Re: On 1/4*Pi in Coulomb's law

[John Denker <jsd@MONMOUTH.COM>]

At 07:51 PM 1/20/01 -0600, Doug Craigen wrote:
>  So why in this case do we have a constant and a 4*pi?  Its because the
> electric field is so prevalent as the quantity to study in
> electrodynamics, and electric fields find their source in charges via
> Gauss' Law.

I think Doug touched on the key point, but this statement was buried in a
discussion of non-key points.

Similarly, at 04:55 PM 1/20/01 -0500, David Bowman hit the nail on the head
by saying:
> It all depends on *which version* (the local differential versions or the
> integrated bulk versions) of the laws of electromagnetism whose
> appearance you are most interested in simplifying.

Since some people still haven't fully got the message, let me pound on this
nail one more time.  Consider the mathematical identity:
        Laplacian(1/r) = 4 pi delta(r)                  (1)

And consider the relevance of this identity to electromagnetism, where we
might want a point charge to be a delta-distribution of charge density, and
we might want it to be the source term for a potential that falls of like 1/r.

That's right, folks:
 -- EITHER you can have a "nice" 1/r potential
    in which case the differential equations have a 4 pi in them
 -- OR you can have a "nice" unit-strength delta-distribution of charge
    in which case the integral equations have a 1/(4 pi) in them
but you can't have both.  You can't remove the 4 pi from both sides of
equation (1) above.  You just can't.

This is not even physics.  It's mathematics.

That 4 pi is not in there for some "historical" reason.  It is not in there
because Coulomb and Maxwell and Stokes and Green made some "mistake".  It
cannot be removed in any meaningful sense by fiddling with the units of
measurement (although one can play hot potato with it, shifting it from one
place to another in the system of equations).