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

[John Denker <jsd@MONMOUTH.COM>]

At 11:18 AM 1/21/01 -0500, Ludwik Kowalski wrote:
> 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.

OK, here's the no-calculus conceptual version.

Here is the flux line diagram for a charge inside a small sphere:

.                   |
.                \  |  /
.                 \ | /
.                  \|/
.             ------Q------
.                  /|\
.                 / | \
.                /  |  \
.                   |
.                    AAAA
.                    AAAAA   <== Unit area
.                    AAAAAA

And here it is for the same charge inside a larger sphere:

.                   |
.                   |
.              \    |    /
.               \   |   /
.                \  |  /
.                 \ | /
.                  \|/
.         ----------Q----------
.                  /|\
.                 / | \
.                /  |  \
.               /   |   \
.              /    |    \
.                   |
.                   |
.                    AAAA
.                    AAAAA   <== Unit area
.                    AAAAAA

And here are some relevant formulas:

  Total # of field lines = (1/epsilon_0) Q                              [1]
      (which is independent of the size of the sphere)

  # of field lines per unit area =  (1/epsilon_0) Q / Area              [2]

  # of field lines per unit area =  (1/epsilon_0) Q / (4 pi r^2)        [3]


Each of these expresses an important bit of physics.  One of these has a
factor of 4 pi in it.  The others don't.

There is no logical basis for saying that equation [3] is more fundamental
than the others and therefore "deserves" to be freed of the 4 pi.

Let's not blame SI units for this 4 pi.  There will be a relative factor of
4 pi here, no matter what units are used.

Equation [2] is familiar;  it comes up in the analysis of a parallel-plate
capacitor.  Are you surprised that the same formula applies to spherical
capacitors?  The unifying idea is to express the denominator in terms of
area (not radius).  This is obviously mandatory for a parallel-plate
capacitor, and optional for the spherical capacitor.

==========

Possibly constructive suggestion:  If your students are so math-impaired
that they can't cope with a factor of 4 pi in the Coulomb force law,
perhaps the following would help:  Rather than starting with the force
between two test charges, start with the force between the plates of a
parallel-plate capacitor.

For what it's worth:
 *) I've never actually measured the electrical force in the spherical
geometry suggested by the usual form of Coulomb's law.
 *) In contrast, on several occasions I've measured it in the
parallel-plate geometry -- usually not because I wanted to, but because I'm
fond of building capacitive position sensors with femtometer resolution,
and the electrostatic force is a perturbation on the system being measured.

==================
Graphics test pattern, please ignore:
0
 1
  2
 1
0
        t0
         t1
          t2
         t1
        t0
===================
PS:  Nitpickers who don't like my use of inexact concepts such as "field
lines" should disregard this note and refer to my earlier discussion of
Lapacian(1/r) = 4 pi delta(r).