Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

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



At 11:18 AM 1/21/01 -0500, Ludwik Kowalski wrote:
I still claim that unnecessary complications, directly
connected with SI, are real in physics at the introductory level.

Is there any evidence to support this claim? I haven't seen any.

The only evidence I've seen supports the opposite position, namely that SI
does _not_ cause any unnecessary complications in electromagnetism.

In particular, let's consider Maxwell's equations, in the form that appears
at the top of Chapter 4 of Volume II of _The Feynman Lectures on Physics_.

del dot E = rho / epsilon_0

del cross E = - (partial / partial t) B

c^2 del cross B = (partial / partial t) E + j / epsilon_0

del dot B = 0

a) Specifically: I see no factors of 4 pi here.

b) More generally: I do not see any way to simplify this in a way that is
suitable for an introductory course.

I recognize that you could, for instance, simplify the third equation by
setting c^2=1, but that would be distinctly unhelpful at the introductory
level -- and it wouldn't address the 4 pi issue that Ludwik raises.

c) Similarly there are other formulas, such as the capacitance of parallel
plates, that depend on epsilon_0 without any factors of 4 pi.

d) OTOH there are formulas, such as Coulomb's law, (Feynman's eq. 4.9)
|F| = 1 / (4 pi epsilon_0) Q q / r^2
which do have factors of 4 pi in them. But the laws of physics (indeed the
laws of mathematics) do not permit you to simplify this equation without
complexifying the Maxwell equations, the capacitor equations, et
cetera. For details see my previous note.

e) The factor of 4 pi doesn't have anything to do with SI. These equations
are consistent with using SI units for all quantities, but they don't
require it. Pick any units you like, however elegant or inelegant, and this
same factor of 4 pi will still show up, one place or another.

f) There is no logical basis for giving Coulomb's law priority over all the
other laws, and insisting that it be simplified no matter what.

There are at least two issues here:

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

We might also contemplate the distinction:
-- real complexities that are understood
-- real complexities that are misunderstood,
i.e. mis-assigned to imaginary causes.
-- real complexities that have real solutions
-- real complexities that have only imaginary solutions

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

I have seen no evidence of any removable complexity. The mathematical
tells us that no solution of the type Ludwik seeks is obtainable.

Are the difficulties real or am I imagining them?

There are lots of real difficulties in the world. Electromagnetism is not
a trivial subject. The 4 pi in question seems to be a 100% real complexity
that is in danger of being mis-assigned to a bogus cause. The fact that
the area of a unit sphere is not equal to unity is a difficulty; you'll
just have to deal with it. The fact that the constant that appears in the
first Maxwell equation is different from the constant that appears in
Coulomb's law is a perfectly real difficulty; you'll just have to deal with it.

Is it worth complaining or not?

It is not worth complaining about mathematical facts.