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F series S series More precisely, with Hodge’s star operator “ ” (see 5,6 ∗ (φ, cA) Appendix), it can be written as d Y0 E B H D ↓ ( ,c )= Z0( ,c ). (9) (E, cB) (H, cD) ∗ ←∗→ It should be noted that the electric relation and the mag- d Z0 d ↓ ↓ netic relation are united under the sole parameter Z0. 0 (J, c̺) c. Plane wave For linearly polarlized plane waves in d vacuum, a simple relation E = cB holds. If we introduce ↓ −1 0 H (= µ0 B) instead of B, we have E = Z0H. The latter relation was introduced by Schelkunoff1 in 1938. The TABLE I: Relativistic pairs of quantities are arranged as reason why H is used instead of B is as follows. The a diagram, the rows of which correspond to the orders of boundary conditions for magnetic fields at the interface tensors (n = 1, 2, 3, 4). In the left column, the quantities related to the electromagnetic forces (the F series), and in of media 1 and 2 are H1t = H2t (tangential) and B1n = the right column, the quantities related to the electromag- B2n (normal). For the case of normal incidence, which is netic sources (the S series) are listed. The exterior deriva- most important practically, the latter condition becomes tive “d” connects two pairs of quantities. These differential trivial and cannot be used. Therefore H is used more relations correspond to the definition of (scalar and vector) conveniently. The mixed use of the quantities (E and H) potentials, the Maxwell’s four equations, and the charge con- of the F and S series invite Z0 unintentionally. servation. Hodge’s star operator “ ” connects (E, cB) and ∗ d. Let us compare the force (H, cD) pairs. This corresponds to the constitutive relations between charges q and that between the magnetic for vacuum. Here appears the vacuum impedance Z0 = 1/Y0. D monopoles g ( Vs = Wb). If these forces are the same ∼ 2 2 2 2 for equal distances r, i.e., q /(4πε0r )= g /(4πµ0r ), we find that this table is very helpful to overview the struc- have the relation g = Z0q. With this relation in mind, ture of electromagnetism.3,4 the Dirac monopole g0, whose quantization condition is g0e = h, can be beautifully expressed in terms of the e as III. ROLES OF THE VACUUM IMPEDANCE h h −1 g0 = = 2 (Z0e)=(2α) Z0e (10) e Z0e In this section we show some examples for which Z0 where h = 2π~ is Planck’s constant. The dimensionless plays important roles. 2 2 parameter α = Z0e /2h = e /4πε0~c 1/137 is called a. Source-field relation We know that the scalar po- ∼ tential ∆φ induced by a charge ∆q = ̺∆v is the fine structure constant, whose value is independent of unit systems and characterize the strength of the elec- 1 ̺∆v tromagnetic interaction. The use of Z helps to keep ∆φ = , (5) 0 4πε0 r SI-formulae in simple forms. e. The F series versus the S series Impedance (re- where r is the distance between the source and the point sistance) is a physical quantity by which voltage and cur- of observation The charge is presented as a product of rent are related.7 In the SI system, the unit for voltage charge density ̺ and a small volume ∆v. Similarly a is V(= J/C) () and the unit for current is A(= C/s) current moment (current by length) J∆v generates the (). We should note that the latter is proportional vector potential to and the former is inversely proportional to the unit of µ J∆v charge, C (). We also note in Table I that the ∆A = 0 . (6) 4π r units for quantities in the F series are proportional to the volt and those in the S series are proportional to the am- Considering the relations (3) and (4), these equations can pere. After all, the vacuum impedance Z0 plays the role be united as to connect the quantities in the F and S series. In the ∆φ Z0 c̺ above cases we have found that the use of Z0 (together = ∆v. (7) ∆(cA) 4πr J  with c) instead of ε0 or µ0 simplifies equations.

We see that the vacuum impedance Z0 plays the role to relate the source (J,c̺)∆v and the resultant fields IV. THE MAGNITUDE OF THE UNIT OF ∆(φ, cA) in a unified manner. RESISTANCE b. Constitutive relation A similar role can be seen for the constitutive relations for vacuum; D = ε0E and Here we consider a hypothetical situation where we H −1B = µ0 can be combined as are allowed to redefine the magnitudes of units in elec- tromagnetism. E D c The product of the unit of voltage and that of current B = Z0 H . (8) c    should yield the unit of power, 1 V 1A=1W=1J/s, × 3

(φ, A˜ )V (φ, A˜ )V (φ,ˆ A´ ) √N (φ,ˆ A´ ) √N (E, B˜ )V/m (E, B˜ )V/m (Eˆ , B´ ) √N/m (Eˆ , B´ ) √N/m ∗ ∗ (H, D˜ )A/m (H , D˜ )V/m (H´ , Dˆ ) √N/m (H´ , Dˆ ) √N/m ∗ ∗ − (J, ̺˜)A/m2 (J , ̺˜ )V/m2 (c 1Jˆ, ̺ˆ) √N/m2 (J´, ̺ˆ) √N/m2 (a) c-normalization (b) (c, Z0)-normalization (c) Gaussian (d) modified Gaussian

TABLE II: Pairs of quantities in electromagnetism and their units (dimension). Quantities X normalized with c and Z0 ∗ are marked as X˜ = cX and X = Z0X, respectively. As seen in (a) and (b), the variety of units is reduced owing to the normalization. Gaussian unit system (c) and the modified Gaussian unit system (d) are presented with normalized variables: Sˆ = S/√ε0, Fˆ = F √ε0, S´ = S/√µ0, F´ = F √µ0, where S (F ) represents a quantity in the S (F) series. We notice an irregularity in the fourth row of the Gaussian system.

−1 2 which is a fixed quantity determined mechanically, or where E = (4πε0) (q1/r ) is the electric field induced ′ outside of electromagnetism. Thus, a new volt V and by q1. We see a new ampere A′ must be defined so as to satisfy ′ −1 ′ ′ ′ − q2 = k q2 , E = k E , . (16) A = kA, V = k 1V, (11) { } { } { } { } in terms of the currently used V and A, where k (= 0) and find that the numerical value for the charge q and ′ 6 that for the electric field E will be changed reciprocally. is a scaling factor. Accordingly a new ohm Ω must be ′ −2 ′ 2 redefined as We also note ε0 = k ε0 and µ0 = k µ0 . In the SI, the{ } ampere is{ defined} { in terms} of{ the} force ′ − Ω = k 2Ω. (12) F between the parallel two wires carrying the same am- plitude of current I. We have F/l = µ I2/(2πr), where We denote the numerical value as A A/u, when 0 { } ≡ r is the separation and l is the length of wires. Sub- we measure a physical quantity A with a unit u. For − stituting F = 2 10 7 N, r = l = 1 m, I = 1 A, we example, for l = 1.3 m we write l = l/m=1.3. We −×7 { } ′ ′ get µ0 = 4π 10 H/m. Thus the magnitude µ0 (or can have another numerical value A A/u , when we × { } ′ Z ) are fixed. measure the same quantity A in a{ different} ≡ unit u . Now 0 { We} could determine ε by the force between charges we have the relation 0 with the same magnitude.{ } In Giorgi’s unit system (1901), ′ ′ A = A u= A u . (13) which is a predecessor of the MKSA unit system or the { } { } SI system, k was fixed by determining the magnitude of It should be stressed that the physical quantity A itself the ohm. The way of determination Z0 has been and is independent of the choice of units. What depends on will be changed according to the development{ } of high the choice is the numerical value A . precision measurement technique. In the SI system, from the{ definition} of c and µ0, the vacuum impedance is represented as Z0 = −7 µ0/ε0 = cµ0 = (299792458m/s) (4π 10 H/m) = × × V. TOWARD NATURAL UNIT SYSTEMS p(119.9169832 π)Ω 377Ω. In our new system, with (12) we have × ∼ As shown in Table II (a), by introducing a new set of 2 ′ ′ ′ Z0 = Z0 Ω= k Z0 Ω = Z0 Ω . (14) normalized quantities, X˜ = cX, derived from SI quanti- { } { } { } ties X, we can reduce the number of fundamental dimen- The numerical value must be changed from Z0 = 377to ′ 2 { } sions. In this case, we only need three; the ampere, the Z0 = 377k . For example, we could choose a new ohm { ′ } ′ volt, and the meter. Ω so that Z0 =1Ω is satisfied by setting k 1/√377. ∼ Further, as seen in Table II (b), when we introduce a Conversely, to fix Z to a particular number implies ∗ 0 set of normalized quantities, X = Z X, by multiplying the determination{ of the} magnitude of units (Ω, V, A, 0 Z , only the volt and the meter are required. By nor- and others) in electromagnetism. 0 malizing the quantities with the fundamental constants, Once k, or Z is fixed, the numerical values for quan- 0 c and Z , we have a simplified set of Maxwell’s equations: tities in the F{ series} are multiplied by k and those in the 0 ∗ S series are divided by k. The sole parameter k or Z0 { } ∗ ∗ ∗ ∂D˜ ∗ determines the numerical relation between the F and S ∇ D˜ =̺ ˜ , ∇ H˜ = + J series. · × ∂τ ∂B˜ Coulomb’s law for charges q1 and q2 can be rewritten ∇ B˜ =0, ∇ E = . (17) as · × − ∂τ 1 q q ∗ 1 2 D˜ E H∗ B˜ F = 2 = q2E = q2 As E V/m with = and = . Considering τ = ct, this 4πε0 r { } ×{ } ′ ′ ′ ′ set of equations resembles to the Maxwell’s equations in = q2 A s E V /m (15) { } ×{ } the Gaussian system of units except for the rationalizing 4 factor 4π [See Eq. (29)]. However there is a significant series, each of which is inversely proportional to C, can difference; the factor 1/c is missing in the current density be normalized by multiplication with √ε0. For example, term. We will return to this point later. It should be E, D, B, and H, are normalized as stressed that a natural system of units can be reached from the SI system by normalizations without detouring D √N D D √N Eˆ = E√ε0 , Dˆ = , via the Gaussian system. ∼ m √ε0 ∼ m The number of basic units has been reduced from four D √Ns H D √N (m, kg, s, A) to two (m, V) by introducing the quantities Bˆ = B√ε0 , Hˆ = , (20) m2 ε s normalized with c and Z0. For further reduction toward ∼ √ 0 ∼ a natural unit system,8,9 ~ and the gravitational constant respectively. We have the constitutive relation G can be used for example. Dˆ = Eˆ , Hˆ = c2Bˆ , (21) VI. GAUSS AND HEAVISIDE-LORENTZ and the normalized permittivityε ˆ0 = 1 and permeability SYSTEMS OF UNITS 2 µˆ0 =1/c . The normalized vacuum impedance is

The SI and the cgs (esu, emu, Gaussian) systems differ ˆ µˆ0 1 D s in three respects. First, in the cgs unit systems, no fun- Z0 = = . (22) r εˆ0 c ∼ m damental dimensions are supplemented to the three fun- damental dimensions for mechanics; length, mass, and For the cgs electromagnetic system of units (emu), S- time. On the other hand in the SI (MKSA) system, a series quantities are multiplied by √µ0 and F-series quan- new fundamental dimension that for electric current is tities are divided by √µ0. With this normalization, µ0 is introduced. The cgs systems contain three basic units, eliminated from the magnetic Coulomb law or the law of while the SI system contains four. magnetic force between currents. The fields are normal- Secondly, the cgs systems are irrational systems; the ized as factor (1/4π) is erased from Coulomb’s law but the fac- B tor 4π appears in the source terms of Maxwell’s equations D √N D √N B´ = , H´ = H√µ0 , instead. The SI is a rational system, which has the op- √µ0 ∼ m ∼ m posite appearance. E √ √ E´ D N D´ D D Ns Thirdly, the base mechanical system for the cgs sys- = , = √µ0 2 . (23) tems is the cgs (centimeter, gram, and ) mechan- √µ0 ∼ s ∼ m ical system. That for the SI system is the MKS (meter, The constitutive relations are , and second) system. − In order to focus all our attention on the first respect, H´ = B´ , D´ = c 2E´ , (24) i.e., the number of basic units, we will ignore the differ- ences in the last two respects. From now on, we pre- and we have the normalized permeabilityµ ´0 = 1 and per- 2 tend that all the cgs systems (esu, emu, and Gaussian) mittivityε ´0 =1/c . The normalized vacuum impedance are constructed rationally on the MKS mechanical sys- is tem. (Actually the Heaviside-Lorentz system is an MKS version of the Gaussian system, namely, a three-unit, ra- µ´0 D m Z´0 = = c . (25) tional system based on the MKS system.) r ε´0 ∼ s To go from the SI system to the cgs systems, we have to reduce the number of basic units by normalization with The Gaussian system of units is a combination of the a universal constant. esu and emu systems. For electrical quantities the esu In the cgs electrostatic system of units (esu), normalization is used and for magnetic quantities the Coulomb’s law is expressed as emu normalization is used. Namely we use Eˆ , Dˆ , B´ , and H´ , all of which have the dimension √N/m. The 1 q1q2 1 qˆ1qˆ2 constitutive relations are simplified as F = 2 = 2 . (18) 4πε0 r 4π r Dˆ = Eˆ , H´ = B´ , (26) Thus, the normalized charge10 and we haveε ˆ0 =1 andµ ´0 = 1. So far it looks nice be- q D C cause electric and magnetic quantities are treated sym- qˆ = = √Jm = √Nm (19) √ε0 ∼ F/m metrically. This appearence is the reason why the Gaus- p sian system has been used so widely. However, there is is a quantity expressed by mechanical dimensions only. an overlooked problem in the normalization of current The quantities in the S series, each of which is propor- density. It is normalized as Jˆ = J/√ε0 in the Gaussian tional to the coulomb, C, can be normalized by division system. The current density is the quantity primarily with √ε0. On the other hand, the quantities in the F connected to magnetic fields and therefore it should be 5 normalized as J´ = J√µ0 as for the emu system. Be- the emu system when he found that light is an electro- cause of this miscasting, we have an irregularity in the magnetic disturbance propagated according to electro- fourth row of the column (c) of Table II. The Gaussian magnetic laws.12 For the emu system the dimensions of normalization happens to make the pairs of quantities resistance and velocity are degenerate. For the Gaus- relativistic with exception of the (J,̺) pair. sian system, the vacuum impedance reduces to unity, The relativistic expression for the conservation of µ´0/εˆ0 = 1. charge should be pThus there is no room for the vacuum impedance in the cgs systems, which contains only three basic units. ∗ ∂̺˜ ∗ However when we move to a unit system with four basic + ∇ J =0, (27) ∂(ct) · units, the vacuum impedance Z0 should be recognized as a fundamental constant as important as the velocity of as for the cases of c- or (c,Z0)-normalization. In the light c. It has been underestimated or ignored for long Gaussian system, however, the non-relativistic expression time. It is due to the fact that the Gaussian system, for which Z0 is dimensionless and of unit magnitude, has ∂̺ˆ + ∇ Jˆ = 0 (28) been used for long time even after the introduction of the ∂t · MKSA and the SI systems. As has been pointed out by Sommerfeld,3 the Gaus- is adopted. As a practical system of units, it is a reason- sian system tends to veil the significance of D and H in able (and perhaps unique) choice. vacuum. Sometimes it is told that in vacuum only E This quirk can clearly be seen, when we compare the and B have their significance and D and H lose their Maxwell’s equations (17) in the natural system of units meaning. This argument is strongly misled by the use and that for the Gaussian system: of Gaussian system of units. Considering the tensorial nature of quantities as in (9), the constitutive relations 1 ∂Dˆ 1 ∇ Dˆ =̺, ˆ ∇ H´ = + Jˆ, for the Gaussian system are expressed as · × c ∂t c Eˆ B´ H´ Dˆ 1 ∂B´ ( , )= ( , ). (31) ∇ B´ =0, ∇ Eˆ = . (29) ∗ · × − c ∂t with Hodge’s star operator. This relation represents im- portant geometrical relations of electromagnetic fields, The factor 1/c in the current density term is a seam in- which can hardly be suggested by the simple vector rela- troduced when the esu and emu systems are joined into tions (26). the Gaussian system. Now we have understood that without the help of The common belief that the Gaussian system is supe- Gaussian system, we can reach natural systems of units rior to the SI system because of the similarity to a natural directly from the SI system. We believe it’s time to say unit system or because of the compatibility with relativ- goodbye to the Gaussian system of units. You won’t miss ity is almost pointless. We should remember that the its simpleness if you have the vacuum impedance Z0 as Gaussian unit system was established in 1870s, when the a key parameter. relativity or the Lorentz transformation were not known yet. 11 The modified Gaussian system, in which J´ is adopted Acknowledgments and the above seam is eliminated, has been proposed but is rarely used. Column (d) of Table II contains quantities We thank K. Shimoda for helpful discussions. This in the modified Gaussian system. They differ uniformly work is supported by the 21st Century COE program by a factor ε from the (c,Z )-normalized quantities in √ 0 0 No. 14213201. Column (b).

APPENDIX: TENSOR NOTATIONS VII. SUMMARY AND DISCUSSION In this appendix, we will explain mathematically the The important expression (1) for the velocity of light entities of Table I. With basis vectors e0, e1, e2, e3 , a also holds for the esu and emu systems: space-time vector can be represented as{ } 1 1 1 3 c = = = , (30) i (ct, x) = (ct)e0 + x, x = x ei. (A.1) √µ0ε0 √µˆ0εˆ0 √µ´0ε´0 Xi=1 but not for the Gaussian system; 1/√εˆ0µ´0 =1 = c. The basis vectors satisfy eµ eν = gµν (µ, ν =0, 1, 2, 3). 6 · The expression (2) for the vacuum impedance, is The nonzero elements of g are g00 = 1, g11 = g22 = − rewritten as Zˆ0 = 1/c and Z´0 = c for the esu and g33 = 1. We also introduce the dual basis vectors: 0 1 2 3 0 i emu systems, respectively. Maxwell himself worked with e , e , e , e , with e = e0, e = ei (i =1, 2, 3). { } − 6

The quantities in electromagnetism are expressed by and the application to an n-form results in an (n + 1)- antisymmetric tensors of rank n (n-forms) in the four- form. For example, we have d(φ, cA) = (E,cB), which is dimensional space.4,5,6 For scalar fields α, β and 3- equivalent to E = ∇φ ∂A/∂t and B = ∇ A. The dimensional vector fields X, Y , n-forms (n = 1, 2, 3) successive applications− of−d always yield zero× (dd = 0), are defined as therefore we have d(E,cB) = 0, which corresponds to ∂B/∂t + ∇ E = 0, ∇ B = 0. Furthermore, 0 × · (α; X)1 = αe + X, d(H,cD) = (J,c̺) yields ∂D/∂t + ∇ H = J, 0 ∇ D d J − × (X; Y )2 = e X + Y = ̺, and ( ,c̺) = 0 yields the conservation ∧ · ∇ J 0 of charge: ∂̺/∂t + = 0. (Y ; β)3 = e Y + βσ, (A.2) · ∧ where σ = e1 e2 e3 is the 3-form representing the Another important notation is Hodge’s star operator volume element∧ and∧ “ ” represents the antisymmetric “ ”, which converts an n-form into a (4 n)-form; ∧ 3 ∗ − tensor product. Y = ǫijkYiej ek is a 2-form j,k=0 ∧ (in three dimensional space)P derived from Y . ǫijk is the Levi-Civita symbol. With these, the pair quantities in Table I are defined as (α; X)1 = (X; α)3, ∗ (φ, cA) = ( φ; cA) , (E,cB) = ( E; cB) , (X; Y )2 = ( Y ; X)2, 1 2 ∗ − − − Y Y (H,cD) = (H; cD)2, (J,c̺) = ( J; c̺)3. (A.3) ( ; β)3 = (β; )1. (A.5) − ∗ The differential operator d is defined as

3 0 ∂ ∂ d = e + ei . (A.4) From the second relation and Eq. (A.3), the constitutive ∧ ∂x0 ∧ ∂xi Xi=1 relations are represented as (E,cB)= Z0(H,cD). ∗

∗ D Electronic address: [email protected] the resistance happen to share the same dimension Ω 1 S. A. Schelkunoff, “The Impedance Concept and Its Ap- but their physical meaning is rather different. Dimension-∼ plication to Problems of Reflection, Refraction, Shield- ally the vacuum impedance resembles to the former be- ing, and Power Absorption,” Bell System Tech. J. 17, 17 cause it is introduced in terms of the fields of plane wave D (1938). 0 2 as Z = E/H (V/m)/(A/m) = Ωm/m. See Reference 1. Usually, in order to avoid the use of a specific unit sys- 8 M. J. Duff, L.∼ B. Okun, and G. Vemeziano, “Trialogue tem, a dimension is represented universally in terms of L on the number of fundamental constants,” J. High Energy (length), M (mass), T (time), and I (current), such as −2 Phys. PHEP03, 023 (2002); arXiv:physics/0110060 (2002). [f] = LMT . Here, for simplicity, we use SI units repre- 9 F. Wilczek, “On Absolute Units, I: Choices,” Physics To- D senting dimensions as f N=kgm/s2. day, October, 12 (2005). 3 ∼ 10 A. Sommerfeld, Electrodynamics — Lectures on Theoreti- To be precise the real esu chargeq ˆesu is cal Physics (Academic Press, New York, 1952), pp. 45–54, 212–222. 4 G. A. Deschamps, “Electromagnetics and Differential qˆ √dyncm qˆ qˆesu = = , Forms,” Proc. IEEE 69, 676 (1981). √4π √Nm √4π 105 5 T. Frankel, The Geometry of Physics: An Introduction × (Cambridge, 2004), 2nd Ed., pp. 118–123. 2 6 F. W. Hehl and Y. N. Obukhov, Foundations of Classical where dyn = g cm/s is the cgs unit of force. We will · −5 −1 2 Electrodynamics, Charge, , and Metric (Birkh¨auser, hereafter ignore factors such as (4π 10 ) / . 11 × Boston, 2003), pp. 143–162. J. D. Jackson, Classical Electrodynamics (John Wiley and 7 Consider a rectangular, uniform sheet with size w l. Two Sons, New York, 1998), 3rd Ed., pp. 782–784. See footnotes electrodes are attached along the edges of length×w. The for Tables 3 and 4 in Appendix. 12 resistance R between the electrodes is proportional to l and J. C. Maxwell, “A Dynamical Theory of Electromagnetic inversely proportional to w: R = σl/w. The proportional- Field,” Royal Society Transactions 155, 526 (1864); T. K. ity constant σ is called the surface resistivity of the sheet Simpson, Maxwell on the Electromagnetic Field (Rutgers D and σ = Rw/l Ωm/m = Ω. The surface resistivity and University Press, New Brunswick, 1997), p. 284. ∼