"A mathematically correct statement of a physical relation is not necessarily a true statement, because at least some of the terms of that relation must have physical dimensions (otherwise it would be merely a mathematical statement, not a physical statement), and if those dimensions are wrong, the statement itself is physically wrong, regardless of its mathematical accuracy."
[This is the same reasoning that my Calculus professor used when deducting points from an exam for deriving the incorrect sign in my answer; "the space shuttle still burns up...," he suggested. He was right. -The TActivist.]
"The truth is that we do not have this option, because the dimensions are inherent in the physical relations. In any instance where two different derivations lead to different dimensions for a physical quantity, one of the derivations is necessarily wrong. The case cited by McCaig is a good example. The conventional dimensional interpretation of the gravitational equation is obviously incompatible with the accepted definition of force based on Newton’s second law of motion. Force cannot be proportional to the second power of the mass, as required by the prevailing interpretation of the gravitational equation, and also proportional to the first power of the mass, as required by the second law. And it is evident that an interpretation of the force equation that conflicts with the definition of force is wrong. Furthermore, this equation, as interpreted, is an orphan. The physicists have not been able to reconcile it with physical theory in general, and have simply swept the problem under the rug by assigning dimensions to the gravitational constant."
[I wield a rather mighty broom, myself! -The TAcivist]
clarifies some hitherto obscure aspects of the energy picture
Magnetic Quantities and Units
One of the major issues in the study of magnetism is
the question as to the units in which magnetic quantities should be expressed,
and the relations between them. “Since the first attempts to put its study
on a quantitative basis,” says J. C. Anderson, “magnetism has been bedevilled
by difficulties with units.”83 As
theories and mathematical methods of dealing with magnetic phenomena have
come and gone, there has been a corresponding fluctuation in opinion as
to how to define the various magnetic quantities, and what units should
be used. Malcolm McCaig comments that, “with the possible exception of
the 1940s, when the war gave us a respite, no decade has passed recently
without some major change being made in the internationally agreed definitions
of magnetic units.” He predicts a continuation of these modifications.
“My reason for expecting further changes,” he says, “is because there
are certain obvious practical inconveniences and philosophical contradictions
in the SI system as it now stands.”84
Actually, this difficulty with units is just another
aspect of the dimensional confusion that exists in both electricity and
magnetism. Now that we have established the general nature of magnetism
and magnetic forces, our next objective will be to straighten out the
dimensional relations, and to identify a consistent set of units. The
ability to reduce all physical quantities to space-time terms has given
us the tool by which this task can be accomplished. As we have seen in
the preceding pages, this identification of the space-time relations plays
a major part in the clarification of the physical situation. It enables
us to recognize the equivalence of apparently distinct phenomena, to detect
errors and omissions in statements of physical relationships, and to fit
each individual relation into the total physical picture.
The basic magnetic quantity, magnetic charge, is
not recognized in current physical thought, but an equivalent quantity,
magnetic flux, is used instead of charge, as well as in other applications
where flux is the more appropriate term. The space-time dimensions of
this quantity are the dimensions of electric charge, t/s, multiplied by
the factor t/s that relates magnetism to electricity: t/s x t/s = t2/s2.
In the cgs system, magnetic flux is expressed in maxwells, a unit equivalent
to10-8 volt-sec. The SI unit is the
weber, equivalent to the volt-sec. The justification for deriving the
basic magnetic unit from an electric unit, the volt, can be seen when
this derivation is expressed in space-time terms: t/s2
x t = t2/s2.
However, a few empirical relations
did indicate the existence of such a quantity. For example, one of the
important relations discovered in the early days of the investigation
of magnetism is Ampére’s Law, which relates the intensity of the
magnetic field to the current. The higher permeability of ferromagnetic
materials had to be recognized in the statement of this relation. Permeability
was originally defined as a dimensionless constant, the ratio between
the permeability of the ferromagnetic substance and that of “free space.”
But in order to make the mathematical expression of Ampére’s Law
dimensionally consistent, some additional dimensions had to be included.
The texts that define permeability as a ratio assign these dimensions
to the numerical constant, an expedient which, as pointed out earlier,
is logically indefensible. The more recent trend is to assign the dimensions
to the permeability, where they belong. In the cgs system these dimensions
are abhenry/cm. The abhenry is a unit of inductance, t3/s3,
and the dimensions of permeability on this basis are t3/s3
x 1/s = t3/s4,
which agrees with the previous determination. The SI units henry/meter
and newton/ampere2 (t/s2
x t2/s2
= t3/s4)
are likewise dimensionally correct. The unit farad/meter has been used,
but this unit is dimensionless, as capacitance, of which the farad is
the unit, has the dimensions of space. Using this unit is equivalent to
the earlier practice of treating permeability as a dimensionless constant.
McCaig is quite critical of the unit henry/meter. He makes this comment:
Most books now.quote the units of m0 as
henry per metre. Although this usage is now almost universal, it seems
to me to be a howler...The henry is a unit of self or mutual inductance
and it seems quite incongruous to me to associate a metre of free space
with any number of henries. If one wishes to be silly, one can invent
numerous absurdities of this kind, e.g., torque is measured in Nm or
joule!86
The truth is that these two examples of what McCaig calls
dimensional “absurdities” are quite different. His objection to coupling
inductance with length is a purely subjective reaction, an opinion that
they are incompatible quantities. Reduction of both quantities to space-time
terms shows that his opinion is wrong. As indicated above, the quotient
henry/meter has the dimensions t3/s4,
with a definite physical meaning. On the other hand, if the dimensions
of torque are so assigned that they are equivalent to the dimensions of
energy, there is a physical contradiction, as a torque must operate through
a distance to do work; that is, to expend energy.
Torque is a product of force and distance, t/s2
x s = t/s. The space-time dimensions of the product mH are t2/s
x 1/t = t/s. The equation T = mH is thus dimensionally correct. The space-time
dimensions of the product mB are s3/t
x t2/s4
= t/s. So the equation T = mB is likewise dimensionally correct. The only
difference between the two is that in the Kennelly system the permeability
is included in m, whereas in the Sommerfeld system it is included in B.
This situation emphasizes the importance of a knowledge of the space-time
dimensions of physical quantities, particularly in determining the nature
of the connection between one quantity and another. A mathematically correct
statement of a physical relation is not necessarily a true statement,
because at least some of the terms of that relation must have physical
dimensions (otherwise it would be merely a mathematical statement, not
a physical statement), and if those dimensions are wrong, the statement
itself is physically wrong, regardless of its mathematical accuracy.
The dimensions constitute a description of the physical nature of the
quantities to which they apply, and give the mathematical statement of
each relation a physical meaning.
As matters now stand, this is not recognized by everyone.
McCaig, for example, indicates, in his discussion, that he holds an alternate
view, in which the dimensions are seen as merely a reflection of the method
of measurement of the quantities. He cites the case of force, which, he
says, could have been defined on the basis of the gravitational equation,
rather than by Newton’s second law, in which event the dimensions would
be different.
The truth is that we do not have this option, because
the dimensions are inherent in the physical relations. In any instance
where two different derivations lead to different dimensions for a physical
quantity, one of the derivations is necessarily wrong. The case cited
by McCaig is a good example. The conventional dimensional interpretation
of the gravitational equation is obviously incompatible with the accepted
definition of force based on Newton’s second law of motion. Force
cannot be proportional to the second power of the mass, as required by
the prevailing interpretation of the gravitational equation, and also
proportional to the first power of the mass, as required by the second
law. And it is evident that an interpretation of the force equation that
conflicts with the definition of force is wrong. Furthermore, this equation,
as interpreted, is an orphan. The physicists have not been able to reconcile
it with physical theory in general, and have simply swept the problem
under the rug by assigning dimensions to the gravitational constant.
Reduction of the dimensions of all physical quantities
to space-time terms, an operation that is feasible in a universe where
all physical entities and phenomena are manifestations of motion, not
only clarifies the points discussed in the preceding pages, but also accomplishes
a similar clarification of the physical situation in general. One point
of importance in the present connection is that when the dimensions of
the various quantities are thus expressed, it becomes possible to take
advantage of the general dimensional relation between electricity and
magnetism as an aid in determining the status of magnetic quantities.
Field Intensity or Flux Density
E = V/s = t/s2 x 1/s
= t/s3
Potential per unit space
E = R/t = t2/s3
x 1/t =t/s3
Resistance per unit time
Magnetic
B = A/s = t2/s3
x 1/s = t2/s4
Potential per unit space
µH = m/t = t3/s4
x 1/t = t2/s4
Permeability per unit time
Ordinarily the electric field intensity is regarded as
the potential per unit distance, the manner in which it normally enters
into the static relations. As the tabulation indicates, it can alternatively
be regarded as the resistance per unit time, the expression that is appropriate
for application to electric current phenomena. Similarly, the corresponding
magnetic quantity B or µH, can be regarded either as the magnetic
potential per unit space or the permeability per unit time.
A dimensional issue is also involved in the relation
between magnetization, symbol M, and magnetic polarization, symbol P.
Both are defined as magnetic moment per unit volume. The magnetic moment
entering into magnetization is s3/t,
and the dimensions of this quantity are therefore
s3/t x 1/s3
= 1/t, making magnetization dimensionally equivalent to H. The magnetic
moment entering into the polarization is the one that is generally called
the magnetic dipole moment, dimensions t2/s.
The polarization is then t2/s x 1/s3
= t2/s4.
Magnetic polarization is thus dimensionally equivalent to field intensity
B. To summarize the foregoing, we may say that there are two sets of these
magnetic quantities that represent essentially the same phenomena, and
differ only in that one includes the permeability, t3/s4,
while the other does not. The following tabulation compares the two sets
of quantities:
Magnetic moment
s3/t
Dipole moment
t3/s4
x s3/t = t2/s4
Magnetization
1/t
Polarization
t3/s4
x 1/t = t2/s4
Vector H
1/t
Field Intensity
t3/s4
x 1/t = t2/s4
A point to be noted about these quantities is that the
magnetic polarization is not the magnetic quantity corresponding to the
electric polarization. The magnetic polarization is a magnetostatic quantity,
with dimensions t2/s4,
and its electric analog would be an electrostatic quantity with dimensions
t/s3. This what electric polarization
would be on the basis of the conventional theory of storage of electric
charge in capacitors. But, as we saw in Chapter 15, the capacitor stores
electric current, not electric charge. It has therefore been found necessary
to introduce a term with the dimensions s2/t
into the mathematical relations, eliminating the electrostatic quantities;
that is, reducing coulombs (t/s) to coulombs (s). The need for this mathematical
adjustment is a verification of our conclusion that the electrical storage
process does not involve any polarization in the electrostatic sense.
The magnetic quantities identified in the discussion
in this chapter–the principal magnetic quantities, we may say–are listed
in Table31, with their space-time dimensions and their units in the SI
system.
Table 31: Magnetic Quantities
Quantity
SI Units
Dimensions
dipole moment
weber x meter
t2/s
flux
weber
t2/s2
pole strength
weber
t2/s2
vector potential
weber/meter
t2/s3
MMF
t2/s3
flux density
tesla
t2/s4
field intensity
t2/s4
polarization
tesla
t2/s4
inductance
henry
t2/s3
permeability
henry/meter
t2/s4
magnetization
ampere/meter
1/t
vector H
ampere/meter
1/t
magnetic moment
ampere x meter2
s3/t
reluctance
1/henry
s3/t3
The mathematical treatment of magnetism has improved
very substantially in recent years, and the number of dimensional inconsistencies
of the kind discussed in the preceding pages is now relatively small compared
to the situation that existed a few decades earlier. But the present-day
theoretical treatment of magnetism tends to deal with mathematical abstractions,
and to lose contact with physical reality. The conceptual understanding
of magnetic phenomena therefore lags far behind the mathematical treatment.
This is graphically illustrated in Table 32. The upper section of this
tabulation shows the “corresponding quantities in electric and magnetic
circuits,”89 according to a current
textbook, with the space-time dimensions of each quantity, as determined
in the present investigation. The lower section shows the correct analogs
(magnetic = electric x t/s) in the three cases where a magnetic analog
actually exists. Only two of the seven identifications in the textbook
are correct, and in both of these cases the dimensions that are currently
assigned to the magnetic quantity are wrong. As brought out in the preceding
discussion, the permeability, which belongs in both the MMF and the magnetic
field intensity, is omitted from these quantities in the SI system.
Table 32: Corresponding Quantities
Magnetic
From reference 89, with space-time dimensions
added
s/t
current
t2/s2
magnetic flux
1/st
current density
t2/s4
magnetic induction
s2/t2
conductivity
t3/s4
permeability
t/s2
EMF
t2/s3
MMF
t/s3
electric field intensity
t2/s4
magnetic field intensity
s3/t2
conductance
t3/s3
permeance
t2/s3
resistance
s2/t3
reluctance
Correct analogs (magnetic = electric x t/s)
s/t
current
no magnetic analog
1/st
current density
no magnetic analog
s2/t2
conductivity
no magnetic analog
t/s2
EMF
t2/s3
MMF
t/s3
electric field intensity
t2/s4
magnetic field intensity
s3/t2
conductance
no magnetic analog
t2/s3
resistance
t3/s4
permeability
When the dimensions of the various magnetic quantities are assigned in
accordance with the specifications in the preceding pages, these quantities
are all consistent with each other, and with the previously defined quantities
of the mechanical and electric systems. This eliminates the need for employing
illegitimate artifices such as attaching dimensions to pure numbers. The
numerical magnitudes of the existing valid magnetic relations have already
been adjusted in previous practice to fit the observations, and are not
altered by the dimensional clarification.
This dimensional clarification in the magnetic area completes
the consolidation of the various systems of measurement into one comprehensive
and consistent system in which all physical quantities and units can be
expressed in terms that are reducible to space and time only. There are,
of course, many specialized units that have not been considered in the
pages of this and the preceding volume–such as the light year, a unit
of distance; the electron-volt, a unit of energy; the atmosphere, a unit
of pressure; and so on–but the quantities measured in these units are
the basic quantities, or combinations thereof, and their units are specifically
related to the units of space and time, both conceptually and mathematically.
No comments:
Post a Comment