane to any point on the locus is a constant.

Now consider the locus of points in a plane such that the *quotient*
of the distances between two fixed points to a point on the locus is
a constant. What plane figure does this describe? The answer seems
to be not nearly as widely known as for the other cases mentioned.
But the answer is nevertheless very simple. The figure described is
a *circle* that surrounds one of the fixed points ('foci') but not
the other one. Just which 'focus' is surrounded depends on the
value of the fixed ratio of the distances to the 'foci'. The 'focus'
surrounded is the one with the shortest distance to that focus for
the locus. For instance let K = r_A / r_B where K is a constant
positive value, r_A is the distance from 'focus' A to the locus
point, and r_B is the distance from 'focus' B to the locus point
where A & B are just two arbitrary labels distinguishing the two
'foci'. If K > 1 then the circle surrounds 'focus' B. If K < 1 then
the circle surrounds 'focus' A. Let R be the distance between the
two fixed 'foci'.

It is important to realize that the surrounded 'focus' is *not* at
the center of the circle. The location of the circle's center is on
the line connecting the foci, but it is shifted a distance equal to
R/(max(K^2,1/K^2) - 1) from the surrounded focus in the direction on
the connecting line that is *away* from the non-surrounded 'focus'.
The radius of the circle is R/|K - 1/K|. In the limit of K --> 1
the circle's center shifts by an infinite distance, and its radius
also becomes infinite, resulting in a circle which opens up into
the straight line which is the perpendicular bisecting line that
bisects the line segment connecting the 'foci'. In the limit
K --> 0 the circle shrinks to zero size right on top of 'focus' A.
In the limit K --> [infinity] the circle also shrinks to a point, but
this time it shrinks to the point on top of 'focus' B. For each K
value larger than 1 the corresponding family of circles surrounding
'focus' B are all nested but *not* concentric. Similarly, for each
K value smaller than 1 the corresponding family of circles
surrounding 'focus' A are also all nested but *not* concentric
either. In no case does a circle defined by any value of K ever
intersect any other circle defined by any other value of K.

What does any of this stuff have to do with the problem of circular
silver dots or parallel conducting cylinders? Good question. All
will become clear shortly.

Now consider the simple electrostatics problem of a point charge in a
2-dimensional world or a uniformly charged (infinitely long) line in
3 dimensions. From Gauss' law it is simple to conclude that the
E field from such a charge distribution is inversely proportional to
the distance from the (2-d point/3-d line) charge. This means that
the potential due to this charge is *logarithmic* in the distance
from it. Now consider a second such charged point/line which happens
to be oppositely charged as the first one. The total potential from
both charge distributions is simply the superposition of the
potential from each one separately. Let's place one of the charges
on 'focus' A (from our previous discussion) and place the opposite
charge on 'focus' B. The potential from the sum of these charges
has the superposed structure of: V = V_0 - C*(ln(r_A) - ln(r_B)).
Here V_0 is just some convenient zero level for the potential,
and C is proportional to the magnitude of the charge (or charge per
unit length in 3-d) on each focus. In particular,
C = Q'/(2*[pi]*[epsilon]_0) where Q' is the 2-d charge or the
3-d charge per unit length for the charge on 'focus' A. 'Focus' B
is correspondingly charged to -Q'.

Now since the "difference of the logarithms is the logarithm of the
quotient" we can re-write the potential as V = V_0 - C*ln(r_A / r_B).
Notice that an equipotential locus (surface in 3-d) is a
locus/surface of constant r_A / r_B. In particular the K value of
our previous discussion is simply related to the potential by
K = exp(-(V - V_0)/C) = r_A /r_B. This means that our previously
discussed families of nested circles are the equipotential sets
(i.e. loci/surfaces) for our current electrostatics problem. Since
each such circle defined by a given K value is an equipotential set
we can have the problem give the exact same potential solution for a
modified problem where we place all the +Q' charge (that was formerly
on 'focus' A) on some conducting equipotential circle (cylinder in
3-d) that surrounds 'focus' A, and place all the -Q' charge (that was
formerly on 'focus' B) on some other conducting equipotential circle
(cylinder in 3-d) surrounding 'focus' B. As far as the world outside
these conducting circles(/cylinders) is concerned, it can't tell
whether the charges are spread out on the circular conductors or
whether all the charge is concentrated on the 'foci'.

We thus have the problem essentially solved. All we have to do now
is solve some matching conditions that relate the distance R between
the 'foci' to the distance L between the centers of the conducting
circles, relate the potentials (K-values) on the conducting circles
to their radii and charge, etc. The results are the formulae I
mentioned in my earlier post.

David Bowman
David_Bowman@georgetowncollege.edu