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



At 09:34 AM 1/23/01 -0500, Robert A Cohen wrote:
It seems the only reason why students might be confused by the 4*pi in
Coulomb's law is because there is no 4*pi in the gravitational law.

Good point.

Perhaps we should introduce the 4*pi in the gravitational law. Would it
be any less abstract for students to have gravity inversely proportional
to surface area?

Maybe. Or we could go the other way.

Here is the parallel between gravity and electricity:

0) |force| = CONST source1 source2 / r^2

1a) In one case the sources are masses, and the CONST has a particular value.
1b) In the other case the sources are charges, and the CONST has another value.

The numerical value of the CONST depends on what system of units you are
using, but the physical meaning of the equation is invariant.

So all there is left to quibble about is
-- the name, and
-- what you have to do to look up the value of the CONST.

2a) In one case, you have to go to a book and look up something named G.
2b) In the other case, you have to go to a book and look up something named
epsilon_0, take its reciprocal, and divide by 4 pi.

Is the name epsilon_0 required by the laws of physics? No, it is just a
conventional, agreed-upon name. This is not worth throwing a fit about;
almost all communication depends on using words and symbols in conventional
ways.

Does the inconsistency (2a versus 2b) make it harder to remember the form
of Coulomb's law? Yes, to some extent it does. But who cares? The basic
form of the equation remains
|force| = CONST source1 source2 / r^2
and it's not worth remembering it in much more detail than that. I
certainly don't. Anybody who needs the next level of detail can figure out
by dimensional analysis (or by order-of-magnitude considerations) that the
epsilon_0 goes in the denominator, not the numerator of this CONST. And
anybody who needs help remembering where the 4 pi goes can cross-check
against the parallel-plate capacitor formula.

Does this inconsistency mean we should restate Coulomb's law? No, because
of the following dilemma: Simplifying Coulomb's law would complexify the
parallel-plate capacitor formula (or would require tabulating two different
electrostatic constants). The capacitor law is more important. It is the
only one I bother to remember. I'm glad it's simple.

Does this inconsistency mean we should restate the gravitational
formula? No, because there is no gravitational technology comparable to
capacitors. The gravitational law is not just the most important
gravitational formula, it is practically the only important gravitational
formula.

Should these naming conventions be considered dogma? I don't think
so. It's not the sort of thing most people consider dogma. Dogma is
something you are asked to believe given zero (or less) evidence. Dogma is
something that is imposed by The Authorities. But using names in a
conventional way hardly rises to this level. Furthermore, if we want to
write Coulomb's law as
|force| = K Q q / r^2
we are free to do so. We will not be hauled before the inquisition and
burned at the stake. The only penalty is the penalty you impose on
yourself, namely the burden of pencilling into your reference books the
value of K in your favorite units.

I would hope students could be taught not to worry about this. A lot of
them can't tell the difference between logic and dogma anyway. The ones
who are sophisticated enough to care about the difference ought to be
sophisticated enough to know that renaming something doesn't change its
meaning. K is just a name. Inverse 4 pi epsilon_0 is just a name.