## CHAPTER 20## Magnetic Quantities and UnitsOne 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. Furthermore, the verification process operates in both
directions. The fact that The conventional treatment of magnetic phenomena employs
the units of the mechanical and electrical systems so far as they are
appropriate, and also, in some specialized applications, utilizes the
same quantities under different names. For example, F = -L dI/dt In space-time terms, the inductance is then L = t/s These are the dimensions of mass. Inductance is therefore equivalent to inertia. Because of the dimensional confusion in the magnetic area the inductance has often been regarded as being dimensionally equivalent to length, and the centimeter has been used as a unit, although the customary unit is now the henry, which has the correct dimensions. The true nature of the quantity known as inductance is illustrated by a comparison of the inductive force equation with the general force equation, F = ma. F = ma = m dv/dt = m d The equations are identical. As we have found, I (current) is a speed, and q (electric quantity) is space. It follows that m (mass) and L (inductance) are equivalent. The qualitative effects also lead to the same conclusion. Just as inertia resists any change in speed or velocity, inductance resists any change in the electric current. Recognition of the equivalence of inductance and inertia
clarifies some hitherto obscure aspects of the energy picture. An equivalent
mass L moving with a speed I must have a kinetic energy¹/2LI The inverse of inductance, t In addition to the quantities that can be expressed in terms of the units of the other classes of phenomena, there are also some magnetic quantities that are peculiar to magnetism, and therefore require different units. As brought out in the preceding chapter, these magnetic quantities and their units are analogous to the electric quantities and units defined in Chapter 13, differing from them only by reason of the two-dimensional nature of magnetism, which results in the introduction of an additional t/s term into each quantity. The basic magnetic quantity, The natural unit of magnetic flux is the product of
the natural unit of electric potential, Since conventional physical science has never established the nature of the relation between electric, magnetic, and mechanical quantities, and has not recognized that an electric potential is a force, the physical relations involving the potential have never been fully developed. Extension of this poorly understood potential concept to magnetic phenomena has then led to a very confused view of the relation of magnetic potential to force and to magnetic phenomena in general. As indicated above, the vector potential is the quantity
corresponding to electric potential. As indicated earlier, the magnetic quantities thus far defined are derived from the quantities of the mechanical and electrical systems. The units derived from the electrical system are related to the corresponding units of that system by the dimensions t/s, because of the two-dimensional nature of magnetism. Most of the other magnetic quantities in common use are similarly derived, and all quantities of this set are therefore dimensionally consistent with each other and with the mechanical and electrical quantities previously defined in this and the preceding volume. But there are some other magnetic quantities that have been derived empirically, and are not consistent with the principal set of magnetic quantities or with the defined quantities in other fields. It is the existence of inconsistencies of this kind that has led to the conclusion of some physicists, expressed in a statement quoted in Chapter 9, that a consistent system of dimensions of physical quantities is impossible. Analysis of this problem indicates that the difficulty,
as far as magnetism is concerned, is mainly due to incorrect treatment
of the dimensions of One of the empirical results that has contributed to
the dimensional confusion is the experimental finding that Dimensional consistency in magnetomotive force and related
quantities can be attained by introducing the permeability in those places
where it is applicable. Recognition of the broad field of applicability
of this quantity has been slow in developing. As noted earlier, in most
substances the permeability has the same value as if no matter is present,
the reference level of unity, generally called the “permeability of free
space.” Because of the relatively small number of substances in which
the permeability must be taken into account, the fact that the dimensions
of this quantity enter into many magnetic relations was not apparent in
most of the early magnetic experiments. 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
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 Returning now to the question as to the validity of
the empirical relation MMF = nI, it is evident from the foregoing that
the error in this equation is the failure to include the permeability,
which has unit value under the conditions of the experiments, and therefore
does not appear in the numerical results. When the permeability is inserted,
the equation becomes MMF = µnI, the space-time dimensions of which
are t In our previous consideration of a magnetic quantity currently measured in amperes, the magnetic scalar potential, we found that the assigned dimensions are correct, but that the quantity has no physical significance. In the case of the magnetomotive force, also measured in amperes in current practice, the magnetic quantity called by this name actually does exist in a physical sense, and it is a kind of force, but the dimensions currently assigned to it are wrong. As in the electric system, the The permeability issue also enters into the question
as to the definition of An alternate definition of the magnetic moment introduces
the permeability. This quantity, which is called the The relation between B and H has supplied the fuel for
some of the most active controversies in magnetic circles. McCaig discusses
these controversial issues at length in an appendix to his book Torque is a product of force and distance, t/s 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
McCaig’s comments about the dimensions of torque emphasize
the need to bear in mind that a numerically consistent relation does not
necessarily represent physical reality, even if it is also consistent
dimensionally. Good mathematics is not necessarily good physics. The definition
of torque is Fs, the product of the force and the lever arm (a distance).
The work of rotation is defined as the product of the torque and the angle
of displacement . The work is
thus Fs. But work is the product
of a force and the distance through which the force acts. This distance,
in rotation, is not , which is
purely numerical, nor is it the lever arm, because the length of the lever
arm is not the distance through which the force acts. The effective distance
is s. Thus the work is not Fs
x (torque x angle), but F x
s (force x distance). Torque
is actually a force, and the lever arm belongs with the angular displacement,
not with the force. Its numerical value has been moved to the force merely
for convenience in calculation. Such transpositions do not affect the
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. For instance, an examination in the light of this relation makes it evident that identification of the vector H as the magnetic field intensity is incorrect. The role of this quantity H in magnetic theory has been primarily that of a mathematical factor rather than an expression of an actual physical relation. As one textbook comments, “the physical significance of the vector H is obscure.”88 (This explains why there has been so much question as to what to call it.) Thus there has been no physical constraint on the assignment of dimensions to this quantity. The unit of H in the SI system is the ampere per meter, the dimensions of which are s/t x 1/s = 1/t. It does not necessarily follow that there is any phenomenon in which H can be identified physically. In current flow, the quantity 1/s appears as power. Whether the quantity 1/t has a role of this kind in magnetism is not yet clear. In any event, H is not the magnetic field intensity, and should be given another name. Some authors tacitly recognize this point by calling it simply the “H vector.” As noted earlier, the magnetic field intensity has the
dimensions t ## Field Intensity or Flux Density
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
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 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. The magnetic scalar potential has been omitted from the tabulation, for the reasons previously given, together with a number of other quantities identified in the contemporary magnetic literature in connection with individual magnetic phenomena that we are not examining in this volume, or in connection with special mathematical techniques utilized in dealing with magnetism. The dimensionally incorrect SI units for MMF and magnetic field intensity are likewise omitted. ## Table 31: Magnetic Quantities
There is a question as to how far we ought to go in attaching
different names to quantities that have the same dimensions and are therefore
essentially equivalent. It would appear that the primary criterion should
be usefulness. It is undoubtedly useful to distinguish clearly between
electric quantity (space) and extension space, but it is not so clear
that this is true of the distinction between the various quantities with
the dimensions t 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
From reference 89, with space-time dimensions
added
Correct analogs (magnetic = electric x t/s)
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. |