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Re: IONS metals/dielectrics



Bob wrote:

Apropos' to this discussion (specifically the Earnshaw theorem),
I always include the following problem in any E&M course:

A point charge Q1 = ___coulombs and a second point charge
Q2 =___coulombs are separated by ___ meters. (Fill in any
numbers and signs [of charges] - there will be a solution.)

It is desired to add a third point charge so that the three charge
system will be in equilibrium under only electrostatic forces -
ie; the Coulomb force on each charge (due to the other two) is
to be zero.

1) Where should this third charge be placed?
2) What sign should this third charge be?
3) Calculate the required magnitude of this third charge.
4) Is this system in stable equilibrium? Prove your answer.

The theorem was clearly described by Bob in another message.
It states (in brief): --> No stability for a probe charge when
ONLY electric forces are involved.

Consider two very small identically charged spheres firmly
mounted on plexiglas columns (think of them as two
point charges, if you wish). The electric field created is zero
at the midpoint C, between the two sources. This is an
unstable equilibrium; E is not zero at points surrounding the
center. But suppose we have a very large number of identical
negative sources uniformly distributed over a sphere (or a
continuos distribution).

In that case E is zero not only at the point C but also at points
surrounding C. The probe charge will be at stable (neutral)
equilibrium. Can we say that the Earnshaw's Theorem is
contradicted? A defender of the theorem could say that
sources, creating the field, repel each other and that
mechanical (rather than electric ONLY) forces are involved
to keep them in place. The stability of the point charge does
not contradict the theorem which insists on "ELECTRIC
ONLY".

A critic of the theorem, however, could say that the so-called
"mechanical" forces are ultimately electric in nature. The
probe charge is stable under the influence of electric forces
ONLY; neither nuclear nor gravitational forces play a
significant role. According to him the "experimentally
observed" stability of the probe charge contradicts the
theorem. Who is right and who is wrong?

I know it is silly to argue about something that was
already clarified (a Maxwellian F is not the same thing
as an electric F). But I can not resist.

Let us consider an ideal student in an introductory physics
course. He learned mechanics in the first semester and he
is learning electricity for the first time now. He knows
that a net force must be zero when a particle is at rest. The
textbook says that like charges repel. It has a picture which
shows how a net charge is distributed on the surface of a
metallic sphere. Therefore, the student would conclude,
some attractive forces must exist to compensate for the
repulsive Coulomb forces. The surface must attract
charges as soon as they come out of metal. What is the
nature of this attraction?

How to deal with this hypothetical situation? It is a
pedagogical issue. I have no problem in pretending that
Galilean kinematics is exact, but I will now face a problem
of pretending that electrostatics I teach is logically
consistent with mechanics. Galilean kinematics is at least
approximately correct in common situations. Something
is missing in our ways of introducing e&m. What is it?



Ludwik Kowalski