Re: Electrostatic shielding [long!]

[John Mallinckrodt <ajmallinckro@CSUPOMONA.EDU>]

On Thu, 1 Feb 2001, Bob Sciamanda wrote:

> You have shown that IF the imposed charge and the inner surface charge are
> to effect the required E=0 without the help  of the outer surface charge,
> then there is a unique solution, etc.  But this begs the very question:
>
> The raw physics only requires that the E=0 condition (in conducting
> material) be wrought by the superposition of the effects of ALL charges...
> it needs to be proven that the combination of the imposed charge and the
> inner surface charge must ALONE effect the required E=0 result (without
> taking into account the effects of the outer surface charge) .

But again, "existence-uniqueness" to the rescue!

For a given total charge on the outer surface, there is one and
only one arrangement of charges that produces E = 0 everywhere
inside the outer surface.  Also, for a given charge distribution
within the cavity, there is one and only one arrangement of
charges on the inner surface that produces E = 0 everywhere
outside the inner surface.  The superposition of these two
solutions satisfies all boundary conditions and is, therefore, THE
solution.  (BTW, I should have appealed to the linearity of
Laplace's equation to support the superposition principle.)

If I move the charges inside the cavity (without altering the net
charge) I will have to rearrange the charges on the inner surface
(without altering their net charge either), but since that *new*
distribution on the inner surface and the *same old* distribution
on the outer surface satisfies all boundary conditions it is,
therefore, THE solution.  In other words, the existence-uniqueness
theorem tells us that, since nothing NEEDS to change on the outer
surface, nothing CAN change.

John Mallinckrodt      mailto:ajm@csupomona.edu
Cal Poly Pomona          http://www.csupomona.edu/~ajm