IEEE TRANSACTIONS ON EDUCATION, VOL. E-15, NO. 1, FEBRUARY 1972

Electrical Units and

Electromagnetic Field Vectors

ALBERT GUISSARD, SENIOR MEMBER, IEEE

Abstract-This paper attempts to present more clearly theprinciples underlying the definition of a system of electricalunits. First, it recalls briefly how fundamental and derivedquantities are defined and how fundamental quantities arechosen. The main laws of electromagnetism are then writtenin a general form using the least number of arbitrary constants,namely three, compatible with the assumption that all theelectrical units are derived in a consistent manner from theunits of charge and current. It is shown how a system of unitsis selected and how this choice determines the values and di-mensions of the three constants, two of which are just eo and1o. These considerations lead us to a discussion of the moreor less fundamental nature of the four vectors describing theelectromagnetic field and of the different ways in which thesevectors can be associated.

1. INTRODUCTIONIn teaching electromagnetism or electrical measurements,

one must face the problem of a clear definition of electricalunits. This problem is a difficult one and is directly related toseveral basic questions: how many electrical quantities may bechosen as fundamental, what is the exact nature of the con-stants eo and po, what is the difference between the field in-tensities and the inductions?In many textbooks, electrical units are introduced in a very

intricate manner on the basis of historical development. Someauthors assert that it is impossible to construct an absolutesystem of electrical units, i.e., a system based entirely onmechanical units; others consider that the introduction of afourth fundamental quantity, electrical in nature, is only aquestion of convenience. For some, the constants c0 and poare independent quantities describing some property of freespace, fof others, they are just constants resulting from thechoice of a particular system of units.These various view-points are quite different from one

another and imply distinct interpretations of the very natureof the field vectors. We felt the need to examine all theserelated questions carefully and we try to throw some light onthem.

Manuscript received March 20, 1971; revised July 20, 1971.This work is based on an earlier paper (On the Definition ofElectrical Units, Revue E - Belgium, vol. 6, pp. 189-196,1970), which however has been revised and completed withrelativistic arguments.The author is with the Department of Electrical Engineering,

Lovanium University, Kinshasa, Rep. Dem. du Congo.

2. FUNDAMENTAL AND DERIVED QUANTITIESOne basic problem with regard to the definition of a system

of units in a specific field is the following: may we choose theunits of some quantities in an arbitrary way and, if so, howmany? The classical answer is [I]: if we are concerned withMquantities, related byN independent physical laws, and ifM isgreater than N, we are free to choose the units ofM - N quan-tities. These quantities are called fundamental. The remainingunits are then defined with the help of theN relations, inwhich arbitrary and purely numerical constants are introduced,usually of simple value. These are the derived units. Such asystem of units is said to be a "consistent system."Length and time are usually considered as fundamental

quantities. The units of velocity and acceleration are derivedunits defined respectively by v = Ilt and a = v/t. We wish insome cases to choose arbitrarily the units of more thanM - Nquantities. This is always possible but automatically intro-duces non-arbitrary and dimensional constants in these rela-tions which otherwise would be used to define these units. Ifwe arbitrarily choose the units ofM - N + A quantities, we canonly simplify N - A relations. If, for instance, we assign anarbitrary unit to velocity the preceding relation becomes v =

kllt, with k a constant determined in value and dimensions bythe three units of length, time and velocity.The fundamental quantities are normally regarded as un-

related; so are length and time as long as we do not enter thefield of electromagnetism. Let us consider here another ex-ample related to the definition of the unit of mass [2]. Theconcept of mass was first introduced in the fundamental lawof dynamicsf= k'ma. This law contains two quantities: massand force; it was therefore natural to consider one of these as afundamental quantity and to define its unit arbitrarily. Themass is usually chosen as fundamental and the unit of force isthen defined by f = ma, with the constant k' numericallyequal to unity. Actually mass and force are not unrelated.They appear indeed in another fundamental law: the universalattraction law f -k"m m2/r2, in which the value and dimen-sions of the constant k" are determined by the already madechoice of the units of mass and force(1). It is the universalgravitational constant. But why not define the unit of force,writingf- m1 m2I/r2 with k" numerically equal to unity andthe same unit of mass as before? With this choice, Newton'slaw would become f = k'ma where k' = 1/k". In MKS units,

(1)In the present discussion, we assume as proved the iden-tity of inertial mass and gravitational mass whose proportion-ality is an experimental fact (see for instance [3]).

41

IEEE TRANSACTIONS ON EDUCATION, FEBRUARY 1972

the values of these constants are with a very small error k=6.670 X 10-1 I m3/kg sec2 and k' = 1.5 X 1010 kg sec2/m3 .This example shows clearly that mass is not a more funda-

mental quantity than force and that the usual choice of massis quite arbitrary. Furthermore we can construct a system ofunits in which mass and force are both derived quantities [211and the constants k' and k" are both numerically equal tounity. Indeed let us imagine an experiment which would con-sist in determining the acceleration of a mass m2 in the gravita-tional field of a mass ml: we must have the equality m2a =m1m2/r2 and consequently ml = ar2 . In this system the unitof mass is a mass of such a value that it gives a unit of accelera-tion to any other arbitrary mass at a unitary distance. Its di-mensions are [m] = L3 /T2 and the dimensions of force are[f] = L4/T4; in these relations the notations L and T standrespectively for the units of length and time.Let us designate this unit of mass byM and evaluate it in a

system based on the meter and the second. We write the defin-ing relation in MKS units, i.e. m2a = k"m1m2/r2 or m1 =k'ar2. By definition m -M if a = 1 m/sec2 and r = 1 m andwe getM = k' = 1.5 X 1010 kilogram. This is of course anenormous mass and the kilogram is certainly more convenientfor current practice. Let us note however that this unit ofmass is very small on an astronomical scale. For instance theearth mass is approximately equal to 6 X 1024 kilogram [3],which is equivalent to 4 X 1014 M.Our example has clearly shown that the choice of fundamen-

tal quantities has no theoretical or physical grounds. Thischoice is essentially determined by practical and historicalconsiderations. One needs fundamental units of practical sizeand fundamental quantities whose standards are easily main-tained and reproduced. On the other hand, some units havebeen in use for such a long period of time that it would bevery difficult to change them; this is the case of the ampere,volt and ohm in electricity. From the point of view of di-mensional analysis, the number of fundamental units mustnot be too low, for it then loses much of its utility. All thesereasons explain why one chooses in practice a greater numberof fundamental quantities than theoretically possible. Thesame conclusion will appear in the analysis of electrical units.

3. GENERAL EXPRESSION OF THE LAWSOF ELECTROMAGNETISM

3.1 Vacuum or free space: To see clearly how the prob-lem arises in the field of electromagnetism, let us write theprincipal laws in a general form with no particular system ofunits specified. We follow the way proposed by Durand [4]and start with Coulomb and Ampere laws written for freespace. For two charges q1 and q2 a distance r apart, the forceis given by equation (1) in which ar is a unit vector pointingfrom q1 to q2 ifF is to be the force acting upon q2. For two

mined only for a particular choice of units; the 4ir factors areintroduced for convenience.

Current and charge are not independent quantities: equation(3) expresses the experimental fact that the current flowingthrough a given section in a conductor is proportional to thecharge crossing that section per unit time. This equation isalso written with an arbitrary constant, for we have no reasonto suppose that it would just equal unity.

o k qlq2 -+F=kl 4Tr2 ar

dF =k2 4rr2 dl2 X (dlI X ar)

dq3 dt

(1)

(2)

(3)

The preceding laws are three independent equations relatingtwo electrical quantities, charge and current, to mechanicalquantities, length, time and force. We can select any two outof the three, with arbitrary values for the corresponding con-stants, to define the charge and the current units in terms ofmechanical units. As a result the constant in the third equa-tion will necessarily be fixed. Actually the study of the elec-tromagnetic field gives the exact form of the relation thatexists between the three constants, as we shall see.At this point we note that a consistent system of units, as

defined at the beginning of section 2, can be constructed forall electrical units without the introduction of any new con-stant, as soon as the units of charge and current have beenselected. We shall make such a choice in the following, butshall remember that other options are quite acceptable.Gelman [5] , for instance, presents the fundamental equationsof electromagnetism with five constants; Sommerfeld ([6],sect. 8) mentions the possibility of selecting the magnetic polestrength as a separate unit.Let us now proceed to introduce the fundamental field

quantities and to write Maxwell's equations in general form.We define the electric field intensity E produced by a chargeq at distance r as

-+ q +E =k1 47rrr 2ar (4)

so that the expression of the force acting on any other chargeq becomes

F=qE. (5)

We define in a similar manner the magnetic inductionB pro-duced by a volumic distribution of currents with density J as

B2 iV 4irr2 (JX ar)dV, (6)

current elements i, dl 1 and i2dl 2, the force is given by equa-tion (2) with the same signification for a, if dF is to be theelementary force acting upon i2d12. These equations are writ-ten with arbitrary constants k, and k2 which will be deter-

so that the expression of the force acting on any current ele-ment idl becomes

dF=idlX B (7)

42

GUISSARD: DEFINITION OF ELECTRICAL UNITS

and consequently the force upon a moving charge q with ve-locity v is

F=k3q( v X B), (8)where use has been made of (3). If we now compare the ex-pressions (5) and (8) of the forces acting on an electric charge,we see that the quantities E and k3(v X B) must be dimen-sionally identical. Making use of equations (3), (4) and (6)we obtain the following dimensional relation

[k1 ] L2 (9)[k3J2[k2J T2

and conclude that the quantity I/k3 \/k77I/2 has the dimen-sions of a velocity. Further analysis will indeed show that itexactly equals the velocity of the propagating electromagneticfield.From the definitions (4) and (6) of the fields, there is no

fundamental difficulty in obtaining the expression of Max-well's equations. They are

- aBcurl E= - k3 , (a) divE=k1p, (c)

curl B = k2 J + ) (b) div B =0. (d)

(10)

The charge and current densities, p and J, are related by

apdivJ=-k3 at. (11)

Consider a domain without sources or dissipation (p = 0, J= 0)and eliminate B from Maxwell's equations; the result is

1a2EAE--- =0 (12)

with the notation

c 3 4 (13)k3 k2

An elimination of E gives an identical equation for the field B.We have just shown that the quantity c has the dimensions ofa velocity. It is well known that the solutions of equation (12)represent waves propagating precisely with the velocity c,which is the velocity of light measured experimentally and ex-pressed in the MKS system of units as (1)

c = 2.99790 X 108 m/sec,

or with an excellent approximation

c= 3 X 108 m/sec. (14)

The velocity c has of course different numerical valuesaccording to the units chosen for the measure of length andtime: the important fact is that it has a quite clear and un-

(')Value quoted by Stratton [7].

questionable physical signification. This is obviously not truefor the three constants k, , k2 and k3: they do not representany physical property of free space but merely appear as con-stants of proportionality in the experimental laws (1) to (3).Their dimensions and numerical values depend specifically onthe system of units. They are however not independent as aconsequence of the fact that current is composed of movingcharges: their interrelation is explicitly given by (13).3.2 Physical medium: Let us now consider what modifica-

tions must be made in the preceding equations if the mediumis not free space. A physical medium is usually assumed to beequivalent to an electric and/or a magnetic polarization (seefor instance [7] or [8] ). In an electric field, the medium isdescribed by a set of equivalent electric dipoles with momentp = qd, a charge multiplied by a distance. The electric polar-ization P is defined as the moment per unit volume; its dimen-sions are thus Q/L2 where Q stands for the unit of charge. IfP is a function of position, there exists an equivalent chargedensity

Pe = - div P . (15)It is assumed that for equation (lO.c) to remain valid, one

must add this equivalent charge density of the medium to thefree charge density p. There results

divE=k (P+Pe).

So as to give to this equation the usual form divD = p, wedefine a new field vector, the electric induction (or electricdisplacement) D, as

-+ 1 -+ -eD=- E+P.

k, (16)

We note that the dimensions ofD and P are identical, namelyQ/L2 ; the dimensions of E on the other hand depend upon thechoice made for the constant kl. If the electric polarizationP is a function of time,there also arises an equivalent currentdensity Je whose value is, if we remember equation (3) or(11),

e aPJe=k3 at. (17)

In a magnetic field, the medium is described by a set ofequivalent magnetic dipoles with moment m = iS, a current

multiplied by a surface. The magnetic polarizationM is de-fined as the moment per unit volume; its dimensions are thus

IIL where I stands for the unit of current. IfM is a functionof position, there exists an equivalent current density

Jm =curlM. (18)It is assumed that for equation (lO.b) to remain valid, onemust add the equivalent current densities of the medium, Jeand Jm, to the free current density J. One obtains

curl B = k2(J +Je+ Jm + = k2( J + JM 3 ak,3t ~ "~ at /

43

IEEE TRANSACTIONS ON EDUCATION, FEBRUARY 1972

where the second form results from the definition (16). Inorder to give its usual form to this equation, we define a newmagnetic vector, the magnetic intensity H, as

-) 1 -* -H=-B-M. (19)

k2

The dimensions ofH andM are identical, namely IIL; the di-mensions of B depend upon the choice made for the constantk2.Maxwell's equations (10) take the following new form

-o aB -ocurl E = -k3-, (a) divD = p, (c)

at

-~~~-~~~aD ~~~(20)curlH=J+k3 at , (b) divB =0, (d)

which is the familiar form in the MKS system of units, butfor the constant k3. Let us note that for more generality wecould have defined D and H in equations (16) and (19) witha constant multiplier [5] . But we decided to construct a con-sistent system of units; therefore we have chosen unity as thevalue of both constants.We have followed here, in order to define the field vectors,

the usual pattern, i.e. the "amperian-current model" as op-posed to the "magnetic charge model," as Fano et al. [91 callthem. This latter model differs in two respects from theformer. Firstly the magnetized matter is represented by anequivalent "magnetic charge" density and an associated"magnetic current" density, in a perfect analogy with therepresentation of polarized matter. Secondly, Maxwell's equa-tions are written with the pair E,H as a starting-point, in sucha way that H appears as the fundamental magnetic quantityand B as the derived one. The final result is however the same:Maxwell's equations take the form (20), with the relations(16) and (19) between the field vectors.From Maxwell's equations in the form (20), we deduce

without difficulty the expression of Poynting theorem; it is

-JA I(E X H) dS =Jy-J-E +E-* +H*---dV,

(21)

where S is a closed surface containing volume V. In order togive the usual interpretation of this relation, we have to defineenergy (or work) and power. This is done in the same manneras for mechanical quantities. Let a charge move in a stationaryelectric field; the force on it is given by (5) and the work Weffected by the charge going from A to B is

w= F*dl =qA 5,

with the usual definition of the potential difference A'F. Thework per unit time, or power, is

q 1P= - ==-i As

and the power density, or power per unit volume,

P 1- =-I JE.

Thus we may interpret Poynting theorem as expressed by (21)in the usual manner and the Poynting vector is

-+1 IEXH)*Hl = - (E X H). (22)

3.3 Linear medium: In the more general case, the electro-magnetic field is thus described by a set of six vectors: the in-tensities E and H, the inductions D and B, and the polariza-tions P andM . These vectors are related by the six equa-tions (20), (16) and (19), of which only four are really inde-pendent for one knows that equations (20.c) and (d) are justconsequences of equations (20.a) and (b). Two more equa-tions are thus needed: these are the state equations of themedium relating the polarizations to the field. Their generalform is

P -=P(E ) and M =M(B), (23)

and they depend on the properties of the medium.Let us restrict the analysis to the important case of a linear

medium: the polarizations are by definition linearly related tothe corresponding fields. If we further assume that the me-dium is isotropic, the corresponding vectors are parallel andtheir magnitudes are proportional. The relations (23) takethe simple form

P=oiE and M=OH, (24)

where M is expressed as a function ofH to agree with usualpractice [7] and a- and ( are constants('). The defining rela-tions (16) and (19) forD and H take the standard form

D=eE and B-=IH, (25)

if we put

e=- (1+akl) and p=k2(1+3) (26)

Denoting by e0 and po the values taken by the constants e andA for free space (a = , = 0), we write

e = COer and j = gogr, (27)

with

co = - Hlo = k2, (a) (28)

and

Cr = 1+oak1, Pr = 1+ 3. (b)

Thus the so-called permittivity (or dielectric constant) andpermeability of free space, i.e. e0 and Po, appear merely to bethe constants of the defining equations (1) and (2) for charge

O')The relations (24) also imply that the medium has no lossesand no dispersion.

44

GUISSARD: DEFINITION OF ELECTRICAL UNITS

and current. With the notation (28.a), the velocity of light(13) becomes

k3=k (29)

The eventuality of more general state equations than (23) isdiscussed in [10] . We also mention here that the constituentrelations (25) are restricted to the case of a linear and isotropicmedium at rest. For a moving medium, these relations losetheir simplicity; indeed, as is shown by Sommerfeld ([6],sect. 34), following Minkowski's line of thought, D and Bdepend on both E and H, the velocity v of the medium, andthe parameters e and p. But this fact has no more direct con-sequences on the question we are concerned with, than, forinstance, the consideration of a non-isotropic medium.

4. CHOICE OF UNITS

We are now ready to investigate how the electrical units canbe chosen. As already pointed out, we dispose of the threerelations (1) to (3) to define two electrical units, namely theunits of charge and current, in terms of mechanical units only.Thus all the electrical units can be defined in terms of the me-chanical units: a system constructed in that manner is calledabsolute [1] . Relations (1) to (3) contain three constants, ofwhich two are thus arbitrary; the third is determined by thechoice of the charge and current units. The form of the rela-tion between these constants appears in the development ofelectromagnetic theory and is given by (13).As an example, let us consider how the fundamental units

are defined in the MKS system which is more and more widelyaccepted owing to its great advantages (see for instance [ 11] ).The choice of the fundamental units in this system is guidedby the care to maintain, for practical reasons, units which arein use for a long time, namely the ampere, coulomb, voltand ohm. This set of units is known as the "practical units":they were first introduced as multiple or submultiple of theCGS units to avoid the impractical size of the latter (for thedefinitions, see for instance Harris [1] on p. 25).The MKS system is based on the meter, the kilogram mass

and the second respectively as the units of length, mass andtime; the corresponding unit of force is the newton. If equa-tion (2) must give a correct answer for the force in newtonwith the currents expressed in ampere, the constant k2 (i.e.po) must have the value

k2 =po = 4r X 10-7, (30)

and the definition of the ampere ([12], [13] ) is a direct con-sequence of this choice. Since 1 ampere = 1 coulomb/I sec-ond, the constant k3 in (3) is numerically equal to unity, ifthe coulomb is selected as the unit of charge. Equation (13)then gives the following value of kI (i.e. 1/co), using theapproximate value (14) for c,

k1=- 36ir X 109. (31)

It is easily shown, by the method of dimensional analysis, thatk, is dimensionally equal to m/F and k2 to H/m, where F and

H stand respectively for the units of capacitance (farad) andinductance (henry) in the MKS system.By introducing the value (31) of k1 in (1) and expressing

the charge in coulombs, one necessarily gets the correct an-swer for the force in newtons, as Giorgi first pointed out(').Thus the MKS system of units can be constructed as a con-sistent system without the need to modify the practical units.Let us add that the ampere (or the coulomb) is usually con-sidered as a fourth fundamental unit in the MKS system(which is then designated as MKSA or MKSC). This causes nospecial difficulty, because in any system of units we may atany time consider a unit, first introduced as a derived unit, asa fundamental unit without any modification of the con-stants provided that we do not change the size of that unit.The principal reason for having a fourth electrical unit is togive a better signification to the dimensions of electrical quan-tities and to dimensional analysis(2).The only disadvantage of the MKS system is to maintain two

constants which have dimensions and differ from unity. Froma theoretical point of view, there is no objection to puttingtwo among the three constants numerically equal to unity orto any other simple value. Let us note that the only constantappearing in Maxwell's equations in the general form (20) is k3and that a choice such as k3 = 1 (numerically) is the more con-venient for the study of the electromagnetic field and at thesame time simplifies the relation (3) between current andcharge. We recall that the choices made in the CGS systemsare the following(3):

- CGS electrostatic system: k, =4ir, k3 = 1;- CGS electromagnetic system: k2 = 4ir, k3 = 1.

But one might also put k1 and k2 numerically equal to unityand define the units of charge and current in terms of mechan-ical units using (1) and (2). The corresponding value and di-mensions of k3 are given by (13) as k3 = 1/c. Such a choicewould lead to rather surprising consequences(4): voltage hasthe same dimensions as current, resistance and impedance be-come pure numbers and the four vectors E, D, B,H have thesame dimensions. The most unusual consequence is perhapsthat current is no more a charge per unit time but a chargeper unit length.There exists one theoretical way of taking the simplification

further and of making all three constants equal to unity(5).

(1) For reference, see Stratton [7] .(2) Dimensional equations written with the three fundamen-

tal units of mechanics only contain many fractional exponents,which is not quite satisfactory [7] . For a further discussion ofthe MKS system, see the reference given in Golding ([14],ch. 2).

(3)The CGS systems are usually of the unrationalized type;in that case relations (28) are replaced by k 1 /4ir = 1/co andk2/47r=p.

(4)Our attention was drawn to this by M. Morren.(')But not all three dimensionless, unless we decide to give

up one of the fundamental units of mechanics by defining c tobe unity and dimensionless. In such units L = T ([8], Appen-dix 1).

45

IEEE TRANSACTIONS ON EDUCATION, FEBRUARY 1972

We have merely to give up the conventional mechanical unitsand adopt a system in which the unit of velocity is the velocityof light [15]; for instance we can maintain the unit of time(the second) and chose as unit of length the distance trav-elled by light in one second. We do not intend to propose theadoption of such a system, because it is certainly not wellsuited to current needs, but we mention it to bring out moreclearly the arbitrariness of the value and dimensions of theconstants ki, and thus of eo and,o. Let us note however thatsuch a system would perhaps be more convenient for astro-nomical needs, as was the system based on mass M.

5. DISCUSSIONWhat has been said so far will now help us to examine a

much debated question in electromagnetism. The electro-magnetic field is described with the help of four vectors: twofor the electric field, namely E and D, and two for the mag-netic field, namely B and H. The question then is twofold:firstly are the intensities (E, H ) of a different nature than theinductions (D, B ); secondly are two of these vectors morefundamental than the other two?To begin with, let us make a remark about the nature of the

constants co and 11o. Owing to the way these two constantsare introduced, we must not consider them as more funda-mental than the universal constant of gravitation. Just like thelatter, they are essentially determined as for their value and di-mensions by the choice of the units. Deprit and Rouche [2]disagree with this point of view, but Stratton [7] recalled thatno experience had been made which would allow us to assigndimensions to eo and 1.o, considered as independent physicalquantities. Although its value also depends on the units, theconstant c has a quite definite physical signification as thevelocity of light and of the electromagnetic field in free space.The constants eo and .Lo on the other hand must certainly notbe regarded as representing some properties of vacuum. Theterminology commonly used to designate them is very con-fusing and should preferably be avoided (see discussion fol-lowing [16] ). Appellations such as [13] "electric constant"for eo and "magnetic constant" for go would be much bettersuited.Let us now try to answer the first question above. As long

as we consider free space or a medium where no polarizationexists, the answer seems clear. Since eo and -o only dependon the choice of the units, there is no fundamental reason to

consider E and D, or B and H, as quantities of different na-ture. In a physical medium where polarizations do exist, thesevectors are not identical. Let us choose a system of units inwhich the electric and magnetic constants are equal to unityand have no dimensions (this is the case for the Heaviside-Lorentz system for instance [8] ). Equations (16) and (19)become simply

D=E+P and H=B-M (31)

and show that the difference between D and E (or H and B )is just the contribution of the medium. Now in any system ofunits we have the relations

- and 1 -+I -

D=eoE+P and H=-B-M,Ao

(32)

which differ from (31) only through the introduction of theelectric and magnetic constants. The interpretation is just thesame, even if the dimensions of the intensities and inductionsare not identical.The form of Maxwell's equations shows however that the

intensities, E and H, do not play the same role as the induc-tions, D and B. The intensities appear in the equationsthrough their circulations, whereas the inductions appearthrough their flux. So we must certainly make a distinction,but this does not necessarily imply a difference in nature, un-less we hold with Langevin [15] that two physical quantitiesare of the same nature only if their measures or the measuresof their components are equal independently of the systemof units or of the system of reference.

In his work quoted above ([6], sect. 2 and 5), Sommerfeldassociates, like many authors, the two vectors E and B, andthe two other vectors D and H: according to him, the first twowould be "entities of intensity," the last two "entities ofquantity." To emphasize this point of view, he introducesnew designations as follows: electric and magnetic "fieldstrengths" for E and B, electric and magnetic "excitations"for D and H. This distinction is put however as a kind ofarbitrary definition with no clear justification on a physicalbasis. An argument of energetic nature should help but seemsto fail: if the variation of electrical energy 6 We = E * 6Dcorresponds to the pattern "intensity entity X change inquantity entity," the variation of magnetic energy 6 Wm =H * 6B does not. Sommerfeld infers from this that "workneed not be expressible in the form intensity X change inquantity entity."In part III of the same work, another argument is provided

from a relativistic point of view: the vectors cB and - iF arecombined to form an antisymmetric tensor of the second rank(or "six-component field vector") F in a four-dimensionalspace. But this study, as well as another classical one [17], islimited to space free from matter where in any case B is thesame vector as MoH. And if we note that cMo =-oU=O0we may define the tensor F from the equivalent pair r?H, -iE.The case of a material medium is considered briefly in part IVof Sommerfeld's work: we understand, however, that the rela-tionship between the field vectors, proved for free space onthe basis of the Lorentz transformation and expressed in termsof the pair E, B, is applied without any particular justificationto a medium with material constants e and ,u differing fromthose for vacuum. This relativistic viewpoint is also discussedby Darrieus in [16] and [ 18].Let us next approach the second question. According to the

classical presentation we have recalled, we might logically con-sider E and B as more fundamental vectors, for their definitionis directly based on charge and current in a similar manner.

The vectors D and H are introduced afterwards in order tosimplify Maxwell's equations and to avoid, in the case of alinear medium at least, the necessity of taking the polariza-

46

GUISSARD: DEFINITION OF ELECTRICAL UNITS

tions explicitly into account: they are something like an inter-mediate calculation means. This is the point of view of manyauthors (see for instance [7] and [8] ). We might further notethat the forces acting upon a moving electric charge are givenby (5) and (8), as functions of the same vectors E and B. Butthis seems a weaker argument: these expressions are valid forfree space where B E,uoH and it is very difficult to experi-ment the exact force acting on a charge in a material medium,because one must first make some kind of cavity and, by sodoing, one completely modifies local conditions.On the other hand, if we consider Maxwell's equations (20)

and the resulting expression (21) of Poynting theorem, we seea clear association between the intensities E and H, and theinductions D and B, an association which has already beennoticed and seems to contradict what we have just said. Thereis no way out of it and authors like Fano et al. [9] whochoose the magnetic charge model of magnetized matter andconsider E and H as the fundamental pair of field vectors areperfectly right. Would it not be better to admit that the pres-ence of matter obliges us to describe the electromagnetic fieldwith four vectors, which are associated in one or other mannerdepending on the approach? Note that we could describe thefield with the four quantities: E, P, B andM as with: E, D, Band H; but the latter choice gives more symmetric equations.In the absence of matter, two vectors are obviously sufficient,from a classical as well as from a relativistic point of view.Let us add that the field vectors have not exactly the same

signification in all cases. Tai [191 discusses the differentforms of Maxwell's equations and shows that, in the case of amoving medium, the field vectors E and H do not have thesame meaning in these different forms. More definitely, let vbe the velocity of the medium as measured in the frame ofreference in which all field vectors are evaluated. Then onegets the following linear transformation between the vectorsin the usual amperian-current model (subscript "a") and in themodel of Chu ([9], ch. 9) (subscript "c")

Ba =BC

Da =Dc(-* -_* -+ -+ (33)Ea = EC +ioM, X V,

Ha=Hc-PcX V,

which imply distinct definitions of the polarizations as follows

Pa =PC - OMcXvMa= <c+X (34)Ma = MC + PC X V.

As a consequence, the classical expression of the Lorentz forceEa + v X Ba is identical with the expression of Fano et al. [91EC + V X uoHcPThe following consideration might contribute to the present

discussion. Since the definition of all electrical units is inde-pendent of mass, as soon as charge, or current, is chosen as afourth fundamental unit, some authors propose that length,

time and two electrical quantities be taken as fundamental.Dimensional equations become particularly simple if, asKalantaroff suggests ([4], [18], [20]), the two fundamentalelectrical quantities are electric charge and magnetic flux. IfQ and 4' stand for the units of charge and flux, we obtain thefollowing dimensional equations for the field vectors:

[El]=-, [H]=QL T' L T

[B] ='y [D = Q.

These four relations clearly show a twofold correspondence be-tween the four field vectors and also a striking similarity iftime is considered as the fourth dimension of a four-dimen-sional space.

6. CONCLUSIONWe shall draw two main conclusions from the above discus-

sion. On the one hand, we consider that the only fundamentalproperty of free space is the light propagation velocity. Thepermittivity and permeability of free space are simply con-stants whose values and dimensions are essentially fixed by thechoice of a particular system of units and which, to avoid con-fusion, could be best designated as the electric and magneticconstants. The properties of a linear medium are described bythe permittivity and permeability of the medium, which is theeasiest way to account for the contributions of electric andmagnetic polarizations in matter. We showed moreover thatas a matter of fact every system of electrical units is absoluteand that the introduction of a fourth fundamental quantityappears to be mainly a question of convenience and not ofnecessity.On the other hand, the electromagnetic field is best con-

sidered as described by a set of four vectors related throughMaxwell's equations and the state equations. These vectorscan be associated in two ways according to the point of viewadopted but there seems to exist at present no firm basis forregarding one pair as more fundamental than the other.Classical electromagnetism provides no definite answer to thequestion of the existence of a fundamental difference in naturebetween the intensities and the inductions, although Maxwell'sequations clearly indicate a distinction as to the vectorialcharacter of these quantities. Moreover, the theory of rela-tivity, although emphasizing the different definitions of thefield vectors for moving mediums, does not seem to provideus with clear arguments to answer the question in hand.Anyway, the four electromagnetic vectors obviously reduceto two distinct vectors in empty space.

BIBLIOGRAPHY[1] F. K. Harris, Electrical Measurements. New York, J.

Wiley, 1952 (chap. 2).[2] A. Deprit, N. Rouche, M6canique rationnelle. Louvain,

Librairie Universitaire Uystpruyst, 1963 (chap. 7, par. 5).[3] F. W. Sears, Mechanics, Wave Motion and Heat, Reading,

47

IEEE TRANSACTIONS ON EDUCATION, VOL. E-15, NO. 1, FEBRUARY 1972

Massachusetts, Addison-Wesley Publ. Co., 1958 (p. 649).[4] E. Durand, Electrostatique et magn6tostatique. Paris,

Masson et Cie., 1953 (chap. 20).[5] H. Gelman, Generalized conversion of electromagnetic

units, measures and equations. Amer. J. Phys., vol. 34,no 4, pp. 291-5 (1966).

[6] A. Sommerfeld, Electrodynamics (Lectures on Theoreti-cal Physics, part III), New York, Academic Press, 1964.

[7] J. A. Stratton, Electromagnetic Theory, New York,McGraw-Hill, 1941 (chap. 1).

[8] W. K. H. Panofsky, M. Phillips, Classical Electricity andMagnetism. Reading, Massachusetts, Addison-Wesley,1962.

[9] R. M. Fano, L. J. Chu, R. B. Adler, ElectromagneticFields, Energy and Forces. New York, J. Wiley, 1960(par. 5.3, 5.4, 7.1, 7.10).

[10] D. T. Paris, L. Padulo, Maxwell's Equations and Deter-minate Systems. Amer. J. Physics, vol. 33, n° 5,pp. 410-1 (1965); vol. 34, no 7, pp. 617-9 (1966).

[11] P. Grivet, Apologie du systbme MKS Giorgi. Bull. Soc.Fr. Electriciens, vol. 1 n° 10, pp. 630-44 (1951).

[12] P. Poincelot, Pr6cis d'6lectromagn6tisme thdorique.Paris, Dunod, 1963 (pp. 67, 69).

[13] J. Terrien, The Work of the Bureau International desPoids et Mesures Concerning Electromagnetic Units

and Measurements. IEEE Trans. on Instr. and Measure-ments, vol. IM-15, n° 4 (Dec. 1966).

[14] E. W. Golding, Electrical Measurements and MeasuringInstruments. London, Pitman and Sons, 4th edition,1955; 5th edition, 1963, with F. C. Widdis.

[15] P. Langevin, Sur les grandeurs champ et induction. Surla nature des grandeurs et le choix d'un systWmed'unites electriques. Oeuvres scientifiques, Paris, CNRS,1950, pp. 491-505.

[16] M. G. Darrieus, Champ et induction magnetiques. Bull.Soc. Fr. Electriciens, vol. 4, no 43, pp. 381-400 (1954).

[17] L. D. Landau, E. M. Lifshitz, The Classical Theory ofFields, Oxford, Pergamon Press, 1962.

[18] M. G. Darrieus, La rationalisation dans le systdmed'unites Giorgi et l'enseignement elementaire de l'elec-trotechnique. Bull. Soc. Fr. Electriciens vol. 1,no 10, pp. 614-629 (1951).

[19] C. T. Tai, A Study of Electrodynamics of Moving Media,Proc. IEEE, vol. 52, no 6, pp. 685-689 (June 1964).

[20] A. Ferry, Grandeurs et unites-Expose critique desprincipaux systWmes-SystWme Giorgi. Paris, Gauthier-Villars, 1956.

[21] M. G. Darrieus, Les unites de la mecanique et de l'elec-tricite et le systWme Giorgi. L'industrie Nationale, 1949,pp. 163-178.

Microwave Experiments Using

a Nanosecond Pulse Generator

and a Sampling Oscilloscope

K. E. LONNGREN

Abstract-This paper describes four experiments using ananosecond pulse generator and a sampling oscilloscope whichare useful in an introductory microwave course.

I. INTRODUCTIONThe rapid development of subnanosecond pulse generators

and high frequency (> 1 GHz) sampling oscilloscopes can alsochange some pedagogical techniques in an "Introductory Mi-

Manuscript received June 26, 1970. This work was sup-ported in part by the National Science Foundation.The author is with the Department of Electrical Engineer-

ing, University of Iowa, Iowa City, Iowa 52240.

crowave Theory and Techniques" course usually taught at thesenior-graduate level. The "State-of-the-art Computer" coursemust also cope with some of the problems previously asso-ciated only with microwaves in transmitting a pulse betweenvarious logic circuits. It is the purpose of this paper to suggestfour microwave experiments which we have found useful andquite instructive for either of these two courses.

A) Time-of-flight measurement of propagation velocity.B) Reflection coefficient determination.C) Dispersion in a waveguide.D) Physical interpretation of the Q of a resonant cavity.

In Section II we describe the experiments and present typi-cal results. All equipment used in these experiments is com-

48