Re: quantum of electric flux?

[John Denker <jsd@MONMOUTH.COM>]

At 04:16 PM 3/10/00 -0500, Ludwik Kowalski wrote:
> I understand that Gauss's law leads you to the idea that
> the total E flux is quantized (because each part of Q is
> quantized).

OK.

> And I can visualize a loop on the surface
> pierced by the total flux. For example, a loop of one
> steradian on a sphere centered on a point charge.

OK, right, that's the picture I had in mind, too.

> But the idea that the total flux is quantized while any
> part of it is continuos (not quantized) is hard to
> swallow.

But still it is true;  see below.

>  It is like saying that the total charge in a
> room is quantized but in one half that room it is not.

That is an absurd statement as applied to charge, but charge is not the
same as flux.

> How can it be? Is it supposed to be a paradox?

It's not a paradox.  It's not even hard to visualize.  Consider Gauss's law
applied to a cubical box with a point charge moving around outside.  If the
box is just outside one face, there will be a large flux through that face
and smaller opposite-sign fluxes through the other faces.  As you move the
charge around you can redistribute the fluxes in lots of ways.  Everything
is nice and continuous.  But the total flux over the whole surface remains
zero.  Only when by moving the charge itself into the box can you change
the flux on the closed surface.  The only discontinuity occurs when the
charge approaches the surface, whereupon the flux calculation involves an
infinitely large field acting on an infinitely small area.

You may also find it helpful to consider a surface that is _almost_
closed.  Imagine cutting a tiny hole in the aforementioned cube.  Then the
flux through the leaky cube is almost always very close to one of the
canonical quantized values, but you can produce intermediate values by
moving the charge near the hole.