On 1/4*Pi in Coulomb's law

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

1) In old days, Coulomb's law was used for two purposes,
(a) to express the observed proportionality between F and
Q1*Q2/r^2 and (b) to define the unit of electric charge.
After writing F=Q1*Q2/r^2 a teacher would say:

Two charges of equal magnitude interact with a force of
one unit from a distance of one unit. What can be more
simple and more natural to somebody who is just starting
to learn electricity? 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).

2) The absence of the 4*Pi in Coulomb's law resulted in the
appearance of this factor in some of the derived formulas,
for example, in Gauss's law and in the formula for C of the
parallel plate capacitor. After discovering this student were
introduced to Heaviside's idea of rationalization. It went like
this:

> The capacitance formula is used more frequently than
> Coulomb's law. Therefore, for the sake of economy of
> computations, we can eliminate the cumbersome 4*Pi
> factor from the often used formulas by introducing it
> artificially into the Coulomb's law.

In the spirit of defining the unit of charge by Coulomb's
law the new formula:

     F = (1/4*Pi) *(Q1*Q2/r^2)

leads to a new unit of charge (for which F=1/4*Pi newtons
when r=1 m, provided Q1=Q2=1 unit).

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.

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.

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.

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.
...................................................................

(*) 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).
...................................................................
Ludwik Kowalski