Download Unit Systems for Electrical and Magnetic Quantities

Measurement Techniques, Vol. 44, No. 7, 2001
MEASUREMENT WORLD
UNIT SYSTEMS FOR ELECTRICAL AND
MAGNETIC QUANTITIES
G. M. Trunov
Systems of units for electrical and magnetic quantities have passed through a complicated and somewhat conflicting history on account of changes in our knowledge of electrical and magnetic phenomena. Historically, electrical and magnetic phenomena were researched independently up to 1820, when Oersted discovered the magnetic actions of an electric current. Therefore, two independent systems of electrical and magnetic units arose, which were considered by Maxwell, and
which in 1881 received the names CGSE (electrostatic system) and CGSM (electromagnetic system) at the first International
Congress of electricians in Paris.
At the end of the 19th century, the practical units for electromagnetic quantities became widely used: the volt,
ampere, ohm, and so on. These units and those in the CGSE and CGSM systems were not related to the units for mechanical quantities, and a need arose for a single system of units for mechanical and electromagnetic quantities. Various approaches were used to set up such a system by physicists and electrical engineers.
Physicists did not consider it necessary to introduce new basic quantities and instead preferred to consider electrical and magnetic quantities as derived mechanical ones. The Gauss system of units arose (the CGS system), in which the
basic units are the centimeter, gram, and second, while the electromagnetic units were formed by combining the electrical
ones from the CGS system and the magnetic ones from CGSM.
Electrical engineers preserved the units for mechanical quantities and added the practical electrical units to them:
ampere, volt, and so on. In 1901, the Italian engineer Giorgi proposed the MKS system, in which the basic units were the
meter, kilogram, and second. Those units were chosen because the units formed from them for energy and power coincided
with the practical units of the joule and watt. Also, one can then add a fourth basic unit as one of the electrical quantities to
obtain a system of electrical and magnetic units in which the derived units are all practical units for electromagnetic quantities widely used in engineering. Later, the fourth basic unit was taken as the ampere, and the system obtained the name MKSA
(meter–kilogram–second–ampere). With the passage of time, the electromagnetic units in the MKSA system were transferred
completely to the SI system [1].
Here I consider the design principles for systems of electromagnetic units together with the advantages and shortcomings of the CGS and SI systems in the area of electromagnetism, and I propose a new system for electromagnetic units.
Unit System Design Principles [2]. By unit system is meant a set of basic and derived units for physical quantities.
The basic units are selected arbitrarily, while the derived units are defined from equations relating them to the basic ones. The
basic units do not necessarily have to be independent. The important point is that they should be units for basic quantities,
i.e., independent ones. For example, the basic unit in SI, which is determined in terms of other basic units, is the unit of electric current, the ampere.
There are two distinct approaches to unit system design. Tamm, the author of the fundamental text “Principles of
Electricity Theory” proposed the following principle: any unit system is arbitrary, and the criterion in choosing a particular
one is only its internal consistency, convenience, and fit with current physical concepts (p. 220 of [3]). The other approach
uses the practical desirability of the units in industry, science, trade, and so on [4].
Translated from Izmeritel’naya Tekhnika, No. 7, pp. 66–70, July, 2001.
0543-1972/01/4407-0771$25.00 ©2001 Plenum Publishing Corporation
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When the electromagnetic units were constructed in SI, use was made of the principle of practical desirability, which
rules out Tamm’s proposal of correspondence with current physical concepts. This has had adverse consequences, which are
considered below.
Quantities and their units are characterized by dimensions. The dimensions are an expression in the form of a single term in powers compiled from the symbols for the basic units raised to various degrees, which reflects the connection of
the quantity with the basic ones, and where the coefficient of proportionality is 1.
This definition of dimensions is supplemented with the following:
1) the dimensions of a physical quantity should be denoted by dim, e.g., dim F = LMT–2;
2) the concept of dimensions is extended to the basic quantities, i.e., the formula for the dimensions of a basic quantity coincides with the symbol, e.g., dim L = L, and dim M = M.
The dimensions of a given electrical or magnetic quantity differ as between systems of units, e.g., CGS, and SI.
Units for CGS Electromagnetic Quantities. The gaussian system of units as a rigorous system was set up earlier
than others. It was never legalized by any international agreement and never obtained official recognition at international congresses of electricians or at general conferences on weights and measures. In the USSR, GOST 8033-56 applied from 1 January
1957 “Electrical and magnetic units,” which allowed the CGS system to be used (the name gaussian system is not mentioned
in that standard).
The basic units in the CGS system are the centimeter, gram, and second, while the units for electrical and magnetic quantities are considered as derived from these mechanical ones. The initial equation for constructing the CGS electromagnetic units is provided by Coulomb’s law, which expresses the force between two point charges Q1 and Q2 separated by
a distance r in a medium with dielectric constant ε:
F=
Q1Q2
εr 2
.
(1)
As ε is defined as a dimensionless quantity (ε = 1 for vacuum), from (1) we get the dimensions of electric charge:
dim Q = L3 / 2 M1 / 2 T −1
and this defines the CGS (Q) unit of electric charge as the charge that interacts with an equal charge at a distance of 1 cm
from it in vacuum with a force of 1 dyne. The dyne is the unit of force in the gaussian system and is the force that produces
an acceleration of 1 cm/sec2 with a mass of 1 g.
The unit for electrical current CGS (I) is defined by
I = dQ/dt
(2)
as the current that causes an electric charge of 1 CGS (Q) to pass through the cross section of a conductor in 1 sec. The dimensions of electric current are
dim I = L3/2M1/2T–2.
The units of charge and current in the CGS system completely coincide with the corresponding CGSE units.
The units for CGS magnetic quantities coincide with the corresponding CGSM ones, which may be defined from
Coulomb’s law for magnetic masses:
mm
F = 1 22 ,
µr
where m1 and m2 are the magnetic masses, and µ is the magnetic permeability of the medium, which is dimensionless by definition (µ = 1 for vacuum).
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TABLE 1. Dimensional Formulas and Relation between Units of Electrical and Magnetic Quantities in Various Unit Systems
Dimensional formula
Physical quantity
Relation between units
SI
CGS
ST
SI and ST
ST and CGS
Electric charge Q
TI
L3/2M1/2T–1
Q
1 Cu = 3 10 9 Tm
1 Tm = 10 9 CGS (Q)
Electric current I
I
L3/2M1/2T–2
T–1Q
L2MT–3I–1
L1/2M1/2T–1
L2MT–2Q–1
Potential ϕ, voltage U
1 A = 3 10 9
1V =
1
3 10
1Ω=
9
Tm
sec
J
Tm
1
ST ( R)
9 ⋅ 10 9
1
Tm
= 10 9 CGS ( I )
sec
1
J
= 10 5 CGS (ϕ)
Tm
Electrical resistance R
L2MT–3I–2
L–1T
L2MT–1Q–2
Electrical capacitance C
L–2M–1T4I2
L
L–2M–1T2Q2
1 F = 9·109 ST (C)
1 ST (C) = 102 cm
Inductance L
L2MT–2I–2
L
L4MT–2Q–2
1 H = 107 ST (L)
1 ST (L) = 102 cm
Electric field strength E
LMT–3I–1
L–1/2M1/2T–1
LMT–2Q–1
Electrical displacement D
L–2TI
L–1/2M1/2T–1
LMT–2Q–1
MT–2I–1
L–1/2M1/2T–1
LMT–2Q–1
L–1I
L–1/2M1/2T–1
LMT–2Q–1
L2MT–2I–1
L3/2M1/2T–1
L3MT–2Q–1
Magnetic induction B
Magnetic field strength H
Magnetic flux Φ
1
1
V
1
N
=
9 Tm
m
3 10
Cu
m2
= 12π 10 9
1
N
= 10 CGS (E)
Tm
1
N
= 10 CGS (D)
Tm
N
Tm
1
N
= 10 G
Tm
A
4π N
=
m
10 7 Tm
1
N
= 10 Oe
Tm
1 T = 10 7
1
N
Tm
1 ST (R) = 10–2 CGS (R)
1 Wb =
10 7 ST (Φ)
1 ST (Φ) = 10 9 Mx
Although the concept of magnetic mass is a fiction, the CGSM system is a logically rigorous system of units.
However, the CGSE and CGSM systems would not be in conflict if a current did not generate a magnetic field. As it does,
the current appears not only in the definition of (2) but also in the equation for the force of interaction between two parallel
conductors carrying currents I1 and I2 and placed a distance r apart in a medium with magnetic permeability µ. This equation is as follows in the CGSM system (Ampere’s law for parallel currents):
F 2µI1 I2
=
,
l
r
(3)
where F /l is the force acting on unit length l of each conductor.
The unit for current strength in CGSM (I) as defined by (3) differs from the unit of current in CGSE (I) as defined
by (2). Various experiments, in particular Eichenwald’s experiments of 1901, show that the ratio of the current in CGSE units
to the current in CGSM units is numerically and dimensionally equal to 3·1010 cm/sec, i.e., the velocity of light in vacuum c.
Consequently,
CGSM (I) = c·CGSE (I) = c·CGS (I).
Then Ampere’s law for parallel currents is written in CGS as
F 2µI1 I2
.
=
l
c2r
(4)
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TABLE 2. Equations for Electromagnetism Written in Various Unit Systems
SI
ε0 = 8.85·10–12 F/m,
µ0 = 1.26·10–6 H/m
CGS
c = 3·1010 cm/sec
ST
c = 3·108 m/sec
k0 = 1 N·m2/Tm2
Coulomb’s law
F = (1 /4πε0)(Q1Q2 / εr2)
F = Q1Q2 / εr2
F = k0(Q1Q2 /εr2)
Ampere’s law for two parallel currents
F/l = (µ0 /4π)(2µI1I2 /r)
F/ l = 2µI1I2 /c2r
F/l = k0(2µI1I2 / c2r)
Law or definition
Electric field strength E
Electrical displacement D
Lorentz force
Magnetic field strength H
E = F/Q
D = εε0E
D = εE
F = QE + Q[v × B]
F = QE + Q[(v/ c) × B]
H = B /(µµ0)
H = B/µ
Magnetic flux Φ
Φ = BS
Electrical capacitance C
C = Q/U
Capacitance of a planar capacitor
C = (εε0S)/ d
Thomson’s formula for a tuned circuit
T = 2π(LC)1/2
Poynting vector S
Faraday’s law
S = [E × H]
C = (εS)/(4πd)
C = (εS)/(4πk0d)
T = (2π/ c)(LC)1/2
S = (c/ 4π)[E × H]
S = (c/4πk0)[E × H]
curlE = –dB /dt
curlE = –(1/ c)dB / dt
Total current law
curl H = J + dD/ dt
curlH = (4π/c)J + (1/c)dD/dt curlH = (4πk0 /c)J + (1/c)dD/dt
Gauss’s theorem
divD = ρ
Continuity in magnetic induction lines
div D = 4πρ
div D = 4πk0ρ
divB = 0
Table 1 gives the units for other electrical and magnetic quantities in CGS and the dimensional formulas for them.
Deficiencies of CGS Electromagnetic Units. Table 1 shows that many electromagnetic units have fractional dimensional parameters. The fractional parameters obtained originally for electric charge were then extended to most other electrical and magnetic quantities. This deficiency of the electromagnetic units in CGS hinders the establishment of relationships
between electrical and magnetic quantities by dimensional analysis.
Also, in CGS, the dimensional equations for certain electromagnetic quantities coincide with the formulas for the
dimensions of mechanical quantities. For example, electrical conductivity has the dimensions of velocity, and resistivity has
the dimensions of time, while the dimensions of two distinct physical quantities (inductance and capacitance) are not only
identical but also coincide with the dimensions of length.
If we adopt the view taken by Max Planck, namely that the dimensions of any physical quantity are not a property
related to the essence of it but instead represent something nominal defined by the choice of the measurement system [5],
then these deficiencies of the CGS system must be perceived as inevitable and caused by the construction of the CGS electromagnetic units solely on the basis of free mechanical units.
Table 2 gives the basic laws and equations of electromagnetism written in the CGS system.
SI units for Electromagnetic Quantities. The main role was played in setting up the electrical and magnetic units
in the international system by the circumstance that electrical engineering had long and universally used what are called the
practical units: the coulomb, volt, ampere, ohm, and so on. As the fourth basic unit in SI, the unit adopted was that of electric current I, the ampere. The ampere was defined as the strength of an unvarying current that on passage through two parallel conductors of infinite length and negligibly small area with circular cross section placed in vacuum at a distance 1 m
apart would produce on each part of a conductor of length 1 m an interaction force of 2·10–7 N.
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Correspondingly, the fourth element in the dimensional formulas for electromagnetic quantities in the SI system
became the symbol for the dimensions of current strength I.
Introducing a fourth basic unit (the ampere) to describe electromagnetic phenomena in SI required an additional
dimensional coefficient µ0 in Ampere’s law for two parallel conductors carrying currents, in order to balance the dimensions
of the two parts of the equation
F µ 0 2µI1 I2
.
(5)
=
l
4π c 2 r
The value of it is µ0 = 4π·10–7 H/m = 1.256·10–6 H/m; dim µ0 = LMT–2I–2, and it is called the magnetic constant and is determined by the following factors.
Historically it had occurred that the numerical value of the fourth basic SI unit, the ampere, was taken as equal to
the numerical value of the international ampere, which is equal to the strength of the current that on passage through a solution of AgNO3 in water deposits 0.0011180 g of silver in 1 sec. The international ampere in turn by decision of the international conference of 1908 was defined as 0.1 units of the electric current in the CGSM system of units, where the relationship was assumed to have been established once and for all (p. 222 of [3]). Then from (5) one can get a relation between the
current units in SI and CGS:
1 A = 0.1 CGSM (I) = 0.1 c· CGS (I) = 3·109 CGS(I).
Also, in writing Ampere’s law in SI, use was made of the rationalized style proposed in 1892 by the British physicist Heaviside, which consists in introducing a factor 1/4π into this equation (and also into Coulomb’s law for point charges).
The main conception in this Heaviside rationalization is that this mathematical technique eliminates the factors 4π and 2π
from the most frequently used electromagnetism formulas, and in particular from Maxwell’s equations [8].
These factors mean that the formula for Ampere’s law for parallel currents takes different forms: in CGS, it is (4),
and in SI, it is (5).
Coulomb’s law for point charges is also written in rationalized form in SI:
F=
1 Q1Q2
,
4 πε 0 εr 2
(6)
where ε0 is a dimensional coefficient called the electrical constant, with ε0 = 1/µ0 c2 = 8.854·10–12 F/m; dimε0 = L–3M–1T4I2.
Table 1 gives the units and dimensional formulas for the derived electric and magnetic SI quantities, while Table 2
gives the basic laws and equations in electromagnetism.
Deficiencies of SI Electromagnetic Units. The derived units for electrical and magnetic SI quantities have been
taken from Giorgi’s electrotechnical system, in which the electrical and magnetic units were based on the theory of a mechanical ether on the assumption that between the ether (vacuum) and ordinary material bodies there is no essential difference.
Then the force characteristics of the electric and magnetic fields (the vectors E and D or B and H) in vacuum differ one from
another not only in numerical factors but also in essence: D = ε0E and B = µ0H. Then the dimensional coefficients ε0 and µ0
in the Giorgi system, and also originally in SI, were called the dielectric and magnetic permeabilities of vacuum. However,
that division of the electric field in vacuum into E and D, and of the magnetic field into B and H, is artificial and an unnecessary complication (p. 336 of [9]). On current concepts, the electromagnetic field in vacuum is described completely if one
is given the single electrical vector E and the single magnetic vector H. Therefore, in vacuum one has D = E and B = H, but
these relationships are not obeyed in SI. Also, in SI the dimensions of the vectors E, D, B, H are different, although their
entire essence is that these quantities are forces referred to unit charge and should have identical dimensions.
In accordance with the standard [1], the instruction in all teaching institutions should be based on SI, but in the most
authoritative physics textbooks [3, 6, 7], the science of electricity is presented in a gaussian system of units CGS, although
this is not even one of those permitted for teaching and relative to SI has various shortcomings, the main one being the fractional dimensional parameters. The writers of those texts were guided by the argument that the laws and basic definitions of
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electromagnetism in the CGS system do not contain the dimensional coefficients µ0 and ε0, and the vectors E, D, B, H characterizing the electrical and magnetic components of the single magnetic field have identical dimensions:
dimE = dimD = dim B = dim H = L–1/2M1/2T–1,
while the basic equations of electromagnetism are distinguished by exceptional simplicity and rigor, and one should not hesitate to meet the price of converting electromagnetic units from CGS to SI and vice versa (p. 352 of [10]).
Theoretical System of Electromagnetic Units. There is no doubt that one should use the SI electromagnetic units
in practice (ampere, coulomb, volt, ohm, farad, and so on), and interest thus attaches to a new four-dimensional system of
electrical and magnetic units [11], which does not have the above shortcomings in the CGS and SI systems and which I have
called the theoretical system (abbreviation ST). This system has been constructed in accordance with Tamm’s criteria for unit
systems [3].
In ST, the first three basic units are length, mass, and time, which are respectively the meter, kilogram, and second.
The fourth basic unit is taken as the unit of electrical charge, the tamm Tm, which is named in honor of I. E. Tamm, the recipient of the Nobel Prize. This unit in accordance with Coulomb’s law describes the interaction between two point charges Q1
and Q2 a distance r apart in a medium having dielectric permeability ε:
F = k0
Q1Q2
εr 2
,
(6)
where k0 = 1 N·m2/Tm2 is a dimensional coefficient (dimk0 = L3MT–2Q–2), whose occurrence is due to the requirement of
the theory of dimensions.
In accordance with (6), the unit of electrical charge 1 Tm in ST is that charge which when placed on two bodies of
negligibly small size located in vacuum at a distance 1 m apart will interact with a force of 1 N.
The following is the relation between the charge units in SI, ST, and CGS:
1 Cu = 3 10 9 Tm;
1 Tm = 10 9 CGS (Q).
Table 1 gives the units and dimensional formulas for electrical and magnetic quantities in ST, as well as the relations
between the ST, SI, and CGS units. Table 2 gives the basic laws and equations of electromagnetism written in the ST system.
Comparison of Units for Electrical and Magnetic Quantities in ST, SI, and CGS. Table 1 implies that the electromagnetic units in ST do not have internal conflicts and are readily brought into conformity with the corresponding units
in CGS and SI. Table 2 shows that the equations of electromagnetism in ST reflect current ideas on the essence of the electromagnetic field: the dimensional coefficients ε0 and µ0 are absent, but there is the fundamental constant c, the velocity of
light in vacuum; the force characteristics of the electromagnetic field, namely E, D, B, and H, have identical dimensions.
Some of the electromagnetism equations contain the dimensional coefficient k0 equal to 1, but this causes no difficulty in
understanding the essence of the physical quantities appearing in these equations.
The ST units differ from the electromagnetic units in the CGS system in not having fractional dimensions; inductance and capacitance have differing dimensions. Also, in ST the characteristics of the electromagnetic field E, D, B, and H
have dimensions LMT–2Q–1, which clearly emphasizes their force character, and contrasts with the meaningless dimensions
of those quantities (L–1/2M1/2T–1) in the CGS system.
Conclusions. The SI and CGS systems of electromagnetic units have shortcomings that reflect the conflicts in their
formation. In the area of electromagnetism, SI does not meet one of the basic criteria imposed on systems of units: correspondence with current physical concepts on the electromagnetic field. Presenting the science of electricity solely with SI
units (in accordance with [1]) may lead students to obtain incorrect ideas on the essence of electrical and magnetic fields. The
new system of electromagnetic units I have considered here and called the theoretical system (ST) does not have the essential shortcomings of the SI and CGS unit systems. When the science of electricity is taught, it is desirable to use SI and ST.
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The fundamental concepts and basic laws of electromagnetism can conveniently be explained by the use of ST, while applied
disciplines such as electrical engineering and radioengineering can be based on using only SI.
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1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
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