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On Thu, 1 Feb 2001, Bob Sciamanda wrote:charge
One must prove that, regardless of the geometry, the imposed fixed
theand the inside surface charge together produce a net E=0 throughout
surfaceconducting material - **without the help or hindrance of the outside
takecharge**. If this is so, then the outer surface charge must always
yetthat unique configuration which, by itself, contributes E=0 in the
conducting material.
But the first statement must be proven, not merely stated. I have not
seen a rigorous proof of that statement - though I believe it is so.
I think Bob's question is at the very heart of the confusion here.
Proving this isn't easy! Fortunately it's already been done for
us in the form of the "existence-uniqueness theorem" that applies
to solutions of Laplace's equation.
It helps to forget about the hollow conductor qua "conductor" and
simply consider its two surfaces ("inner" and "outer") in the
light of the answers to the following questions:
1 Given a static, but otherwise arbitrary localized distribution
of charges and the boundary condition that the electrostatic
potential be zero at every point on an arbitrary closed surface
that completely encloses the given charge distribution, what is
the required charge distribution on that surface?
The existence-uniqueness theorem guarantees that there *is* one
and *only* one surface charge distribution that meets the boundary
condition.
2 What is the electric field outside the surface considered in
question 1?
Since the solution "V = 0 everywhere outside that surface"
satisfies both Laplace's equation and the boundary condition, the
existence-uniqueness theorem guarantees that it is *the* solution.
(And Gauss' Law now tells us that the total charge *on* the
surface must be equal and opposite to the total given charge
residing *within* that surface.)
3 (New question) Given an amount of charge to distribute on a
closed surface and the requirement that that surface be an
equipotential how does it distribute itself?
The existence-uniqueness theorem guarantees that there *is* one
and *only* one surface charge distribution that meets the boundary
condition.
4 What is the electric field inside the surface considered in
question 1?
Since the solution "V = the potential of the surface everywhere
inside that surface" satisfies both Laplace's equation and the
boundary condition, the existence-uniqueness theorem guarantees
that it is *the* solution. (Therefore, there is no electric field
inside the surface.)
Now put these general answers together to see what happens in the
case of the hollow conductor.
If you specify 1) an arbitrary charge distribution within the
cavity of the conductor and 2) an arbitrary total charge on the
conductor, then: The charge on the inner surface will be equal and
opposite to the net charge inside the cavity, will arrange itself
as specified by the answer to question 1, and will produce no
field outside as indicated by the answer to question 2. The
charge on the outer surface will be whatever it needs to be to
insure the correct total charge on the conductor, will arrange
itself as specified by the answer to question 3, and will produce
no field inside as indicated by the answer to question 4.
Furthermore, since the *amount* of charge on the outer surface
does not change when the charges inside the cavity are moved
around (without changing their net charge), and since we *can*
satisfy the boundary conditions by rearranging *only* the charge
on the *inner* surface, the existence uniqueness theorem insures
that that *will be* what happens. There is no alteration of any
kind beyond the inner surface.
Similar arguments apply to the situation in which the conductor is
"grounded" (rather than specifying its total charge.) We see
that, as long as the *net* charge inside the cavity is fixed, it
may be arbitrarily rearranged with no need to alter the charge on
the outer surface. Therefore, no charge will flow to or from the
"ground."
Conclusion: The *only* way to affect the charge on the outer
surface or the fields outside the conductor is to change the *net*
charge inside the cavity.
John Mallinckrodt mailto:ajm@csupomona.edu
Cal Poly Pomona http://www.csupomona.edu/~ajm