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Earnshaw in brief



A repost of a Sat 9/26 post (lost):
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A home-brew proof of Earnshaw's Theorem (actually a corollary thereof):
If there were a point in empty space (no charge there) where a positive
test charge would be in stable equilibrium under only electrostatic
forces, the electric field would have to be zero at that point, and at
every point on the surface of a neighboring sphere (centered on that
point) the field would have to point inward toward that point. (outward
if you use a negative test charge). Since E lines are continuous and
only start and terminate on charges (Div E(r) = 0 in charge free space),
this is impossible. QED

I believe Earnshaw's original statement concerns the average potential
over the surface of a sphere Vav :

Vav = Q/(4*Pi*epsilon0*r) + Constant, where Q is the charge enclosed
within the sphere and r is the radius of the sphere.

Corollaries of this (as the above statement) are often also called
Earnshaw's theorem.

-Bob

Bob Sciamanda
Physics, Edinboro Univ of PA (ret)
trebor@velocity.net
http://www.velocity.net/~trebor