"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]

Recognition of the equivalence of inductance and inertia

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 = t^{2}/s^{2}.

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/s^{2}

x t = t^{2}/s^{2}.

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, t^{3}/s^{3},

and the dimensions of permeability on this basis are t^{3}/s^{3}

x 1/s = t^{3}/s^{4},

which agrees with the previous determination. The SI units henry/meter

and newton/ampere^{2}(t/s^{2}

x t^{2}/s^{2}

= t^{3}/s^{4})

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 m

_{0}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 t^{3}/s^{4},

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/s

^{2}

x s = t/s. The space-time dimensions of the product mH are t^{2}/s

x 1/t = t/s. The equation T = mH is thus dimensionally correct. The space-time

dimensions of the product mB are s^{3}/t

x t^{2}/s^{4}

= 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 isphysicallywrong, 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

definitionof 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

Electric

E = V/s = t/s ^{2}x 1/s

= t/s^{3}Potential per unit space E = R/t = t ^{2}/s^{3}

x 1/t =t/s^{3}Resistance per unit time

Magnetic

B = A/s = t

^{2}/s^{3}

x 1/s = t^{2}/s^{4}

Potential per unit space

µH = m/t = t

^{3}/s^{4}

x 1/t = t^{2}/s^{4}

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 s^{3}/t,

and the dimensions of this quantity are therefore

s^{3}/t x 1/s^{3}

= 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 t^{2}/s.

The polarization is then t^{2}/s x 1/s^{3}

= t^{2}/s^{4}.

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, t^{3}/s^{4},

while the other does not. The following tabulation compares the two sets

of quantities:

Magnetic moment s ^{3}/tDipole moment

t^{3}/s^{4}

x s^{3}/t = t^{2}/s^{4}

Magnetization 1/t Polarization t ^{3}/s^{4}

x 1/t = t^{2}/s^{4}Vector H 1/t Field Intensity t ^{3}/s^{4}

x 1/t = t^{2}/s^{4}

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 t^{2}/s^{4},

and its electric analog would be an electrostatic quantity with dimensions

t/s^{3}. 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 s^{2}/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

QuantitySI Units

Dimensions

dipole moment weber x meter t ^{2}/sflux weber t ^{2}/s^{2}pole strength weber t ^{2}/s^{2}vector potential weber/meter t ^{2}/s^{3}MMF t ^{2}/s^{3}flux density tesla t ^{2}/s^{4}field intensity t ^{2}/s^{4}polarization tesla t ^{2}/s^{4}inductance henry t ^{2}/s^{3}permeability henry/meter t ^{2}/s^{4}magnetization ampere/meter 1/t vector H ampere/meter 1/t magnetic moment ampere x meter ^{2}s ^{3}/treluctance 1/henry s ^{3}/t^{3}

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

Electric

Magnetic

From reference 89, with space-time dimensions

added

s/t

current

t ^{2}/s^{2}magnetic flux 1/st current density t ^{2}/s^{4}magnetic induction s ^{2}/t^{2}conductivity t ^{3}/s^{4}permeability t/s ^{2}EMF t ^{2}/s^{3}MMF t/s ^{3}electric field intensity t ^{2}/s^{4}magnetic field intensity s ^{3}/t^{2}conductance t ^{3}/s^{3}permeance t ^{2}/s^{3}resistance s ^{2}/t^{3}reluctance

Correct analogs (magnetic = electric x t/s)

s/t current no magnetic analog 1/st current density no magnetic analog s ^{2}/t^{2}conductivity no magnetic analog t/s ^{2}EMF t ^{2}/s^{3}MMF t/s ^{3}electric field intensity t ^{2}/s^{4}magnetic field intensity s ^{3}/t^{2}conductance no magnetic analog t ^{2}/s^{3}resistance t ^{3}/s^{4}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.

Read more at www.reciprocalsystem.comThis 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.

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