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

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

Regarding Ludwik's comments of yesterday 19 JAN 01:
> ...
>          The mechanical units used to be dyne and
> centimeter but nothing would prevent us from using the same
> approach with newton and meter. The corresponding unit of
> charge would be defined by the amount of Q needed to make
> F=1 N when r=1 m (assuming Q1=Q2=Q). It is not difficult
> to see that the new unit of charge would be equal to 94868 C.
> This is nearly the same as the unit of charge still used by
> electro-chemists (* see the footnote at the end).

This is not correct. One of these new units of charge would be

1.054822286... x 10^(-5) C = (1/94802.69926...) C =

= sqrt(10000000)/299792458 C (exact) .

> ...
> 3) I do not like the "rationalization idea" of Heaviside
> because it leads to two undesirable consequences:
>
> (a) Pedagogically, it is not wise to tell students "accept
> Coulomb's law as written; the advantage of the 4*Pi will
> become clear later". The old advice "do not accept anything
> without understanding" is still worth giving.

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.  The differential
form: div(E) = (?)*[rho] is simplest with a rationalized system since
the (?) constant doesn't include any 4*[pi] factor in such a system.
Whether or not the final value of this factor is some other nonunity
factor or not depends on just *which* rationalized system one chooses.
Also the differential form of Ampere's law has no 4*[pi] factor in front
of the current term for a rationalized system, but does have an extra
factor of 4*[pi] for an unrationalized system.

OTOH, the integrated form of Coulomb's law and the bulk integrated form
for the electrostatic field from a static distribution of charged sources
have a 1/(4*[pi]) factor in front of them for a rationalized set of
units, but have no such factor for an unrationalized set of units.  No
matter which kind of set of units is to be chosen, all the 4*[pi] factors
can't be made to vanish simultaneously from *both* the differential and
integrated forms of the electromagnetic laws.  The reason many physicists
prefer a rationalized set of units is they tend to be partial to the
local differential forms of the laws (and consider them to be more
'fundamental' in some sense), and they want *those* forms to be as
'pretty' as possible--even if it means the introduction of a factor of
1/(4*[pi]) in front of the Coulomb force law for a pair of isolated
static charges.  Also, it should be noted that an unrationalized system
causes extra 4*[pi] laden factors to appear in expressions for the
electromagnetic Lagrangian, Hamiltonian, their densities, the Poynting
flux, the EM stress tensor, relationship between the electric field and
electric polarization, the relationship between the magnetic field and
magnetization, etc. that just do not appear using a rationalized system.
Also, if one wanted to generalize electromagnetism to a generic
higher-dimensional spacetime than ours, one would find that an
unrationalized system would not be as simple to generalize cleanly.  The
fact that the weird factor *is* 4*[pi] rather than some other value weird
factor an artifact of ours being a 3+1 dimensional spacetime.

> b) A situation in which the writing of a formula depends on
> the system of units is not desirable. Note that F=m*a, or any
> other formula in mechanics, does not change when we decide
> to use feet and pounds instead of meters and newtons. But in
> electricity the look of a formula (presence or absence of 4*Pi)
> depends on units. Contrary to expectations, SI did not become
> the only system of units used by physicists.

Actually, the form of Newton's 2nd law *can* include an factor of an
extra constant of nature in it for *some* sets of units.  For instance,
consider a set of units where time is measured in seconds, distance is
measured in feet, and *both* mass and force are measured in pounds
(lbm & lbf respectively).  In such a set of units Newton's 2nd law
becomes: F = m*a/g  where the constant of nature g is given by
g = 32.174 f/s^2.

> In my opinion the "computational economy" concept is not
> valid in the era of electronic computations. The 4*Pi in
> Coulomb's law creates unnecessary difficulties for teachers
> of introductory courses. I wrote about this in a letter to the
> editor ("The SI is not ideal for teaching elementary
> electromagnetism". Am. J. Phys., December 1985, p 1131)
> but nobody responded.

Things are not the way they are regarding SI electrical units because
of pedogogical considerations for elementary levels of physics.  They
are the way they are because of a particular complicated confluence of
the exigencies of history and the everyday convenience of most of the
workers (usually experimental physicists, engineers, electricians &
technicians) in fields that heavily use these units.

> 4) Let me add that the incorporation of the dimensional
> constant, epsilon_zero, into Coulomb's law, multiplied
> pedagogical difficulties by one million times, in
> comparison with the damage done by incorporating
> Heaviside's idea into SI.

I agree with this order of magnitude estimate--especially when one
includes the damage the SI [epsilon]_0 (& [mu]_0) does(do) to the
conceptualization of electromagnetism in terms of the relativistic
theory.  However, I do pefer a rationalized system to an unrationalized
one.

> (*) Footnote: That electrochemical unit of charge, called
> Faraday, was defined as 96500 C. It would be 94868 C if
> the old "practical ampere" and the "absolute ampere" of
> SI were identical. Some of you may be interested in my
> old note "A short history of the SI units in electricity".
> (The Physics Teacher, February 1986, p 97-99).

No. the Faraday unit has *nothing* to do with the unit of charge Ludwik
introduced above giving a unit 1 N force at a 1 m distance.  Rather, a
Faraday is the amount of charge in one mole of elementary charges.
One Faraday is Avogadro's number times the charge of a proton, i.e.

F = (6.02214199(97) x 10^23 mol^-1)*(1.602176462(63) x 10^(-19) C) =
  = 96485.3415(39) C/mol

These numbers come from the most recent (1998) revision of the
fundamental constants:
http://physics.nist.gov/cgi-bin/cuu/Category?view=pdf&All+values.x=73&All+values.y=22

It is just a coincidence that the numerical value of the Faraday constant
is close to the number of Ludwik's new charge units in one coulomb.

96485.3415 ~= 94802.69926

David Bowman
David_Bowman@georgetowncollege.edu