Re: 1/r^2, geometry, flux

[John Denker <jsd@AV8N.COM>]

Hugh Haskell wrote:
> I was disabused of this geometrical idea early on in my college
> career when it was pointed out to me that there are all sorts of
> interactions that do *not* follow the inverse square law. Since the
> derivation of Gauss's law explicitly depends on the geometric
> argument, it is easy to show that it only applies to those
> interactions that happen to obey the inverse square law. It's not the
> other way round.

I'm not convinced.  I suspect that like most "derivations"
in science, this argument runs just fine in either direction.

I don't claim that the geometric approach is the only valid
approach, but it appears to be one of the valid approaches,
not requiring any disabusing.  It has elegance and predictive
power.

1) There exists a thing called Gauss's Divergence Theorem:
  http://mathworld.wolfram.com/DivergenceTheorem.html
This exists as a mathematical theorem, quite independent
of any application to electromagnetism, and in particular
independent of Gauss's Flux Law:
  http://hyperphysics.phy-astr.gsu.edu/hbase/electric/gaulaw.html

2) In the usual formulation(s) of the Maxwell equations, e.g.
  http://scienceworld.wolfram.com/physics/MaxwellEquations.html
there is no explicit 1/r^2 dependence.

3) Therefore, if you take the Maxwell equations as a starting
point (which is not the only valid way of doing business, but
is certainly a reasonable option) then you can apply Gauss's
Divergence Theorem
 -- to derive Gauss's Flux Law
    (given any number of spacelike dimensions), and
 -- to derive the 1/r^2 dependence
    (given exactly three spacelike dimensions).

So, given this starting point, Gauss's Flux Law is nicely
independent of the geometry, and indeed independent of the
1/r^2 law, in the following sense:

There is a natural and elegant generalization of the Maxwell
equations that is not restricted to exactly three spacelike
dimensions ... just get rid of the &@#+ cross products and
use wedge products instead.  We then find that the flux
density falls off like
 ++ 1/r   in flatland, i.e. two spacelike dimensions,
 ++ 1/r^2 in familiar space, i.e. three spacelike dimensions,
 ++ 1/r^3 given four spacelike dimensions,
 ++ and so forth.

This approach is useful e.g. if you are trying to figure
out what string theory has to say about electromagnetism.

Bottom line:  The sword cuts both ways:
 a) You can perfectly well start with the 1/r^2 dependence
  in D=3 and derive the Gauss Flux Law, plus the conservation
  of flux lines in charge-free regions; ... and/or ...
 b) You can perfectly well start with the Maxwell equations
  and derive the Gauss Flux Law ... and thence (in charge-free
  regions) the conservation of flux lines and the 1/r^2
  dependence in D=3.

There are plenty of books that take the 1/r^2 dependence as
their starting point ... and there are also plenty of books
(e.g. Feynman volume II) that take the Maxwell equations as
their starting point.