Another Look at Gaussian CGS Units

Transcription

1 Another Look at Gaussian CGS Units or, Why CGS Units Make You Cool Prashanth S. Venkataram February 24, 202 Abstrat In this paper, I ompare the merits of Gaussian CGS and SI units in a variety of different senarios for problems involving eletriity and magnetism. I onlude that in most (though not all) situations, using Gaussian CGS units makes the deeper meanings of most formulas muh more luid than using SI. Eletrostatis In SI units (hereafter referred to simply as SI ), Coulomb s law reads F SI = q q 2 4πɛ 0 r2ˆr () while in Gaussian CGS units (hereafter referred to simply as CGS ), it reads F CGS = q q 2 ˆr. (2) r2 The Gaussian definition is eminently sensible and easily reproduible. It states that two harges of magnitude esu eah separated by m will exert a fore of magnitude dyn on eah other. It an be experimentally dupliated and verified, and from this other quantities like the harge of an eletron an be found. The SI definition is an artifat of 9 th -entury bikering over what to all the prefator in front of Coulomb s law. It was eventually deided that to rationalize Gauss s law whih is more fundamental than Coulomb s law, the prefator should be 4πɛ 0. It is often said that the permittivity of free spae an be justified by quantum field theory in that the oming and going of virtual partiles in a vauum (or something like that I don t really know anything about quantum field theory beyond that) is what really allows the vauum to support an eletri field. But frankly, the hoie is arbitrary, beause it an just as easily be argued that the prefator in front of Coulomb s law being equal to unity rather than zero suggests a nie maximum value for the fore between two point harges. Neither system has an advantage when it omes to Gauss s law: E = ρ ɛ 0 (SI) (3) E = 4πρ (CGS). (4) In either ase, there are prefa-

2 tors before the harge density on the right-hand side of the equation that do not really have muh meaning other than serving as simple proportionality onstants. This is where the Lorentz-Heaviside CGS unit system really shines, beause its form of Gauss s law is E = ρ (5) and its form of Coulomb s law is (in terms of the eletri field of a point harge soure) E = q 4πr2ˆr. (6) This shows learly that the eletri flux through an arbitrary surfae is exatly equal to the harge enlosed in that surfae, and the eletri field of a point harge on a fititious sphere of radius r is simply the average surfae harge density as seen on the surfae of that sphere. There are no other prefators luttering these equations that need to be justified. Setion sore: Gaussian CGS, Lorentz-Heaviside CGS, SI 0. (For the reord, this is one of the few setions where Lorentz-Heaviside units will be disussed. From now on, CGS units will mostly mean Gaussian CGS units.) 2 Capaitane It is important to reognize that while apaitane in SI has dimensions relating to urrent, length, mass, and time all together, apaitane in CGS has dimensions equal to length and nothing more; sometimes, the CGS dimensions of apaitane as length presents an advantage, while other times it presents a disadvantage. For example, let us onsider a onentri spherial apaitor with inner radius r and outer radius r 2. The apaitane of this system is C = 4πɛ 0 r r 2 (SI) (7) C = r r 2 (CGS). (8) The advantage CGS has here is that there are no extra prefators needlessly luttering an already messy equation. The prefators in SI present no new meaning and simply detrat from the ability to do the alulation quikly. The advantage of using CGS with spherial apaitors beomes muh more apparent when onsidering the apaitane of a single harged spherial shell of radius r relative to an outer shell that an be onsidered to be infinitely large: C = 4πɛ 0 r (SI) (9) C = r (CGS). (0) Here, the apaitane of suh a sphere is exatly equal to its radius in CGS; in SI, the prefators one again present no new information. 2

3 The advantage of Gaussian CGS quikly goes away, though, when onsidering a parallel-plate apaitor of area S and separation s: C = ɛ 0S (SI) () s C = S (CGS). (2) 4πs Beause most pedagogial disussions of eletri permittivity begin with apaitors filled with dieletri materials, I feel like the presene of the permittivity of free spae in SI is more meaningful here than in Coulomb s law. Plus, in CGS, the prefator does not add any new meaning. 4π The real winner, though, would be Lorentz-Heaviside CGS units C = S (3) s beause the apaitane would just be the ratio of the area to the length without any other prefators. Lorentz-Heaviside units would not work though for other situations, suh as the aforementioned spherial apaitor. For other apaitors, like a ylindrial apaitor with inner radius r and outer radius r 2, there is no real advantage to using one system or the other dc dl = 2πɛ 0 ( ) (SI) r ln 2 r (4) dc dl = ( ) (CGS) r 2 ln 2 r (5) beause the geometry means that extra fators of or 2 pop in one way 2 or the other. Setion sore: Gaussian CGS, Lorentz-Heaviside CGS, SI 0. 3 Relativity Relativity is really the area where CGS shines ompared to SI. In relativity, when onsidering eletriity and magnetism, the key idea is that those fields are equivalent to one another in different frames of referene. In CGS, it is in fat possible to freely add and subtrat eletri and magneti fields, whereas in SI, prefators of the speed of light must be used too. 3

4 The eletromagneti field tensor is (E) 0 x (E) y (E) z F µν (E)x 0 (B) = z (B) y (E)y (B) z 0 (B) x (SI) (6) (E)z (B) y (B) x 0 0 (E) x (E) y (E) z F µν = (E) x 0 (B) z (B) y (E) y (B) z 0 (B) x (CGS). (7) (E) z (B) y (B) x 0 In speial relativity, it is usually desirable to have all the omponents of a geometri objet have the same dimensions; with CGS, that is muh easier than with SI beause E and B already have the same dimensions in CGS. This also leads to the invariant quantities assoiated with those fields, 3 ) F µν F µν = 2 (B 2 E2 (SI) (8) µ,ν=0 3 µ,ν=0 2 F µν F µν = 2 ( B 2 E 2) (CGS) (9) and as is immediately apparent, CGS allows for the free addition and subtration of eletri and magneti fields where SI does not, solidifying the notion that they are equivalent fields in all senses of the word as viewed in different frames of referene. In speial relativity, beause the speed of light is the speed limit, the veloity v is eshewed in favor of the quantity β = v. Given that, the transformation laws for the eletri and magneti fields and the Lorentz fore law make it abundantly lear whih unit system is more relativity-friendly out-of-the-box. In SI, the transformation and Lorentz fore laws are E = E (20) B = B (2) E = γ (E + v B ) (22) B = γ (B v 2 E ) (23) F = q(e + v B) (24) 4

5 while in CGS they are E = E (25) B = B (26) E = γ (E + β B ) (27) B = γ (B β E ) (28) F = q(e + β B). (29) One other thing to note is that the relative weakness of the magneti ontribution to the Lorentz fore is immediately apparent in CGS beause usually β is muh smaller than ; this annot really be gleaned from SI. Finally, it is important to note that the eletromagneti four-potential is defined differently in eah system: A µ = ( φ, A) (SI) (30) A µ = (φ, A) (CGS). (3) CGS thus hews loser to the onvention of speial relativity to make all the omponents of a geometri objet have the same dimensions, beause φ and A already have the same dimensions, whereas extra fators of are needed in SI to ahieve this. The greater number of symmetries is apparent, and it is lear that CGS rules the day as far as eletrodynamis in speial relativity is onerned. Setion sore: CGS 4, SI 0. 4 Magnetism The Biot-Savart law, desribing the magneti field reated by a moving point harge, is B = µ 0qv ˆr (SI) 4πr 2 (32) qβ ˆr B = (CGS). r 2 (33) In either ase, it an be seen that the magneti field is quite weak. But it 5 is far easier to see why in CGS. In SI, the fator µ 0 is quite small, but 4π it is just a prefator so the meaning gets obfusated. In CGS, though, it is obvious that the reason why magneti fields are usually quite small is beause a harged partile would have to be moving very fast for a signifiant magneti field to be produed (at whih point the Biot-Savart law would require relativisti orre-

6 tions), and most problems onsider slower partiles anyway. Plus, it is easy to ompare the weakness of the magneti field to the strength of the eletri field; suh a omparison would be absurd in SI beause those fields have different dimensions, so a omparison would be akin to that of apples and oranges. In Ampère s law, no system has the advantage, beause all of them onvey essentially the same information, and all of them have a few superfluous prefators present: B = µ 0 J + E 2 t B = 4πJ + E t (SI) (34) (CGS). (35) However, when examining Faraday s law, CGS has the advantage beause for all the terms not ontaining, there is a prefator of whih makes for some nie symmetry after having examined Ampère s law that is not present in SI: E = B (SI) t (36) E = B (CGS). t (37) Setion sore: CGS 2, SI 0. 5 Waves Waves are another area where CGS is helpful. When desribing an eletromagneti wave, in CGS statements like E = B (38) an be made, showing the equivalene of the eletri and magneti fields, whereas other fators of would be required in SI. Furthermore, the impedane of free spae is µ0 Z 0 = 377 Ω (SI) (39) ɛ 0 Z 0 = 4π (CGS). (40) It is muh easier to hek for dimensional orretness when working with the impedane of free spae in CGS, beause impedane has the dimensions of the reiproal of veloity. Furthermore, it is muh easier to remember something like 4π than a preise number like 377 Ω. This also plays into the Poynting vetor: S = µ 0 E B (SI) (4) S = 4π E B = Z 0 E B (CGS). (42) In SI, it an be said that the presene of µ 0 in S indiates that the vauum an support magneti fields. But then, why is there not any expliit ɛ 0 to indiate the vauum s ability to support eletri fields? By ontrast, the impedane of free spae speaks of the vauum s ability to support both eletri and magneti fields 6

7 traveling as waves, and this is made all the more lear with CGS. Finally, establishing Z 0 in CGS allows for writing Ampère s law in the following form: B dl = Z 0 I. (43) While the presene of Z 0 does not present any new understanding of this law, it does present a nier symmetry when ompared with the voltage aross a generalized impedane: V = E ds = ZI. (44) The first equation says the irulation of a magneti field around a wire is equal to the produt of the urrent and the impedane of the surrounding region. The seond equation says that the potential differene between ends of an impedane in a wire is equal to the produt of the urrent through that wire and the impedane itself. That kind of symmetry annot be ahieved in SI. Setion sore: CGS 3, SI 0. 6 Material media In CGS, the eletri and magneti fields, eletri displaement and magnetizing fields, and the polarization density and magnetization all have the same dimensions, so they an be ompared freely. This is not possible in SI without other fators. Furthermore, omparing the mirosopi and marosopi Maxwell equations in CGS yields only replaing ρ with ρ f and J with J f and replaing the fields appropriately; in SI, other onstants appear and disappear when going between mirosopi and marosopi versions of Maxwell s equations in material media. When omparing how the equations hange in CGS E = 4πρ (45) beomes D = 4πρ f (46) B = 4π J + E t (47) beomes H = 4π J f + D t (48) versus SI E = ρ ɛ 0 (49) beomes D = ρ f (50) B = µ 0 J + E 2 t (5) beomes H = J f + D t (52) the greater self-onsisteny of CGS is obvious. Also, when onsidering the polar- 7

8 ization density 7 Words of Caution D = ɛ 0 E + P (53) where P = ɛ 0 χ e E (SI) (54) D = E + 4πP (55) where P = χ e E (CGS) (56) versus the magnetization B = µ 0 (H + M ) (57) where M = χ m H (SI) (58) B = H + 4πM (59) where M = χ m H (CGS) (60) there is symmetry in the definitions of the polarization density and the magnetization in CGS, and that symmetry is preserved in the definitions of the eletri displaement and magneti fields. In SI, though, for some reason the fator of ɛ 0 is embedded in the definition of the polarization density while µ 0 is not embedded in the definition of the magnetization, so there is an asymmetry there that is also preserved in the definitions of the eletri displaement and magneti fields. It seems rather arbitrary to inlude the permittivity of free spae in one definition but not the permeability of free spae in another, and CGS does not need to worry about that at all. Setion sore: CGS 3, SI 0. It is important to know that it is not possible to blindly fore-fit µ 0 and ɛ 0 in CGS based on symmetries with laws in SI. For example, it is tempting based on Gauss s law to define ɛ 0, and it is tempting based 4π on Ampère s law to define µ 0 4π. However, this will yield the equality µ 0 ɛ 0 =, whih is different from the equality µ 0 ɛ 0 2 = as is required by SI. This will also lead to an inorret onlusion about the value of Z 0 as well. The aforementioned paper asserts that Z 0 = in CGS, whih is patently false, beause it also laims that µ 0 = ɛ 0 =, so Z 0 = µ0 ɛ 0 =. In fat, if µ 0 = 4π, as is true in Gaussian CGS, then Z 0 = µ 0 itself. Furthermore, this implies that Z 0 and an replae fators of 4π in the CGS versions of Maxwell s equations just as well as they replae µ 0 and ɛ 0 in the SI versions of Maxwell s equations. For example, here is what Maxwell s equations in CGS would look like using Z 0 M. Kitano, The vauum impedane and unit systems, arxiv:physis/ v2 (2008) 8

9 and : E = Z 0 ρ (6) B = 0 (62) E = B t (63) B = Z 0 J + E t. (64) In SI, this would be E = Z 0 ρ (65) B = 0 (66) E = B t (67) B = Z 0 J + E 2 t. (68) It is immediately apparent that there is now a greater degree of similarity between the two unit systems when replaing 4π, µ 0, and ɛ 0 with Z 0 and. Thus, the myth that the CGS annot produe a nie version of Maxwell s equations with the quantities and Z 0 like SI an, perpetuated by the paper by Kitano, an be put to rest. Setion sore: CGS, SI. 8 The Narrative The traditional narrative regarding the unifiation of eletriity and magnetism along with the study of light is that ɛ 0 was meant to just deal with eletrial phenomena, µ 0 was meant to just deal with magneti phenomena, and was only supposed to be the speed of light. Therefore, the equation µ 0 ɛ 0 2 = was supposed to be a great triumph of siene. The problem is that is only half the story, beause SI (or MKS, as it was known before) was not the only unit system being used. Simultaneously, Gaussian CGS (beause Lorentz-Heaviside CGS had not been developed yet) was in ommon urreny. At that time, the fator was just a proportionality onstant, and while it had the same dimensions as veloity, it was not until the late 9 th entury that the onnetion was made in CGS as well. Relating the present as a prefator in Ampère s and Faraday s laws to the known as the speed of light in CGS was just as great an ahievement as the relation of µ 0 and ɛ 0 to in SI; sadly, beause of SI s prevalene over CGS, this other story is too often overlooked. Furthermore, now that the present in Maxwell s equations in CGS is known to be the same as the speed of light, it an be more easily justified that CGS an be adapted more easily to show this unifiation between eletriity and magnetism. In SI, the residual presene of µ 0 and ɛ 0 as separate prefators still shows that SI impliitly onsiders eletriity and magnetism to be separate. Thus, CGS, whih is an older system than SI, has aged far better through the unifiation of eletriity and mag- 9

10 netism and optis. This aging also relates to the higher ompatibility of CGS with relativisti eletrodynamis. The paper by Kitano posits that the point is moot beause CGS was developed long before the development of the Lorentz transformation and the rest of speial relativity. Yet, the fat remains that CGS is undoubtedly more relativity-friendly than SI out-of-thebox. Therefore, this statement by Kitano, rather than denigrating CGS as intended, only serves as a testament to how well CGS an handle new developments in eletrodynamis like speial relativity and how poorly the newer SI handles it. Finally, the reason for fators like µ 0 and ɛ 0 in SI stem mostly from onveniene: the people setting up the SI standards found it easier to measure the fore due to urrents in wires upon other wires rather than the fore of one harge upon another. The definition of the ampere omes from where df dl = µ 0I I 2 2πr (69) µ 0 4π 0 7 N A 2 (70) so that the ampere an be a fundamental unit independent of other mehanial units (kg, m, s). Thus, the ampere is defined as the amount of urrent suh that two wires eah with ampere of urrent separated by meter will exert a fore of newtons per meter. This is atually a fairly reasonable definition, and it does make sense to make a quantity of eletrodynamis independent of other mehanial quantities. However, the fat remains that ultimately, eletri and magneti fields arise from harged partiles; all harged partiles are soures of eletri fields, but only moving harged partiles (in a given frame of referene) are soures of magneti fields, so there is no single partile that an produe only a magneti field without an eletri field. Given that, it makes more sense to make the unit of harge in eletri phenomena the fundamental unit rather than the unit of urrent as per magneti phenomena, and this is exatly what CGS does. Setion sore: CGS 2, SI 0. 9 Conlusion Thus, after all this, Gaussian CGS units present a more self-onsistent, relativity-friendly, dieletri-friendly, symmetrial set of equations than SI units do, while mathing any and all benefits that SI may have. Total sore: CGS 7, SI. 0