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

Very good, Ludwik. Now we can <begin> to discuss.

On Mon, 22 Jan 2001, Ludwik Kowalski wrote:

Jack Uretsky wrote:

Hi all-
It will be amusing to see if I can get Ludwik to read the Chapter
on forces in <The Mechanical Universe>, the edition of which
Olenick is the first author. This is not the 95th time that I have
made the suggestion, but we're getting there.

Jack is probably referring to Chapter 11 entitled
"Gravity, Electricity and Magnetism". It comes
after the law of universal gravitation and Kepler
relations were discussed (Ch. 8 and 9) but before
electricity topics are systematically covered in
volume 2.
Exactly. While teaching mechanics we have an
opportunity to show the that the topic we are about to
introduce - fields - has broad applicability. So we are
showing relationships rather than isolated facts.


Coulomb's law first appears in Ch. 10 where it is
written next to the low of gravity. The assumption
is that students are already familiar with the concept
of charge q. (How else can it be? The q1 and q2 appear
in F=Ke*q1*q2/r^2 without any explanation).

How else can it be? It can be that this is the place
where the idea of "charge" is introduced. It helps to read
the paragraph preceding Eq 10.2 (the law of force between
charges with no \eps_0 present).


The first relation of Ch. 11 is the force law for magnetic
poles, F=Km*p1*p2/r^2. Once again p1 and p2 are not
explained, students are expected to know what the
magnetic pole strength p is.
The idea of a magnetic pole is introduced two paragraphs
before the equation in question. Give the notion of a "pole",
it is not a great jump to talk about its strength.

The second relation of the
Ch. 11 is F=p*B where F is the force acting on a pole
while B is a quantity called magnetic field. It was not
introduced as a physical quantity up to this point;
magnetic field was described so far only in terms of
lines from N to S near a bar magnet.
So now we have a picture to help our intuition when
we get into the mathematics.


The third relation of the Ch. 11 is F=q*E where F is the
force acting on a charge q and E is a quantity called
electric field. It was not introduced as a physical quantity;

Very, very good! And quite to the point, since E is
not a <physical> quantity. It is Faraday's mathematical construct,
which is what Chapter 11 is all about.

up to this point the electric field was described only in
terms of lines surrounding wires or radiating from charges.
No connection between E and Coulomb's law was made.

Not true! Look at page 222. Or didn't you recognize
Coulomb's law when it is expressed in gravitational terms?

A little later B is defined as a physical quantity by the
usual cross product formula.

You missed the point, and Eq. 11.2, with all the preceding
words, where B is introduced. The (magnetic) <force on a moving
charge is stated in the equation before the cross-product formula,
and <restated> as a cross-product because students are not yet
fluent with vectors.

Note that q has not yet been
defined as a physical quantity. In other words B is defined
before q. [Stop it, Ludwik, you wanted to be a reporter, not
a critic. OK, I will control myself. ] A little later the 10.1
formula (Coulomb's law) is shown again but this time Ke is
said to be an experimental constant equal to 9*10^9 N*m^2/C.
The unit of charge, C, has not yet been introduced.

And here is the answer to your original question, which
you have not yet recognized.

Next to it formula 10.1 (F=Km*p1*p2/r^2.) is presented
again and the unit pole is defined as the product q*v in
the F=q*v*B. The unit of p, 1 pole, is said to be C*m/s;
the unit of charge, C, has not yet been introduced.

It has, on the contrary, just been introduced. It
has not been given a numerical value.
After
that the authors write: "If we measure the force between
two unit poles, then we can determine the constant Km. In
practice this is not done. Instead we measure the force
between two wires carrying current which we discussed
earlier. The idea, however, is the same: we measure the
force to determine a constant, and the result is
Km=1.00*10^-7 N*s^2/C^2."

Exactly. However we define "charge", or its equivalent,
"pole strength", we need to "measure the force to determine a
constant." What the constant is called is immaterial. There is history,
connected with the use of other units and the introduction of different
fields (D as well as E) for speaking of the "permittivity of free space."
There is, by the way, more insight to me gained by looking
up the dictionary definition of "permittivity".

Yes, there was a picture of two currents on the previous
page but the concept of current is not introduced until
the second volume.

On page 222 you will find: "Suppose we have a charge q,
moving with a velocity v [in bold face] (that constitutes an
electric current)."
What more do you want for a first introduction during
a mechanics course?

The derivation culminates by showing
that the unit of Ke/Km is m^2/s^2. The authors conclude
that by inserting the above values of Ke and Km one finds
that the square of the Ke/Km is the speed of light,
3*10^8 m/s. The chapter ends with a poem written by
James Clerk Maxwell

*******************************************

What do you like in this chapter, Jack? I like the idea of
trying to follow the real history but I do not like the
way in which it was actually done. I see that neither 4*Pi
nor epsilon_zero are included in Ch 11. But they are
used in volume 2, as in any other text based on SI units.
Ludwik Kowalski
They are used in volume 2 (unfortunately now out of print)
but NOT as in any other text based on SI units. The use of "k" or "K"
seems to be preferred in most texts today. See the current Sears (or
Young, or whoever) and Jackson.

What I like is illustrated by some of the points that I
was able to make in the context of your polemical remarks. The
chapter stresses unifying concepts. So, we are advising the students,
as you learn to deal with gravitational problems you are at the same
time learning how to deal with the next topics that you encounter.
The language will be a little different, but the mathematics will be
identical.

I also welcome the opportunity to discuss esthetic ideas in
a physics course. Many of us find great esthetic appeal in the fact
that the three classical forces can all be cast in the same form. This
fact alone, applied to electricity and magnetism, led Dirac to his
prediction of the existence of magnetic poles (he, of course, utilized
all of Maxwell's equations in making his prediction).

The approach fits in well with Arons' admonishment that we
should introduce ideas from many different viewpoints and then circle
back and reintroduce them. The main topic of Chapter 11 is, as the
title indicates, fields. So we get to start talking about fields in
the context of gravity. I introduced, at this stage, an elementary
discussion of Gauss' law in connection with Newton's observation
that there is no gravitational field inside of a spherical mass-shell.
I used only elementary arguments (as in Denker's sketches). The
subject will be reintroduced more rigorously when we get to E&M -
but the student will probably not master the concept until well
into advanced mechanics and a good vector analysis course.

This is a subtle book, and one that is hard to appreciate
until you have taught from it.

Regards,
Jack