D=2 versus cylinders versus dots on paper

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

Ludwik Kowalski wrote:
> Why is it so?
>
> A standard student experiments with Pasco sheets is to
> trace equipotential lines. But how well do these lines
> agree with the theory? And which theory should they
> agree with? I see two options:
>
> a) Filed is like that of a dipole (two spheres in 3-D)
> b) Field is like that of two long cylinders (dots are
>     the cross sections of these cylinders).
>
> The second theory can be objected on the ground that
> there are no long cylinders in this setup.
>
> The first can be objected because the medium in which
> the current is flowing is 2-dimentional.

Similarly Bernard Cleyet wrote:
> It's not intuitive to me that confining the conductivity to a sheet is the
> same as using guard cylinders.

So......  Here's the deal.............

We have two or three different physical situations
that turn out to be described by the same
equations.  We can then apply the rule:  The same
equations have the same solutions.  This rule applies
even when the physics is radically different.

The equation is
  (d/dx)^2 (something) + (d/dy)^2 (something) = (sources)
and we are asked to explain why there is no (d/dz)^2 contribution.

The explanations are as follows:

1) For dots on paper, which is a D=2 piece of paper
embedded in a D=3 world, the (d/dz)^2 contribution
vanishes by symmetry, namely the z -> -z mirror
symmetry, reflection in the xy plane.

2) For long cylinders in real D=3 world, the (d/dz)^2
contribution vanishes by symmetry also.  The argument
is actually the same as in case (1), but students
often don't recognize it as such.  Students are
quick to notice that the cylinders have _translational_
symmetry up and down the z axis -- but that doesn't
directly serve the purpose.  What we really need is
this:  the symmetry group of the cylinder has a
subgroup, namely reflection in the xy plane.  (Indeed
it has many reflection subgroups, reflection in any
plane parallel to the xy plane, but the xy plane itself
is the one we really need.)

To repeat:  The cylinder (case 2) has the same reflection
symmetry as the plane (case 1), plus some other symmetries
that are just a distraction.

To make this brutally explicit:  The symmetry implies
that (d/dz) is equal to (-d/dz) and the only way
something can be equal to its negative is if it is zero.

3) The third case is flatland, a genuine D=2 universe.
In that case there is no (d/dz)^2 contribution because
there is no z axis at all.

===============================

Bottom line:

 -- Often people refer to a D=2 piece of paper embedded
in the D=3 world as a two-dimensional problem.
 -- Often people refer to cylinders embedded in the D=3
world as a two-dimensional problem.

The physics looks different.  The physics _is_ different.
But the same equations have the same solutions.

OK?