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 bedeviled 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 all physical phenomena and relations can be expressed in terms of space and time not only enables identifying the correct relations, but is also an impressive confirmation of the validity of the basic postulate which asserts that the physical universe is composed, in its entirety, of units of motion, an entity defined as a reciprocal relation between space and time.
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, inductance, symbol L, is the term applied to the quantity involved in the production of an electromotive force in a conductor by means of variations in the current. The mathematical expression is
F = -L dI/dt
L = t/s2 × t/s × t = t3/s3
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 d2s/dt2
F =L dI/dt = L d2q/dt2
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 energy ½LI2. We find experimentally that when a current I flowing through an inductance L is destroyed, an amount of energy ½LI2 does make its appearance. The explanation on the basis of conventional theory is that the energy is “stored in the electromagnetic field,” but the identification of L with mass now shows that the expression ½LI2 is identical with the familiar expression ½mv2, and that, like its mechanical analog, it represents kinetic energy.
The inverse of inductance, t3/s3, is reluctance, s3/t3, the resistance of a magnetic circuit to the establishment of a magnetic flux by a magnetomotive force. As can be seen, this quantity has the dimensions of three-dimensional speed.
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, 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 × 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 × t = t2/s2.
The natural unit of magnetic flux is the product of the natural unit of electric potential, 9.31146×108 volts, and the natural unit of time, 1.520655×10-16 seconds, and amounts to 1.41595×10-7 volt-sec, or webers. The natural units of other magnetic quantities can similarly be derived by combination of previously evaluated natural units.
Magnetic flux density, symbol B, is magnetic flux per unit area. The space-time dimensions are t2/s2 × 1/s2 = t2/s4. The units are the gauss (cgs) or tesla (SI). Magnetic potential (also called vector potential), like electric potential, is charge divided by distance, and therefore has the space-time dimensions t2/s2 × 1/s = t2/s3. The cgs unit is the maxwell per centimeter, or gilbert. The SI unit is the weber per meter.
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. The investigators working in this area also recognize what they call a magnetic scalar potential, which they define as B ds/m, where s is space and m is a quantity with the dimensions t3/s4 that will be defined shortly. The space-time dimensions of the scalar potential are thus
t2/s4 × s × s4/t3 = s/t. The so-called scalar potential is therefore a speed, equivalent to an electric current, a conclusion that agrees with the units, amperes, in which this quantity is measured. W. J. Duffin comments that it is not easy to put a physical interpretation on magnetic scalar potential.85The space-time dimensions of this quantity explain why. A potential (that is, a force) equivalent to a speed is a physical contradiction. The scalar potential is merely a mathematical construction without physical significance.
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 permeability, symbol m, a quantity that enters into these and other magnetic relationships. The permeability of the great majority of substances is unity, or a close approximation thereto. The numerical results of magnetic measurements on these substances therefore give no indication of its existence, and there has been a tendency to overlook it, except where some collateral relation makes it clear that there are missing dimensions. But its field of application is actually very wide, as our theoretical development indicates that permeability is the magnetic equivalent of electrical resistance. It has the space-time dimensions of resistance, t2/s3, multiplied by the factor t/s that relates magnetism to electricity, the result being t3/s4.
One of the empirical results that has contributed to the dimensional confusion is the experimental finding that magnetomotive force (MMF), or magnetomotence, is related to the current (I) by the expression MMF = nI, where n is the number of turns in a coil. Since n is dimensionless, this empirical relation indicates that the dimensions of magnetomotive force are the same as those of electric current. The SI unit of MMF has therefore been taken as the ampere. It was noted in Chapter 9 that the early investigators of electrical phenomena attached the name “electromotive force” to a quantity that they recognized as having the characteristics of a force, an identification that we now find to be correct, notwithstanding the denial of its validity by most present-day physicists. A somewhat similar situation exists in magnetism. The early investigators in this area identified a magnetic quantity as having the characteristics of a force, and gave it the name “magnetomotive force” . The prevailing view that this quantity is dimensionally equivalent to electric current contradicts the conclusion of the pioneer investigators, but here, again, our finding is that the original conception of the nature of this quantity is correct, at least in a general sense. Magnetomotive force, we find, is the magnetic (two-dimensional) analog of the one-dimensional quantity known as force. It has the dimensions of force, t/s2, multiplied by the factor t/s that relates electricity to magnetism.
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 Ampère’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 Ampère’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 × 1/s = t3/s4, which agrees with the previous determination. The SI units henry/meter and newton/ampere2 (t/s2 × 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. This situation will be given further consideration later in the present chapter.
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 t2/s3= t3/s4 × s/t. The dimensions t2/s3, which are assigned to MMF on this basis, are the appropriate dimensions for the magnetic analog of electric force, as they are the dimensions of force, t/s2, multiplied by t/s, the dimensional relation between electricity and magnetism.
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 magnetic field intensity is the potential gradient, and should therefore have the dimensions t2/s3 × 1/s = t2/s4, the same dimensions that we found for the flux density. The cgs unit, the oersted, is one gilbert per centimeter, and therefore has the correct dimensions. However, the unit in the SI system is the ampere per meter, the space-time dimensions of which are s/t ×1/s = 1/t. These dimensions have been derived from the ampere unit of MMF, and the error in the dimensions of that quantity is carried forward to the magnetic field intensity. Introducing the permeability corrects the dimensional error.
Magnetic pole strength is a quantity defined as F/B, where F is the force that is exerted. Again the permeability dimensions should be included. The correct definition is µF/B, the space-time dimensions of which are t3/s4 × t/s2 × s4/t2 = t2/s2. Pole strength is thus merely another name for magnetic charge, as we might expect.
The permeability issue also enters into the question as to the definition of magnetic moment. The quantity currently called by that name, or designated as the electromagnetic moment (symbol m), is defined by the experimentally established relation m = nIA, where n and I have the same significance as in the related expression for the magnetomotive force, and A is the area of the circuit formed by each turn of a coil. The space-time dimensions are s/t × s2 = s3/t. The moment per unit volume, the magnetization, M, is s3/t × 1/s3 = 1/t.
An alternate definition of the magnetic moment introduces the permeability. This quantity, which is called the magnetic dipole moment to distinguish it from the moment defined in the preceding paragraph, has the composition mnIA. The space-time dimensions are t3/s4 × s/t × s2 = t2/s. (The distinction is not always effective, as some authors—Duffin, for example—apply the dipole moment designation to the s3/t quantity.) The dipole moment per unit volume, called the magnetic polarization, has the dimensions t2/s4. This quantity is therefore dimensionally equivalent to the flux density and the magnetic field intensity, and is expressed in the same units. The question as to whether the permeability should be included in the “moment” affects other magnetic relations, particularly that between the flux density B and a quantity that has been given the symbol H. This is the quantity with the dimensions 1/t that, in the SI system, is called the field intensity, or field strength. Malcolm McCaig reports that “the name field for the vector H went out of fashion for a time,” and says that he was asked by publishers to use “magnetizing force” instead. But “the term magnetic field strength now seems to be in fashion again.”87
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 Permanent Magnets in Theory and Practice. He points out that there are two theoretical systems that handle this relationship somewhat differently. “Both systems have international approval,” he says, “but there are intolerant lobbies on both sides seeking to have the other system banned.” The two are distinguished by their respective definitions of the torque of a magnet. The Kennelly system uses the magnetic dipole moment (t2/s), and expresses the torque as T = mH. The Sommerfeld system uses the electromagnetic moment (s3/t) and expresses the torque as T = mB.
Torque is a product of force and distance, t/s2 × s = t/s. The space-time dimensions of the product mH are t2/s × 1/t = t/s. The equation T = mH is thus dimensionally correct. The space-time dimensions of the product mB are s3/t × 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.
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 θ. The work is thus Fsθ. But work is the product of a force and the distance through which the force acts. This distance, in rotation, is not θ, 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 sθ. Thus the work is not Fs × θ (torque × angle), but F × sθ (force × 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 mathematical validity, but it should be understood that the modified relation does not represent physical reality, and physical conclusions drawn from it are not necessarily valid.
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 × 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 t2/s4, and is therefore equivalent to mH (t3/s4 × 1/t) rather than to H. This relation is illustrated in the following comparison between electric and magnetic quantities:
|Electric||E = V/s = t/s2 × 1/s = t/s3||Potential per unit space|
|E = R/t = t2/s3 × 1/t =t/s3||Resistance per unit time|
B = A/s = t2/s3 × 1/s = t2/s4
|Potential per unit space|
µH = m/t = t3/s4 × 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 × 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 × 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 × s3/t = t2/s4|
|Magnetization||1/t||Polarization||t3/s4 × 1/t = t2/s4|
|Vector H||1/t||Field Intensity||t3/s4 × 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 Table 31, 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.
|dipole moment||weber × meter||t2/s|
|magnetic moment||ampere × meter2||s3/t|
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 t2/s4, for example. The magnetic field intensity can be identified with these dimensions, by analogy with the electric field intensity. Perhaps there is some justification for distinguishing it from magnetic polarization, which has the same dimensions. Whether this is also true of the other t2/s4 quantities such as flux density and magnetic induction is somewhat questionable.
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 × 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.
|From reference 89, with space-time dimensions added|
|1/st||current density||t2/s4||magnetic induction|
|t/s3||electric field intensity||t2/s4||magnetic field intensity|
|Correct analogs (magnetic = electric × t/s)|
|s/t||current||no magnetic analog|
|1/st||current density||no magnetic analog|
|s2/t2||conductivity||no magnetic analog|
|t/s3||electric field intensity||t2/s4||magnetic field intensity|
|s3/t2||conductance||no magnetic analog|
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.