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Rationalization in the light of quantity equations
BrezinSCak, 1953
Marij an
Physica XIX 599-604
RATIONALIZATION
IN THE LIGHT EQUATIONS
OF QUANTITY
by MAR1 JAN BREZINSCAK Zagreb, Yugoslavia
*)
synopsis It is stated that so-called rationalization, as met in common practice, is to be considered as a transition from the numerical equations of electromagnetism, adjusted to CGSG, CGSM and CGSE units, to quantity equations, which are valid for all systems of units. minor importance, viz. the magnetic polarization lities and the corresponding electric quantities transition.
Only a few quantities of and magnetic susceptibiare to be changed in this
1. Introduction. It is a well known point of view that physical phenomena exist independently of units of measure and that the equations of physics, as symbolic expressions of theories describing these phenomena, should aso be independent of the system of units chosen. In physics and technology three types of equations are in use: 1) “quantity equations” (G. Grossengleichungen) ‘) “); in these equations the letter symbols express “quantities”, which are often represented as a product, composed of a “numerical value” and a “unit” **) ; 2) “numerical equations” (G. Zahlenwertgleichungen) r) “) ; in these equations the letter symbols represent “numerical values of the quantities”; 3) “adjusted quantity equations” (G. zugeschnittene Grossengleichungen) 1) “) ; the letter symbols in these equations represent quantities. *) Writer’s address: Zagreb (Yugoslavia), KrajiSka 13. The idea of this article was first given (in Croatian) in the Yugoslavian review “Elektrotehnicki vjesnik” 21 (1953), 81-86 (No. 3/4). l *) We refrain, however, from using this as a d e f i n i t i o n of a quantity.
-
599 -
600
MAR1 JAN BREZIN SCAK
Only the first type, the quantity equations, are intended to be universal, i.e. valid for all systems of coherent *) or non-coherent units. Only these have the property, desired by J. C. M a x w e 11: “physical equations should be valid in all countries, with application of national or international units”. They have, as is well known, a number of practical advantages. Theoretically they are correct : with quantity symbols we can calculate as with pure numbers “). The second type of equations, the numerical equations, are not universal. They have the same form for all coherent systems of units, but by transitions from a coherent system e.g. to a noncoherent system, some of the numerical equations must be changed. They may be complete but are not always (e.g. “Gaussian numerical equations” ignore the quantities permeability and permittivity of empty space). The third class of equations is sometimes used in technology and in experimental research. It does not interest us here and will not be considered in this study. Of late quantity equations have obtained favour in the eyes of many scientists. The secretariat for the first meeting of the IS0 committee also proposed the use of this class of equations “). Being convinced that quantity equations will in due time be internationally adopted, the writer would like here to express his view on the pro. blem of “rationalization” in the light of quantity equations. 2. Oersteditis. What is t.he meaning of the unit name Oersted (Oe) ? In most handbooks, conversion tables, text books, etc. we find that the “classical unit of magnetic field strength” is related to the unit “ampere per meter” (A/m) by Oe = (1000/47r) A/ m or Oe A
(1000/47c) A/m,
(1)
the symbol r\ meaning that the unit Oe “corresponds to” 1000/4n A/m. In some recent articles, however, we find another relation, viz. Oe = 1000 A/m.
(2)
This relation is given by the outstanding Sovjet metrologist Kalantarov4), bythe Swissmetrologists Haberli5) and Landolt”), by De Boer6) and by Lalou’). The Swiss Electrotechnical Committee s) and IS0 “) are of the same opinion t). *) coherent = in one-to-one relationship. t) Expressed implicitely in “) and “).
RATIONALIZATION
.IN
THE
LIGHT
OF QUANTITY
EQUATIONS
60 1
T w o Oersteds, according to (1) and (2) respectively, can, however, not exist under the same style at the same time! L an d o 1 t, K a 1 a n t a r o v etc. suppose that the so-called Electromagnetic CGS-system (CGSM), the Electrostatic CGSsystem (CGSE) and the “mixed” Gauss system (CGSG) are coherent systems of units. But this is not the case [see ‘) “) “)I! Equation (1) having been so long in use is the best argument for this; the “Oersted” in eq. (2) is new! The way in which eq. (2) may be arrived at is the following. Many physicists are of opinion that the CGSM and the CGSG unit of magnetic field strength is Oe = [&/cm = cm-*g*s-‘, in which [I],” is the “CGSM unit of magnetic current”. As [I],,, = 10 A, they obtain Oe = [I],Jcm = 10 A/cm = 1000 A/m, i.e. equation (2). They forget, however, that with equations of the type cm+g’s’, a relation of the type “second = 3.10” cm” lo) may also be obtained. This process, however, is historically false. C a r 1 F r i e d r i c h G a u s s, in his fundamental work “Intensitas vis magneticae terrestris ad mensuram absolutam revocata” (1832), founded the CGS system of units. In this work 11) he uses equations of the type *) HJH, = L-“M”T-’ , in which H, and H, are the two numerical values of the magnetic field strength (“Erdmagnetische Kraft”, according to Gauss). The symbols L, M and T, however, do not represent “fundamental units”, but r a t i o s of the corresponding members of two series of units! Gauss’s own words (see rl), sect. 26, p. 48) are very clear : “So lange wir den Weg beibehalten den wir bei unseren Beobachtungen verfolgt haben, mussen wir die erste Formel (i.e. our above mentioned equation) gebrauchen; wenn wir z.B. an stelle des Millimeters und des Milligramms das Meter und das Gramm als Einheiten annehmen, so wird L = l/1000, M = l/ 1000, also Hz = H, sein ; wenn die Pariser Linie und das Berliner Pfund, so werden wir haben : L = l/2,255829, M = l/4677 11,4, folglich H, = 0,002 196161 H , . . . . “.
Further Gauss speaks about “magpetische Fhissigkeit”, “free magnetism”, considering them as independent quantities, in contradiction to the definition now often found (S t i 11 e 12): “electricity” = “length 2/ force” or “electricity” = “time d force”). *) Writer’s
notation.
602
MARIJAN BREZIN%AK
From the above two conclusions may be drawn: 1) Gauss did not establish electromagnetic units of the type cm’gys’ and he did not calculate with such units ; 2) Gauss acknowledged the independent existance of magnetic quantities. It is evident that “units” cmxgyszand conclusions drawn from their use are of no value, and, consequently equation (2) neither.Therefore in the following we shall consider as Oersted the unit defined by eq. (1). 3. Rationalization. Generally two sets of equations of the electromagnetic field are in common use, the “rational quantity equations” (as used by S t r a t t o n, S o m m e r f e 1 d, P o h 1 etc.), and “classical (or Gaussian) non-rational equations” (as used by P a g e, etc.). The equations of the second set are J o o s, Becker numerical equations, not quantity equations, as supposed by Kalantarov4). Metrologists often start with two forms of Maxwell’s second integral equation $Hds=I
(3)
and
9;H’ds = (4n/c)lL
(4)
in which H denotes the quantity “rational magnetic field strength”, ds a differential of the quantity Jength”, I the quantity “electric current”, H’ is the numerical value of the q.uantity “non-rational magnetic field strength H’, expressed in the CGSM unit Oe, ds the numerical value of the differential of a length, expressed in cm, c = 3.10” and I, the numerical value of the quantity I, expressed in CGSE units *). As I, = cI, g), in which I, is the value of the same current I in CGSM units, we can write eq. (4) in the form $ H’ ds = 4n I,,,
(5)
We have further: H’ = If’/Oe, ds = ds/cm and I,” = I/[I],,,; from eq. (5) we thus obtain I H’ ds --= % Oe cm 47c[II,,, *) Eq. (3) is a quantity equation and eq. (4) a numerical equation. Symbols for quantities are printed in bold italics, symbols for numerical values in ordinary italics.
RATIONALIZATION
IN THE
LIGHT
OF QUANTITY
EQUATIONS
603
As [I],,, = 10 A and as relation (1) is valid, we have Oe.cm -=
[II,,‘
103Am-’ 4x
x lO”m 10A
1 =476’
and, finally, $ w’ ds = I. (6) Comparing the quantity equations (3) and (6) we see that between the quantities Hand w’ there is no difference; they are i d e nt i c a 1. We can easily find a connection between the unit of magnetic field density (magnetic induction), the gauss (G), and the unit of field strength, the Oersted (Oe). In vacuum the classical numerical equations are valid, according to which B = H, in which B = B/G and H = H/Oe (H’ = H), and the quantity equation giving B = I@, in which y, is the permeability of vacuum (pO = 47~.10m7Vs/Am). B/G = H/Oe and B = p,,H From results the well-known relation G = p. Oe = 10m4Vs/m*. The relative permeability ,LI~is the same in both cases. Denoting the quantity “non-rational permeability” with lo’ and the quantity “rational permeability” with p., we can write P, = l4Po = l-4 = P’lb’lThe unit [p’] = G/Oe = p0 is e.g. used by V o n s o v s k y and S h u r r3). From p./l.~~= ,u, = l.~‘/[k’] = ,LJ~results p = p’, that is: the quantities l.~and p. are identical. The classical (numerical) non-rational equation and the rational quantity equation of magnetic polarization are B = H + 4~ J’ = ,uI H (7) and
B = p.,H + J = pH
(8)
B e c k e r 14) and S h u r 13) stipulate that in eq. (7) the units used are G, Oe and G, respectively, i.e. B = B/G, H = H/Oe and J’ = Z/G. Eq. (7) therefore becomes the form B -=G
H
Oe + 4+.
Comparison of this result with eq. (8) gives a quantity equation J= 4n J’
604
RATIONALIZATION
IN THE LIGHT
OF QUANTITY
EQUATIONS
Therefore the “non-rational magnetic polarization” J’ and the “rational magnetic polarization” J are n o t identical quantities. The same is valid for the magnetic susceptibility “=&IL,-1
=4zz’
and for the magnetic mass-susceptibility )( = x/p = 4?cx' in which p represents the quantity mass-density. An analogous situation is found with regard to the equations of the electric field. We see, therefore, that by practising rationalization in the light of quantity equations only six quantities have to be changed (S, K’, x’ and the corresponding three electric quantities) ; all other “nonrational quantities” are identical with the corresponding “rational quantities”. The writer is indebted to Dr. W. d e G r o o t (Eindhoven) for his interest and trouble in the preparation and printing of this paper. Zagreb 2.3.1953. Received 2-5-53. REFERENCES I) W a 11 o t, J., Z. Physik 10 (1922) 329: Elektrotechnische Z. (ETZ) &l (1922) 1329, 1381; Phys. Z. 44 (1943) 17; ETZ G4 (1943) 13;see also DIN 1313. 2) L a n d o 1 t, PI., GrGsse, Masszahl und Einheit (also in French). Rascher, Ztirich 1943. 3) ISO/TC 12 (Quantities, Symbols, Units, Conversion tables), Proposal of the Secretariat for the first Meeting of the Committee (Copenhagen 20-25 Oct. 1952), Dansk Standardiseringsrld, Kobenhavn 1952. 4) K a 1 a n t a r o v, P. L., Elektrichestwo (Moscou) 1948, No. 1, p. 17; 1950, No. 4, 78. 5) H a b e r 1 i, F., Schweiz. Arch. angew. Wiss. Technik 13 (1947) 65, 113, 136. 6) L a n d o 1 t, M. and D e B o e r, J., Rev. g6n. El. (10 (1951) 449. 7) L a 1 o u, J., J. Telecommunications (ITU) 1953 (No. l), 2. 8) Schweiz. Elektr. Kommittee No. 24, Bull. Schweiz. elektr. Vereins 40 (1949) 462. 9) B o d e a, E., Giorgis rationales MKS-Masssystem mit Dimensionskoharenz, 2 ed., Birkhluser, Basel, 1949. IO) P o h 1, R. W., Zur Darstellung der Elektrizitiitslehre, GGtfingen 1950; V e r m e u 1 e II, R., Philips Res. Rep. 7 (1952) 432-441 (No. 6). 1 I) G a u s s, C. F., Die IntensitPt der erdmagnetischen Kraft auf absolutes Maass zurfickgeftlhrt (Ostwald’s Klassiker, No. 53), Engelmann, Leipzig 1894. 12) S t i 11 e, U., Arch. Elektrotechnik d0 (1952) 249. 13) V o n s o v s k i j, S. V. and S h u r, Ja. S., Ferromagnetism, Ogiz, Moscou-Leningrad 1948. 14) B e c k e r, R., Theorie der Elektrizit~t I, 12-13 ed., Teubner, Leipzig 1944.
Marij an
Physica XIX 599-604
RATIONALIZATION
IN THE LIGHT EQUATIONS
OF QUANTITY
by MAR1 JAN BREZINSCAK Zagreb, Yugoslavia
*)
synopsis It is stated that so-called rationalization, as met in common practice, is to be considered as a transition from the numerical equations of electromagnetism, adjusted to CGSG, CGSM and CGSE units, to quantity equations, which are valid for all systems of units. minor importance, viz. the magnetic polarization lities and the corresponding electric quantities transition.
Only a few quantities of and magnetic susceptibiare to be changed in this
1. Introduction. It is a well known point of view that physical phenomena exist independently of units of measure and that the equations of physics, as symbolic expressions of theories describing these phenomena, should aso be independent of the system of units chosen. In physics and technology three types of equations are in use: 1) “quantity equations” (G. Grossengleichungen) ‘) “); in these equations the letter symbols express “quantities”, which are often represented as a product, composed of a “numerical value” and a “unit” **) ; 2) “numerical equations” (G. Zahlenwertgleichungen) r) “) ; in these equations the letter symbols represent “numerical values of the quantities”; 3) “adjusted quantity equations” (G. zugeschnittene Grossengleichungen) 1) “) ; the letter symbols in these equations represent quantities. *) Writer’s address: Zagreb (Yugoslavia), KrajiSka 13. The idea of this article was first given (in Croatian) in the Yugoslavian review “Elektrotehnicki vjesnik” 21 (1953), 81-86 (No. 3/4). l *) We refrain, however, from using this as a d e f i n i t i o n of a quantity.
-
599 -
600
MAR1 JAN BREZIN SCAK
Only the first type, the quantity equations, are intended to be universal, i.e. valid for all systems of coherent *) or non-coherent units. Only these have the property, desired by J. C. M a x w e 11: “physical equations should be valid in all countries, with application of national or international units”. They have, as is well known, a number of practical advantages. Theoretically they are correct : with quantity symbols we can calculate as with pure numbers “). The second type of equations, the numerical equations, are not universal. They have the same form for all coherent systems of units, but by transitions from a coherent system e.g. to a noncoherent system, some of the numerical equations must be changed. They may be complete but are not always (e.g. “Gaussian numerical equations” ignore the quantities permeability and permittivity of empty space). The third class of equations is sometimes used in technology and in experimental research. It does not interest us here and will not be considered in this study. Of late quantity equations have obtained favour in the eyes of many scientists. The secretariat for the first meeting of the IS0 committee also proposed the use of this class of equations “). Being convinced that quantity equations will in due time be internationally adopted, the writer would like here to express his view on the pro. blem of “rationalization” in the light of quantity equations. 2. Oersteditis. What is t.he meaning of the unit name Oersted (Oe) ? In most handbooks, conversion tables, text books, etc. we find that the “classical unit of magnetic field strength” is related to the unit “ampere per meter” (A/m) by Oe = (1000/47r) A/ m or Oe A
(1000/47c) A/m,
(1)
the symbol r\ meaning that the unit Oe “corresponds to” 1000/4n A/m. In some recent articles, however, we find another relation, viz. Oe = 1000 A/m.
(2)
This relation is given by the outstanding Sovjet metrologist Kalantarov4), bythe Swissmetrologists Haberli5) and Landolt”), by De Boer6) and by Lalou’). The Swiss Electrotechnical Committee s) and IS0 “) are of the same opinion t). *) coherent = in one-to-one relationship. t) Expressed implicitely in “) and “).
RATIONALIZATION
.IN
THE
LIGHT
OF QUANTITY
EQUATIONS
60 1
T w o Oersteds, according to (1) and (2) respectively, can, however, not exist under the same style at the same time! L an d o 1 t, K a 1 a n t a r o v etc. suppose that the so-called Electromagnetic CGS-system (CGSM), the Electrostatic CGSsystem (CGSE) and the “mixed” Gauss system (CGSG) are coherent systems of units. But this is not the case [see ‘) “) “)I! Equation (1) having been so long in use is the best argument for this; the “Oersted” in eq. (2) is new! The way in which eq. (2) may be arrived at is the following. Many physicists are of opinion that the CGSM and the CGSG unit of magnetic field strength is Oe = [&/cm = cm-*g*s-‘, in which [I],” is the “CGSM unit of magnetic current”. As [I],,, = 10 A, they obtain Oe = [I],Jcm = 10 A/cm = 1000 A/m, i.e. equation (2). They forget, however, that with equations of the type cm+g’s’, a relation of the type “second = 3.10” cm” lo) may also be obtained. This process, however, is historically false. C a r 1 F r i e d r i c h G a u s s, in his fundamental work “Intensitas vis magneticae terrestris ad mensuram absolutam revocata” (1832), founded the CGS system of units. In this work 11) he uses equations of the type *) HJH, = L-“M”T-’ , in which H, and H, are the two numerical values of the magnetic field strength (“Erdmagnetische Kraft”, according to Gauss). The symbols L, M and T, however, do not represent “fundamental units”, but r a t i o s of the corresponding members of two series of units! Gauss’s own words (see rl), sect. 26, p. 48) are very clear : “So lange wir den Weg beibehalten den wir bei unseren Beobachtungen verfolgt haben, mussen wir die erste Formel (i.e. our above mentioned equation) gebrauchen; wenn wir z.B. an stelle des Millimeters und des Milligramms das Meter und das Gramm als Einheiten annehmen, so wird L = l/1000, M = l/ 1000, also Hz = H, sein ; wenn die Pariser Linie und das Berliner Pfund, so werden wir haben : L = l/2,255829, M = l/4677 11,4, folglich H, = 0,002 196161 H , . . . . “.
Further Gauss speaks about “magpetische Fhissigkeit”, “free magnetism”, considering them as independent quantities, in contradiction to the definition now often found (S t i 11 e 12): “electricity” = “length 2/ force” or “electricity” = “time d force”). *) Writer’s
notation.
602
MARIJAN BREZIN%AK
From the above two conclusions may be drawn: 1) Gauss did not establish electromagnetic units of the type cm’gys’ and he did not calculate with such units ; 2) Gauss acknowledged the independent existance of magnetic quantities. It is evident that “units” cmxgyszand conclusions drawn from their use are of no value, and, consequently equation (2) neither.Therefore in the following we shall consider as Oersted the unit defined by eq. (1). 3. Rationalization. Generally two sets of equations of the electromagnetic field are in common use, the “rational quantity equations” (as used by S t r a t t o n, S o m m e r f e 1 d, P o h 1 etc.), and “classical (or Gaussian) non-rational equations” (as used by P a g e, etc.). The equations of the second set are J o o s, Becker numerical equations, not quantity equations, as supposed by Kalantarov4). Metrologists often start with two forms of Maxwell’s second integral equation $Hds=I
(3)
and
9;H’ds = (4n/c)lL
(4)
in which H denotes the quantity “rational magnetic field strength”, ds a differential of the quantity Jength”, I the quantity “electric current”, H’ is the numerical value of the q.uantity “non-rational magnetic field strength H’, expressed in the CGSM unit Oe, ds the numerical value of the differential of a length, expressed in cm, c = 3.10” and I, the numerical value of the quantity I, expressed in CGSE units *). As I, = cI, g), in which I, is the value of the same current I in CGSM units, we can write eq. (4) in the form $ H’ ds = 4n I,,,
(5)
We have further: H’ = If’/Oe, ds = ds/cm and I,” = I/[I],,,; from eq. (5) we thus obtain I H’ ds --= % Oe cm 47c[II,,, *) Eq. (3) is a quantity equation and eq. (4) a numerical equation. Symbols for quantities are printed in bold italics, symbols for numerical values in ordinary italics.
RATIONALIZATION
IN THE
LIGHT
OF QUANTITY
EQUATIONS
603
As [I],,, = 10 A and as relation (1) is valid, we have Oe.cm -=
[II,,‘
103Am-’ 4x
x lO”m 10A
1 =476’
and, finally, $ w’ ds = I. (6) Comparing the quantity equations (3) and (6) we see that between the quantities Hand w’ there is no difference; they are i d e nt i c a 1. We can easily find a connection between the unit of magnetic field density (magnetic induction), the gauss (G), and the unit of field strength, the Oersted (Oe). In vacuum the classical numerical equations are valid, according to which B = H, in which B = B/G and H = H/Oe (H’ = H), and the quantity equation giving B = I@, in which y, is the permeability of vacuum (pO = 47~.10m7Vs/Am). B/G = H/Oe and B = p,,H From results the well-known relation G = p. Oe = 10m4Vs/m*. The relative permeability ,LI~is the same in both cases. Denoting the quantity “non-rational permeability” with lo’ and the quantity “rational permeability” with p., we can write P, = l4Po = l-4 = P’lb’lThe unit [p’] = G/Oe = p0 is e.g. used by V o n s o v s k y and S h u r r3). From p./l.~~= ,u, = l.~‘/[k’] = ,LJ~results p = p’, that is: the quantities l.~and p. are identical. The classical (numerical) non-rational equation and the rational quantity equation of magnetic polarization are B = H + 4~ J’ = ,uI H (7) and
B = p.,H + J = pH
(8)
B e c k e r 14) and S h u r 13) stipulate that in eq. (7) the units used are G, Oe and G, respectively, i.e. B = B/G, H = H/Oe and J’ = Z/G. Eq. (7) therefore becomes the form B -=G
H
Oe + 4+.
Comparison of this result with eq. (8) gives a quantity equation J= 4n J’
604
RATIONALIZATION
IN THE LIGHT
OF QUANTITY
EQUATIONS
Therefore the “non-rational magnetic polarization” J’ and the “rational magnetic polarization” J are n o t identical quantities. The same is valid for the magnetic susceptibility “=&IL,-1
=4zz’
and for the magnetic mass-susceptibility )( = x/p = 4?cx' in which p represents the quantity mass-density. An analogous situation is found with regard to the equations of the electric field. We see, therefore, that by practising rationalization in the light of quantity equations only six quantities have to be changed (S, K’, x’ and the corresponding three electric quantities) ; all other “nonrational quantities” are identical with the corresponding “rational quantities”. The writer is indebted to Dr. W. d e G r o o t (Eindhoven) for his interest and trouble in the preparation and printing of this paper. Zagreb 2.3.1953. Received 2-5-53. REFERENCES I) W a 11 o t, J., Z. Physik 10 (1922) 329: Elektrotechnische Z. (ETZ) &l (1922) 1329, 1381; Phys. Z. 44 (1943) 17; ETZ G4 (1943) 13;see also DIN 1313. 2) L a n d o 1 t, PI., GrGsse, Masszahl und Einheit (also in French). Rascher, Ztirich 1943. 3) ISO/TC 12 (Quantities, Symbols, Units, Conversion tables), Proposal of the Secretariat for the first Meeting of the Committee (Copenhagen 20-25 Oct. 1952), Dansk Standardiseringsrld, Kobenhavn 1952. 4) K a 1 a n t a r o v, P. L., Elektrichestwo (Moscou) 1948, No. 1, p. 17; 1950, No. 4, 78. 5) H a b e r 1 i, F., Schweiz. Arch. angew. Wiss. Technik 13 (1947) 65, 113, 136. 6) L a n d o 1 t, M. and D e B o e r, J., Rev. g6n. El. (10 (1951) 449. 7) L a 1 o u, J., J. Telecommunications (ITU) 1953 (No. l), 2. 8) Schweiz. Elektr. Kommittee No. 24, Bull. Schweiz. elektr. Vereins 40 (1949) 462. 9) B o d e a, E., Giorgis rationales MKS-Masssystem mit Dimensionskoharenz, 2 ed., Birkhluser, Basel, 1949. IO) P o h 1, R. W., Zur Darstellung der Elektrizitiitslehre, GGtfingen 1950; V e r m e u 1 e II, R., Philips Res. Rep. 7 (1952) 432-441 (No. 6). 1 I) G a u s s, C. F., Die IntensitPt der erdmagnetischen Kraft auf absolutes Maass zurfickgeftlhrt (Ostwald’s Klassiker, No. 53), Engelmann, Leipzig 1894. 12) S t i 11 e, U., Arch. Elektrotechnik d0 (1952) 249. 13) V o n s o v s k i j, S. V. and S h u r, Ja. S., Ferromagnetism, Ogiz, Moscou-Leningrad 1948. 14) B e c k e r, R., Theorie der Elektrizit~t I, 12-13 ed., Teubner, Leipzig 1944.