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Re: peculiar constants in physical laws



At 01:50 PM 1/19/01 -0500, Ludwik Kowalski wrote:
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.

Maybe the law was used that way, but such usage may not have been well
justified. I don't see how to justify it on practical grounds or
historical grounds. AFAIK the unit of charge has been (from early days
until now) defined by integrating the current. For a goodly part of the
history, the integration was performed in an electrochemical cell.

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?

Whoever said that must have been in dreamland. I don't write Coulomb's law
that way. I don't recommend writing it that way. Mr. Coulomb didn't write
it that way. Electrical engineers don't write it that way. If you write
it that way, you mess up the definition of ampere and lose contact with all
the technology in the physics lab and in the real world. Have you ever
seen a current meter calibrated in statamperes? Have you ever tried to buy
a light bulb with a certain statwatt rating?

(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.

That is a noble thought, but overstated. Remember our previous discussion
of arches. Each block in the arch is held in place by others below AND
ABOVE it. No block makes sense in isolation. Because no block makes sense
in isolation, unless you are willing to let some of the blocks lean on the
falsework, you will never get started.

So it is with students facing new ideas. Very commonly, there is a group
of ideas that make sense together, but none of them can be evaluated in
isolation. The student's trust in the teacher plays the role of
falsework: students need to lean on the teacher until the arch is
sufficiently complete that it can stand on its own.

Returning to this case: The notion that superficially simplifying one
thing will only complexify other things is hardly a gargantuan leap of faith.

b) A situation in which the writing of a formula depends on
the system of units is not desirable.

But it is often unavoidable. Often it expresses real physical facts.

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 if we choose to measure force in pounds, mass in Troy ounces, and
acceleration in knots per fortnight, there _will_ be a nontrivial factor Z
that appears in the dynamical law:
F = Z m a

This messy situation is the usual situation. You can remove the messiness
from a few selected equations by fiddling with the units -- but only a few.

-- For instance, perhaps we could redefine the units of mass to remove the
factor of two from the Schwarzchild formula for the radius of a black hole:
r = 2 m G / c^2

-- Similarly, perhaps we could redefine the units of rotation-rate to
remove the factor of 2 from the Coriolis formula:
a = 2 omega x v

-- And perhaps we should declare the elementary charge to be The One True
Unit of charge, and recalibrate all our current meters in Eeps (exa
electrons per second). One amp is about 6.24 Eeps.

At 04:55 PM 1/20/01 -0500, David Bowman wrote:
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.

Exactly so.

Returning to what Ludwik wrote:
4) Let me add that the incorporation of the dimensional
constant, epsilon_zero, into Coulomb's law, multiplied
pedagogical difficulties by one million times,

My opinion differs. Until somebody "upgrades" my lab with statvoltmeters
and statammeters, I will continue to use practical, conventional units. In
these units, epsilon_zero expresses an important fact about the universe we
live in.

At 12:26 PM 1/20/01 -0500, Ludwik Kowalski wrote:
What is epsilon_zero? Why is it called permittivity?
What kind of experiments had to be performed to find
out that epsilon_zero happens to be equal to 8.85*10^-12
SI units?

Somebody had to build a capacitor of a known size and measure its capacitance.

Most students taking an introductory physics
course, either in a high school or a college, never learn
how to answer such questions.

That's a scandal.

As far as they are concerned physics is dogmatic; it asks them to accepts
things without understanding.

Action item: Wrap a couple of sugar cubes in aluminum foil, to form
conducting cubes about 1 centimeter on a side. Bring them to class, and
place them 1 millimeter apart. Announce the dimensions (1 cm cube, 1 mm
gap) so the kids don't need to strain their eyes. Ask them to estimate, to
1 significant digit, the capacitance between the cubes. Tell them they've
got ten seconds.

If you understand what epsilon_zero is, you can do this in your head in a
lot less than ten seconds. The answer is useful directly (if, say, you
need to estimate the parasitic capacitance between circuit elements) and
the mental image of shiny cubes is well-nigh unforgettable.