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Re: B and electric charge



Something seems to be missing in the exposition. There are, after
all, two currents, so the force (before we talk about fields) is
i^xi^ (denoting cross-product of two vectors). Somehow you've
got to introduce a constant to get this to come out right. Since you seem
to have started by defining e.s. units, you will need a \mu. Otherwise
you can go the Gaussian unit route in which case you will need to have
some factors of c.
Regards,
Jack



On Tue, 1 Jan 2002, Ludwik Kowalski wrote:

In a recent message (see below) I tried to introduce the unit
of electric charge without epsilon_o. The units of E and V
follow naturally from the Coulomb’s law and from the idea
of potential energy. But the introduction of the magnetic
induction, B, and of its unit, is conceptually more difficult.
Here is how I plan to do this next time. note that neither
epsilon_o nor mu_o are needed. Is there anything wrong
with this presentation?

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

Electric charges at rest are responsible for the electric fields
E whose magnitude and direction can be determined, at any
given location, by using a tiny probe charges q and measuring
the force acting on it (from F=q*E). Oersted discovered that
charges in motion are responsible for magnetic forces. In the
old days magnetic field, at any given location, would be
defined in terms of its effect on a tiny compass needle. In
other words, in magnetostatics the needle played the same
role as a probe charge plays in electrostatics.

A more modern approach, which we will use, is to define B in
terms of its effect on a tiny current loop. This loop, suspended
by two highly flexible wires is free to rotate about the axis of
suspension. The current I, (delivered and taken away through
the flexible suspension wires) is flowing through the loop.
Experiments show that the loop behaves as if it was a compass
needle; in the terrestrial field one of its side will always face
north while another will face south. It is easy to verify that
the north side of the loop is consistent with the right hand rule.
Curl the fingers of the right hand along the direction of I and
the thumb will point north. It is convenient to represent the
loop area by a vector n whose magnitude is equal to the area A
and whose direction coincides with the direction of the thumb.

Draw a vertical line representing a very long wire carrying a
current I2. Experiments show that the probe loop will orient
itself is such way as to include the vertical wire in its plane.
A reversal of the current, either I or I2, leads to the rotation
of the loop by 180 (after some oscillations). Let us agree that,
by definition, the direction of the magnetic field B coincides
with the vector n (loops normal). Before defining the magnitude
of B we must focus on the properties of the loop itself. The
changing orientation of the loop indicates the presence of a
torque. Experiments show that at any given location, and for
any given current I2, the torque is directly proportional to the
product of I*A. That product, to be named magnetic moment,
p, is clearly the property of the loop. It is not hard to verify
that the maximum torque, Tmax, exerted on the probe loop
by a magnetic field, is directly proportional to the loops
magnetic moment. This can be written as:

Tmax=const*p=const*A*I

The value of the constant depends on the magnitude of the
current I2 (not to be confused with the current I) and on the
location at which the loop is located. It is thus natural to
identify the magnitude of B with the experimentally
determined constant of proportionality. In other words, the
magnitude of B is given by Tmax=B*A*I. Note that,
according to this definition, the unit of B must be N/(A*m).
To honor a famous Serbian scientist this unit is called tesla,
T. How strong is the magnetic field in a given location if
Tmax=0.005 N*m while p=0.1*0.1 A*m^2 ? The answer
is: B=0.005/(0.01)=0.5 T.

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

This introduction is logically acceptable only if students are
already familiar with the unit of electric current, A. That is
why in my sequence the unit of current will be defined as
C*s and C will be defined in terms Coulomb’s law, as shown
below. Another possibility would be to define A in terms of
the heating effect or in terms of the electrolytic effect.
Ludwik Kowalski
*********************************

This is my earlier message (with corrections)

Mass m is a mechanical attribute of an object. It determines
how the object accelerates (F=m*a) and how it is attracted
by another mass (F=G*m*M/d^2). The first systematically
investigated electric phenomenon was mutual attraction and
mutual repulsion of light objects under the influence of
something which was not a mass or magnet. That something
was named charge. A glass rod rubbed with silk, for example,
acquires a property of repelling another glass rod rubbed with
silk. That property, named charge, was initially modeled as
a fluid. An object containing that fluid was said to be electrified.

Likewise, a plastic rod rubbed with wool repels another plastic
rod rubbed with wool. But an electrified glass and an electrified
plastic attract, rather than repel, each other. This, and many
other, observation, and lead to a realization that there are two
kinds of electric fluids, positive and negative. The term charge
used to be interpreted as the "amount of electric fluids" or
"amount of electricity" which an object can acquire or lose.

A modern interpretation is based on the realization that
submicroscopic particles, protons and electrons, are permanently
charged with positive and negative electricity. A macroscopic
object is charged when the number of electrons and the number
of protons are not identical. An excess of protons results in a net
positive charge while an excess of electrons results in a net
negative charge. The net charge, like the total mass, becomes an
attribute of an object. It is an attribute responsible for forces
between electrified objects. Two similar charges (both positive
or both negative) always repel but two dissimilar charges (positive
and negative) always attract. This was the first qualitative
observation about electric forces.

It turns out that the magnitude of an electric force between two
charges (q1 and q2) is proportional to the product q1*q2 and
inversely proportional to the square of the distance (d^2)
between their centers. This observation, made by Coulomb, is
known of Coulomb’s law. It can be written as:

F = k*q1*q2 / d^2

where k is the proportionality constant. The value of that constant
can be chosen arbitrarily in order to define a unit of electric charge.
For the purpose of this introduction the unit of electric charge, one
coulomb, C, we will defined by declaring that k=9,000,000,000.
This is equivalent to saying that the electric charge is one coulomb
if it attracts or repels an identical charge with a force of nine billion
newtons when the distance between the centers of two charges is
one meter. One coulomb is a very large charge; charges produced
on rubbed rods and plates are usually expressed in microcoulombs
or in nanocoulombs. Ignoring sign differences we can say that
the charge of one electron and the charge of one proton are
identical (1.6*10^-19 C).

The so-called "official" SI definition of the unit of charge is
conceptually different from the one presented above. But in
practical terms it is not at all different. In SI the ampere, A, is
the first unit; all other electrical units are defined in terms of kg, m,
s and A. The unit of charge, coulomb, C, is defined as A*s. In
our sequence C is the first electric unit and A will be defined
as C/s. Other nuances associated with electrical and magnetic
SI units will be discussed later.

http://alpha.montclair.edu/~kowalskiL/SI/si_page.html

Note that F in Coulomb’s law is positive when two charges
repel (q1 and q2 have the same sign) and negative when
they attract (signs of q1 and q2 are different).

Here is my first "electricity problem" for the next semester:

1) Two drops of water (0.01 grams each) are separated by a
distance of 1 cm. Calculate the force of their mutual
gravitational attraction. (The answer is 6.67*10^-17N)
2) Suppose that a superman removed 1% of electrons from
one drop (making it positive) and transferred them to
another drop (making it negative). Calculate the force of
mutual electrical attraction. (Answer: 2.35*10^+14 N).
3) What is the weight of one cubic mile of water ?
(Answer: 4.08*10^+13 N)


--
"But as much as I love and respect you, I will beat you and I will kill
you, because that is what I must do. Tonight it is only you and me, fish.
It is your strength against my intelligence. It is a veritable potpourri
of metaphor, every nuance of which is fraught with meaning."
Greg Nagan from "The Old Man and the Sea" in
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