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

[Ludwik Kowalski <KowalskiL@MAIL.MONTCLAIR.EDU>]

Let me say that these good observations would not be
meaningful to students in the first physics course who
do not know what the laplacian or the delta function is.
The term differential equation is also meaningless to
them. Some teachers of elementary physics, including
myself, are also not fluent in using advanced mathematics.

I still claim that unnecessary complications, directly
connected with SI, are real in physics at the introductory
level. There are at least two issues here:

1) Identifying complications and agreeing that they are
real rather than imagined.

2) Finding a solution (or solutions) helping to improve
the situation.

If I were younger I would probably try to develop a
better sequence. Like in mechanics, new topics would
always depend on topics already covered, not on
topics to be covered later. I am aware that this is not
an easy task. Closing our eyes, however, is not the
best possible attitude. I hope to see more comments.
Are the difficulties real or am I imagining them? Is
it worth complaining or not? Is it worth to discuss
attempts to improve the situation or not?
Ludwik Kowalski

John Denker wrote:

> 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).