# Talk:Gaussian units

WikiProject Measurement (Defunct)
WikiProject Physics (Rated C-class, Mid-importance)

## Finger error in a formula

Thanks for this useful article. Please, correct a "finger error" in the formula relating B and H. The correct equation is B = "mu" H. --189.231.190.214 (talk) 17:00, 16 November 2009 (UTC)

Yes, thank you, I have now changed it! --Steve (talk) 20:49, 16 November 2009 (UTC)

## Gaussian or CGS?

If, as the article states, the term "cgs units" is deprecated, why is it used several times, with no further qualification, in the text? For those who know, would it be safe to change existing unqualified references in this article from "cgs units" to "Gaussian units"?

Fixed! :-) --Steve (talk) 22:40, 14 April 2012 (UTC)

## Why?

This article does a pretty comprehensive job of describing how Gaussian units differ from SI ones, but offers no explanation for why factors of 4pi, epsilon_zero etc. differ between the two systems. There's some (not very clear) discussion at Centimetre_gram_second_system_of_units#Derivation_of_CGS_units_in_electromagnetism, but this article doesn't mention that and only links to it in passing in the lead. There needs to be a brief description of why these factors disappear, and a clear link to the relevant place for more information. I'd fix it myself, but frankly I don't follow the reasoning (despite having a degree in physics and working in a field where Gaussian units are still used), so I doubt I could write anything that was correct. Modest Genius talk 00:35, 27 December 2012 (UTC)π

About the 4pi: It's not clear to me exactly what question you're looking to answer:
• "Why is it that if Coulomb's law has an explicit 4pi then Gauss's law does not have an explicit 4pi, and vice-versa?" I think the article answers that well: "The quantity 4π appears because 4πr2 is the surface area of the sphere of radius r. For details, see the articles Relation between Gauss's law and Coulomb's law and Inverse-square law."
• "What are the advantages and disadvantages of rationalized versus non-rationalized units?" This is a subjective question. One person might think that something is an advantage, someone else with different preferences might think the same thing is a disadvantage. It's not impossible to discuss subjective questions in a wikipedia article, but it is difficult. You need to fairly summarize all the major points of view, and find references for them. The latter is very difficult; even if lots of physicists say to each other over coffee "Gaussian units make XYZ unnecessarily complicated", they rarely would write that into a publication that we can reference.
• "What are the historical circumstances that caused SI to be rationalized while Gaussian is not?" It looks like Reference 5 discusses the history and I'm sure you can find the information you're looking for there. If you want to add better historical information to the article, that would be great, and I'm happy to help as I can.
About epsilon0: The main thing is that in Gaussian units,
1 statC = 1 g1/2 cm3/2 s−1
while in SI, there is no analogous relation between Coulombs and the "mechanical" units (length, mass, time). This difference leads to a wide variety of formula differences, especially when epsilon0 and mu0 are involved.
Again, maybe you're wondering what the advantages and disadvantages are to defining charge in terms of mechanical units. Or maybe you're wondering why, theoretically, people have a choice in the matter at all. Or maybe you're wondering why, historically, different systems developed in different ways. Which is it? --Steve (talk) 18:15, 27 December 2012 (UTC)
The analogous relation in SI comes from Ampere's law with the constant 10-7 N/A2. If you define that constant to be unity, then A^2=10^7 N, or
1 A = 107/2 N1/2.
I suspect that the 107/2 discourages its use.Gah4 (talk) 15:04, 25 October 2015 (UTC)

Rationalization just moves the 4π around. It has to go somewhere so that surface integrals work. Other than that, Gaussian units are electrostatic, SI are magnetostatic. Gah4 (talk) 23:38, 20 October 2015 (UTC)

There is a basic idea that isn't well covered here. That electromagnetic units are based either on the force between charges (electrostatic) or force between currents (magnetostatic). Maxwell's equations (and special relativity) connect electricity and magnetism, such that you can't (or shouldn't) specify the units separately. The claim that Gaussian units don't have ε0 isn't quite right. Gaussian units defines the charge unit (ESU) such that ε0 has the value 1/(4π). SI instead defines the charge unit as the ampere second, (with ampere based on the force between currents). That is, the ampere is defined in terms of mechanical units, and the coulomb derived from the ampere. The definition of ampere is strange enough that ampere is always used instead of the mechanical units. Gah4 (talk) 23:38, 20 October 2015 (UTC)

An ampere is indeed "defined in terms of mechanical units" (at the moment; it will eventually be defined as a certain number of electrons per second). But it is not expressible purely in terms of mechanical units, i.e. there is no SI equation for ampere that looks like the gaussian-units equation
1 statC = 1 g1/2cm3/2s−1.
Do you agree? Experienced physicists might dismiss this as a triviality, but for people first learning about different unit systems it is extremely important and often confusing. People expect the "number of independent base units" (mass, length, time, etc.) to have some profound meaning, and are therefore confused by the fact that SI has an extra base unit for electromagnetism but gaussian does not.
The statement "ε0 does not even exist in Gaussian units" is true in the following sense: People who use gaussian units do not have a quantity called ε0 that they put into electromagnetism formulas. I guess you're saying "Well, they don't use the notation "ε0", but they do have "1/(4π)" which plays the same role." But I think the context makes it clear that we are remarking on the fact that ε0 is a dimensionful constant in SI, and gaussian units definitely does not have any dimensionful constant at all analogous to that. (Perhaps the wording in the article could be improved.)
It seems to me that the choice of "C is my base unit and A = C/s" versus "A is my base unit and C = A*s" is not at all important in understanding or using SI. It's in the fine print, something that only super-high-precision physics labs have any reason to care about. Similarly, today amp is defined in terms of forces on wires, and in 2018 it will be defined as a certain number of electrons per second. This is important to some precision physicists but I don't see why the rest of us should care, or why it should affect the basic way we understand SI.
Sorry if I'm misunderstanding your point. --Steve (talk) 16:29, 23 October 2015 (UTC)

Well, for one, the meter is now a derived unit, not a base unit. There is work to make the kilogram a derived unit, though not quite ready yet. (Either from hbar or avogadro's number.) And the ampere is also a derived unit, by defining the force between two current carrying wires. The important difference is that gaussian units are derived through electrostatic, and SI derived through magnetostatics. That difference is important for those learning E&M to know. Gah4 (talk) 00:30, 24 October 2015 (UTC)

What do you mean by "base unit" and "derived unit"? You should use different words, because in the context of SI, "base unit" and "derived unit" have specific universally-agreed-upon definitions (see for example SI base unit and SI derived unit), and according to those definitions the ampere and meter are base units, not derived units. You must be talking about something else and I don't know what. --Steve (talk) 00:44, 24 October 2015 (UTC)
From SI base unit However, in a given realization in these units they may well be interdependent, i.e. defined in terms of each other. For some reason, there is no Derived unit separate from SI derived unit. A base unit should be one that isn't defined in term of other units, but SI cheats. Also, note the fourth paragraph in Dimensional_analysis#Definition Gah4 (talk) 06:05, 24 October 2015 (UTC)
Will SI cease to be "derived through magnetostatics" in 2018, assuming that ampere is redefined as a certain number of electrons per second?
You say gaussian electromagnetic units are defined in terms of electric forces on charges. You can define them that way if you want, there's nothing wrong with that. But I can equally well define gaussian electromagnetic units in terms of magnetic forces on currents. Or heck, I could say that the definition of statC is
1 statC = 1 g1/2cm3/2s−1.
which has no relation to electric forces or magnetic forces. (I actually prefer this last definition if I had to pick one.) So to summarize, I don't believe your statement that gaussian electromagnetic units are defined in terms of electrostatics. --Steve (talk) 14:04, 24 October 2015 (UTC)
Those units come from setting the constant in Coulomb's law to 1. SI units set the constant in Ampère's_force_law from Magnetostatics to ${\displaystyle 10^{-7}}$ N / A2. In both cases, the corresponding constant in the other equation has the appropriate value required by Maxwell's_equations. Gah4 (talk) 14:22, 25 October 2015 (UTC)
And no, if the proposed 2018 changes define the kilogram in terms of e, they are still "derived through magnetostatics", but in the other direction. Instead of defining the electromagnetic units from mechanical units, the mechanical units will be defined through magnetostatics and quantum mechanics.[1] Gah4 (talk) 15:28, 25 October 2015 (UTC)
See Proposed redefinition of SI base units. You say "the mechanical units will be defined through magnetostatics and quantum mechanics", but I think you are mistaken. The mechanical units will be defined through the the hyperfine energy splitting of cesium atoms, the speed of light, and Planck's constant. How is magnetostatics involved in any of those? You linked to a PDF, but that PDF does not mention magnetostatics either. (Indeed, under the proposal, we will not know the exact numerical value of μ0; it will have experimental error bars.)
You say: "[Gaussian electromagnetic] units come from setting the constant in Coulomb's law to 1." OK, well I say: "Gaussian electromagnetic units come from setting the constant in Ampere's force law to 1/c2". There, I just defined gaussian units in terms of magnetostatics. I don't see what rational basis there could be for saying that my definition is wrong and yours is right. The two definitions lead to the exact same system of units, right?
Similarly, the SI authority says: "the ampere is the constant current that will produce an attractive force of 2e−7 N/m between two parallel wires 1m apart in vacuum". But it would be 100% equivalent to say: The ampere is defined so that the electric permittivity of vacuum is exactly 10000000 / (4*pi* 2997924582) A2⋅s4⋅kg−1⋅m−3. If they had used that definition, everything about SI would be exactly the same, but now things would be based around electric permittivity, with no mention of magnetostatics.
So we see in both cases that you can swap electric-based definitions for magnetostatic-based definitions without changing anything whatsoever about the unit system. It's wise (for pedagogical reasons) to pick a definition and stick to it, and some definitions are pedagogically better than others, but they don't reflect any profound truth or understanding of the unit system.
So I continue to think that "SI is based on magnetostatics, gaussian on electrostatics" is not really true and definitely not a profound truth that is worth emphasizing or even mentioning. --Steve (talk) 02:15, 27 October 2015 (UTC)

I am not so sure by now which units will be defined in terms of which. The Watt_balance is used to define the kilogram in terms of electronic units, instead of the other way around. Since 1983, when c was defined as 299792458 m/s, yes, you could as easily define through Coulomb's law as through Ampere's law, but BIPM didn't do that. They could have changed it in 1983, but didn't. I suspect the reason is that it is easier to physically measure current instead of charge. Yes, the difference isn't as profound as before 1983, but it is still the way the units are defined. The CGS units include both stat units, based on Coulomb's law, and ab units based on Ampere's law, and both defined before 1983. The important part is that you choose a consistent set, such that each unit has only one definition. For many years, c was not so well known. In the early years of spectroscopy, wavelength could be measured with smaller relative error than c. It was, then, usual to do physics with wavenumber (inverse wavelength) instead of frequency, avoiding the uncertainty in c. Gah4 (talk) 08:30, 27 October 2015 (UTC)

I was thinking about this one in a discussion in another talk page. I have wondered for a long time how the unit for the volt came out such that it has a convenient value for common electrochemical cells. It seems that it was intentionally chose such that the Daniell_cell was (close to) 1.0, though in the end, slightly off. After that was chosen, the rest of the units were arranged such that it came out that way. Since voltage is more related to charge than current, that isn't quite as obvious as it could be. This is a little more interesting, as it is possible to measure voltage without measuring current.[2] Oh well. Gah4 (talk) 22:09, 12 June 2020 (UTC)

References

1. ^ Roadmap, SI. "Joint CCM and CCU roadmap towards the redefinition of the SI in 2018" (PDF). SI Roadmap. BIPM. Retrieved 25 October 2015.
2. ^ Panofsky. "THE KELVIN ABSOLUTE VOLTMETER and the speed of light" (PDF). www.sophphx.caltech.edu. Caltech. Retrieved 12 June 2020.

I think there is a contradiction between the articles on cgs unit system http://en.wikipedia.org/wiki/Centimetre_gram_second_system_of_units#Electrostatic_units_.28ESU.29 and the one on the Gaussian unit system http://en.wikipedia.org/wiki/Gaussian_units.

In the Table Electromagnetic units in various CGS systems of http://en.wikipedia.org/wiki/Centimetre_gram_second_system_of_units#Electrostatic_units_.28ESU.29 the conversion of magnetic field units between SI unit system (i.e. Tesla) and ESU unit system is give as 10^4 statT/c. This conversion is quite consistent when one begins with the Lorentz force equation in SI units and converts it to ESU. In this way the additional (1/c) term in the denominator of the corresponding equation in ESU is automatically obtained. In the same table the conversion between Tesla and Gauss is 1 T = 10^4 G. In the light of the conversion between SI and USU this implies that whenever one replaces 1 T in the Lorentz equation in SI units by 10^4 G one obtains the corresponding equation in Gaussian units. On the other hand in the table Electromagnetic unit names of http://en.wikipedia.org/wiki/Gaussian_units the conversion between SI and Gaussian units is given as 1 T = 10^4 Gauss. Probably this does not mean that whenever one replaces 1 T in the Lorentz equation in SI units by 10^4 Gauss one obtains the corresponding equation in Gaussian units. I think this point requires clarification. — Preceding unsigned comment added by 193.140.249.2 (talk) 14:11, 2 April 2013 (UTC)

ESU is not the same as Gaussian. But anyway, you're correct that you cannot replace "1 T" in an SI equation by "10^4 G" in a gaussian or ESU equation. There is already some text about this in the article: " The symbol "↔" was used instead of "=" as a reminder that the SI and Gaussian units are corresponding but not equal because they have incompatible dimensions. For example, according to the top row of the table, something with a charge of 1 C also has a charge of (10^−1 c) Fr, but it is usually incorrect to replace "1 C" with "(10−1 c) Fr" within an equation or formula...". There is similar text in the statcoulomb article. I have just now copied the exact same text and notation to the table in Cgs units#Electromagnetic units in various CGS systems. Does that help? --Steve (talk) 17:49, 2 April 2013 (UTC)
It is correct to use = rather than a double-arrow, because a statement like e = 4.802E-10 Fr, is something that does not imply the construction of the franklin from any given theory. A franklin is equal to a charge of 1E10/4.802 electrons, and can be directly related to that. Whether you choose to suppose the franklin is a mechanical unit dyn^½ cm, or a base unit (1947 standard), is irrelevant. Wendy.krieger (talk) 11:54, 22 December 2016 (UTC)
Do you think that "↔" is technically incorrect? You didn't say that, and I definitely don't think so, because "↔" does not have any strict technical definition in this context that we could be violating. Now, an abundance of personal experience and talk-page comments suggests that people will definitely substitute "1C" with "3e9 Fr" unless they are told not to (and then reminded over and over). Even if the "=" were technically correct, so is "↔", and since the latter is pedagogically better, we should stick with it.
But I actually don't think that "=" is technically correct. How do I know? When two things are equal, you can substitute one for the other in an equation, but you can't substitute "1C" with "3e9 Fr" in an equation. It's as simple as that!
Here's an analogy. Suppose that everyone talks about spheres all the time, but Europeans traditionally specify a sphere by stating its radius, and Africans traditionally specify a sphere by stating its volume. A "3cm sphere" in Europe is the same sphere as a "12π cm^3 sphere" in Africa. But it's wrong to say "3cm = 12π cm^3", and it's right to say "3cm ↔ 12π cm^3 when specifying spheres". The descriptions correspond to each other—they are describing the same sphere—but the descriptions are not equal to each other. Similarly, "1C of charge" and "3e9 Fr of charge" are two descriptions with the same referent. But it's wrong to say "1C = 3e9 Fr", and it's right to say "1C ↔ 3e9 Fr when describing amounts of electric charge". --Steve (talk) 13:46, 22 December 2016 (UTC)

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## Uniting the electromagnetic units articles

In reading through the Gaussian units and Lorentz–Heaviside units articles, I noticed a lot of overlap in how each article presented Maxwell's equations across unit systems. This prompted me to pull out the relevant tables into template pages (Template:Table of Maxwell equations and Template:Table of electromagnetic laws) to present the same information the same way in all of the articles where the equations appear. I still think having the both of the Gaussian units and Lorentz-Heaviside units articles is redundant, however---Gaussian units only contains a little more information about historical significance and additional conversion rules that should probably appear in Lorentz-Heaviside units as well. Consequently I'd like to propose merging these two articles into a new "Electromagnetic units" page; I think such an article could present much more information much more concisely, possibly with sections on the history of each unit system, derivations in a system-independent form (as already sort of exists in Lorentz-Heaviside units), and then tables giving all of the equations in common unit systems. To me, at least, this would feel much more "complete."Glosser.ca (talk) 18:18, 14 August 2017 (UTC)

I undid your change. The Lorentz-Heaviside formulas are off-topic in an article about Gaussian units. To my knowledge, Gaussian and Lorentz-Heaviside units are not commonly used in the same situations, whereas Gaussian and SI certainly are.
Also, Wikipedia already has a general discussion of electromagnetic units covering SI, Gaussian, Lorentz-Heaviside, ESU, EMU. See Centimetre–gram–second_system_of_units#Derivation_of_CGS_units_in_electromagnetism.
I think there are a lot of people (in atomic physics, astronomy, etc.) who come across Gaussian units when reading old articles in their field, and these people want to know how Gaussian works, and especially how Gaussian relates to its primary alternative SI, and these people are not particularly interested in getting bogged down in the theory of electromagnetic units in general, or Lorentz-Heaviside units in particular (which are generally not used in the same domains that Gaussian is). This article is just right for those people. It has the descriptions they want, the formulas they're looking for, it's right to the point. If this article were merged with Lorentz-Heaviside (to say nothing of ESU and EMU), it would be less useful to that audience because the information they want is mixed in with lots of other stuff.
If there's some overlap between two articles, no big deal, that happens all the time. --Steve (talk) 01:41, 15 August 2017 (UTC)

Gaussian units are still used often enough that you don't have to go look for old papers. Especially Gauss is often used for magnetic fields. It is a more convenient size, for one. Also, I disagree that an article on Gaussian units shold not mention other unit systems. Often enough, it is convenient and useful to compare and contrast to other systems. One should not overmention them, but just enough to cover the important similarities and differences. Unless I forget, Lorentz-Heaviside units are similar to Gaussian units, but with the ${\displaystyle 4\pi }$ move around. Gah4 (talk) 07:34, 15 August 2017 (UTC)

If Heaviside-Lorentz units are off-topic, shouldn't SI units also be off-topic? Gah4 (talk) 07:39, 15 August 2017 (UTC)

Gah4, what do you think of the status quo as of now? I personally think other unit systems are described to an appropriate extent, and indeed Lorentz-Heaviside is mentioned in a couple places where it's relevant.
SI is not off-topic because it's very very common in practice to need to translate between Gaussian and SI formulas, whereas it is pretty rare to want to translate between Gaussian and Lorentz-Heaviside. SI is ubiquitous in all domains, Gaussian is occasionally used in atomic physics etc, and Lorentz-Heaviside is occasionally used in particle physics. Since Gaussian and Lorentz-Heaviside are each rare and are not especially overlapping in their usages, it's pretty rare to need to go directly back and forth between them.
By the way, I disagree that using "Gauss" or "Oersted" is really an example of using "Gaussian units". It's only really Gaussian units if you switch all of your formulas, stop mentioning ε0 and μ0, give capacitance a unit of length, etc. etc. If you're just using Gauss as a synonym for 1e-4 teslas, but still using SI electromagnetism formulas (which is what most people indeed do), it's still basically SI. (When I've written dimensional-analysis software in the past, I've need separate variables for "Gauss in the context of SI" vs "Gauss in the context of Gaussian units", and those two variables are not dimensionally equivalent.) --Steve (talk) 12:14, 15 August 2017 (UTC)
I'm with Gah4 on this one. It's true that it might be rare to translate between Gaussian and L-H units (and thus of little mechanical utility to include both), but I feel that the ability to compare and contrast the unit systems across the articles where they're discussed more thoroughly adds a lot for the people who aren't necessarily "just" reading old articles and want a bigger/deeper picture. Adding an additional column to two tables does not unduly clutter the article, particularly if the new information occurs far to the right and "away" from the more relevant bits. Glosser.ca (talk) 16:20, 15 August 2017 (UTC)
I once had a professor, at the beginning of a lecture, remind us that the power line voltage in his house was 1/3 Statvolt. That is consistent use of Gaussian units. But yes, statvolt meters are rare to find on the shelf at nearby electronics dealers. (It can't be that hard to put in the option for a DVM, though.) I did my quals in Heaviside-Lorentz units. We were allowed to consistently use any unit system, and declare at the top which one we were using. It was easier to remember where everything goes in H-L units. Engineering is commonly done SI, but theorists like other systems. I believe my first E&M course was taught in Gaussian units, and especially if you want to teach 4-vectors Gaussian units are more obvious. Gah4 (talk) 18:49, 15 August 2017 (UTC)
It seems that in 2013, Purcell was updated to SI units, pretty much confirming my memory that it was Gaussian when I used it. I suspect that there are enough first and second editions around for anyone who wants to teach Gaussian units. [1] One difference from some other books, is that it teaches magnetism through electrostatics and special relativity. (Consider a current in the reference frame of moving electrons.) Gah4 (talk) 19:05, 15 August 2017 (UTC)
Gah4, you could have just read the article! Footnote 3 describes how Purcell switched to SI.

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## Formula-conversion tables

Shown immediately below is the original Gaussian to SI formula-conversion table that appeared in the text.[1]

TABLE A

Name Gaussian units SI units
Speed of light ${\displaystyle c}$ ${\displaystyle {\frac {1}{\sqrt {\epsilon _{0}\mu _{0}}}}}$
Electric field, Electric potential ${\displaystyle \left(\mathbf {E} ,\varphi \right)}$ ${\displaystyle {\sqrt {4\pi \epsilon _{0}}}\left(\mathbf {E} ,\varphi \right)}$
Electric displacement field ${\displaystyle \mathbf {D} }$ ${\displaystyle {\sqrt {\frac {4\pi }{\epsilon _{0}}}}\mathbf {D} }$
Charge, Charge density, Current,
Current density, Polarization density,
Electric dipole moment
${\displaystyle \left(q,\rho ,I,\mathbf {J} ,\mathbf {P} ,\mathbf {p} \right)}$ ${\displaystyle {\frac {1}{\sqrt {4\pi \epsilon _{0}}}}\left(q,\rho ,I,\mathbf {J} ,\mathbf {P} ,\mathbf {p} \right)}$
Magnetic B field, Magnetic flux,
Magnetic vector potential
${\displaystyle \left(\mathbf {B} ,\Phi _{\text{m}},\mathbf {A} \right)}$ ${\displaystyle {\sqrt {\frac {4\pi }{\mu _{0}}}}\left(\mathbf {B} ,\Phi _{\text{m}},\mathbf {A} \right)}$
Magnetic H field ${\displaystyle \mathbf {H} }$ ${\displaystyle {\sqrt {4\pi \mu _{0}}}\mathbf {H} }$
Magnetic moment, Magnetization ${\displaystyle \left(\mathbf {m} ,\mathbf {M} \right)}$ ${\displaystyle {\sqrt {\frac {\mu _{0}}{4\pi }}}\left(\mathbf {m} ,\mathbf {M} \right)}$
Relative permittivity,
Relative permeability
${\displaystyle \left(\epsilon ,\mu \right)}$ ${\displaystyle \left({\frac {\epsilon }{\epsilon _{0}}},{\frac {\mu }{\mu _{0}}}\right)}$
Electric susceptibility,
Magnetic susceptibility
${\displaystyle \left(\chi _{\text{e}},\chi _{\text{m}}\right)}$ ${\displaystyle {\frac {1}{4\pi }}\left(\chi _{\text{e}},\chi _{\text{m}}\right)}$
Conductivity, Conductance, Capacitance ${\displaystyle \left(\sigma ,S,C\right)}$ ${\displaystyle {\frac {1}{4\pi \epsilon _{0}}}\left(\sigma ,S,C\right)}$
Resistivity, Resistance, Inductance ${\displaystyle \left(\rho ,R,L\right)}$ ${\displaystyle 4\pi \epsilon _{0}\left(\rho ,R,L\right)}$

72.251.60.95 (talk) 01:23, 3 November 2017 (UTC)

To Sbyrnes321: The new table appears in an edition of Resnick and Halliday, as indicated in the reference.[2] There are infinitely many ways to convert from SI to Gaussian; this one is probably the most convenient. It cannot contradict the old table, as you claim, since the two tables convert in opposite directions. The comment added in parenthesis ("For the reverse conversion, replace ε0 and μ0 in the table by 1/c.") is necessary if the old table is to be used in reverse; otherwise ε0 and μ0 remain in the formula with their SI meanings rather that their Gaussian meanings, i.e., the number one. Please give an example where the new table gives a wrong answer. For the benefit of those who wish to judge for themselves, here is the new table.

TABLE B

Name SI units Gaussian units
Speed of light ${\displaystyle c}$ ${\displaystyle c}$
Electric field, Electric potential ${\displaystyle \left(\mathbf {E} ,\varphi \right)}$ ${\displaystyle \left(\mathbf {E} ,\varphi \right)}$
Electric displacement field ${\displaystyle \mathbf {D} }$ ${\displaystyle {\frac {1}{4\pi }}\mathbf {D} }$
Charge, Charge density, Current,
Current density, Polarization density,
Electric dipole moment
${\displaystyle \left(q,\rho ,I,\mathbf {J} ,\mathbf {P} ,\mathbf {p} \right)}$ ${\displaystyle \left(q,\rho ,I,\mathbf {J} ,\mathbf {P} ,\mathbf {p} \right)}$
Magnetic B field, Magnetic flux,
Magnetic vector potential
${\displaystyle \left(\mathbf {B} ,\Phi _{\text{m}},\mathbf {A} \right)}$ ${\displaystyle {\frac {1}{c}}\left(\mathbf {B} ,\Phi _{\text{m}},\mathbf {A} \right)}$
Magnetic H field ${\displaystyle \mathbf {H} }$ ${\displaystyle {\frac {c}{4\pi }}\mathbf {H} }$
Magnetic moment, Magnetization ${\displaystyle \left(\mathbf {m} ,\mathbf {M} \right)}$ ${\displaystyle c\left(\mathbf {m} ,\mathbf {M} \right)}$
Permittivity,
Permeability
${\displaystyle \left(\epsilon ,\mu \right)}$ ${\displaystyle \left({\frac {1}{4\pi }}\epsilon ,{\frac {4\pi }{c^{2}}}\mu \right)}$
Electric susceptibility,
Magnetic susceptibility
${\displaystyle \left(\chi _{\text{e}},\chi _{\text{m}}\right)}$ ${\displaystyle 4\pi \left(\chi _{\text{e}},\chi _{\text{m}}\right)}$
Conductivity, Conductance, Capacitance ${\displaystyle \left(\sigma ,S,C\right)}$ ${\displaystyle \left(\sigma ,S,C\right)}$
Resistivity, Resistance, Inductance ${\displaystyle \left(\rho ,R,L\right)}$ ${\displaystyle \left(\rho ,R,L\right)}$

72.251.63.241 (talk) 02:53, 23 October 2017 (UTC)

For example, consider the formula m = IS for a small current loop, valid in SI. The new table converts this into cm = IS, or m = IS/c, which is valid in Gaussian. On the other hand, the old table in reverse yields (4π/μ0)1/2m = (4πε0)1/2IS, which, since ε0 = μ0 = 1 in Gaussian, is equivalent to m = IS, which is invalid in Gaussian. If, however, one first replaces ε0 and μ0 in the table by 1/c, the old table in reverse yields (4πc)1/2m = (4π/c)1/2IS, which is equivalent to m = IS/c, which is valid in Gaussian.

It should also be observed that the new table cannot be used in reverse, unlike the old table. If we apply the new table in reverse to the Gaussian formula D = E in a vacuum, we obtain 4πD = E, or D = E/4π, which is invalid in SI, the correct formula being D = ε0E. The reason for this is that the Gaussian formula contains a hidden ε0 which is not converted. 72.251.63.241 (talk) 07:30, 23 October 2017 (UTC)

I can do the small current loop example using the old table!
The SI formula is m = IA where A is the area of the current loop. (I'm not sure why you wrote it as S ... ?)
First I multiply both sides by sqrt(ε0μ0/4π)=1/(c sqrt(4π)), to get m sqrt(ε0μ0/4π) = IA/(c sqrt(4π)), i.e. m sqrt(μ0/4π) = I/sqrt(4πε0) * A/c.
Now, following the table, I replace m sqrt(μ0/4π) with m, and I/sqrt(4πε0) with I, and the area A stays the same. So the Gaussian formula is m=IA/c, the correct result.
(I think the table used to have some note that you occasionally need to use the identity 1/sqrt(ε0μ0)=c during conversions, but maybe it got deleted at some point?)
I want to think it through a bit more carefully before I reply to the other interesting things you wrote :-D --Steve (talk) 14:36, 23 October 2017 (UTC)

If I deduce a false conclusion from certain premises to show that there is something wrong with them, it is no proper reply for you to show that you can deduce the correct conclusion from the same premises; you must show that there was a flaw in my reasoning. The fact remains that I naively applied the original table in the text to obtain an incorrect result. That you were able to obtain the correct result from the same table is irrelevant; you must show that there was a mistake in my calculation. It would not have occurred to me to do what you have done, and if one must be so clever, then there must be something wrong with the table. What you have done is to use a true formula in SI to transform the given SI formula to a different one which you then convert using the table, but this is an illegal procedure. Truth here is irrelevant; the table is supposed to give an algorithm for translating formulas, true or false, from one language to another. You have not translated the original formula, but a different one. Some time ago the table did not contain a row for the speed of light, but contained the note that you refer to. Such a note is itself illegal, and so I moved it into the body of the table as the row for the speed of light. There is no separate note in Jackson's Classical Electrodynamics; everything is in the table, including the row for the speed of light. Shown below is a table that reflects the reverse of the table in the text, emended as explained before.

TABLE C

Name SI units Gaussian units
Speed of light ${\displaystyle c}$ ${\displaystyle c}$
Electric field, Electric potential ${\displaystyle \left(\mathbf {E} ,\varphi \right)}$ ${\displaystyle {\sqrt {\frac {c}{4\pi }}}\left(\mathbf {E} ,\varphi \right)}$
Electric displacement field ${\displaystyle \mathbf {D} }$ ${\displaystyle {\frac {1}{\sqrt {4\pi c}}}\mathbf {D} }$
Charge, Charge density, Current,
Current density, Polarization density,
Electric dipole moment
${\displaystyle \left(q,\rho ,I,\mathbf {J} ,\mathbf {P} ,\mathbf {p} \right)}$ ${\displaystyle {\sqrt {\frac {4\pi }{c}}}\left(q,\rho ,I,\mathbf {J} ,\mathbf {P} ,\mathbf {p} \right)}$
Magnetic B field, Magnetic flux,
Magnetic vector potential
${\displaystyle \left(\mathbf {B} ,\Phi _{\text{m}},\mathbf {A} \right)}$ ${\displaystyle {\frac {1}{\sqrt {4\pi c}}}\left(\mathbf {B} ,\Phi _{\text{m}},\mathbf {A} \right)}$
Magnetic H field ${\displaystyle \mathbf {H} }$ ${\displaystyle {\sqrt {\frac {c}{4\pi }}}\mathbf {H} }$
Magnetic moment, Magnetization ${\displaystyle \left(\mathbf {m} ,\mathbf {M} \right)}$ ${\displaystyle {\sqrt {4\pi c}}\left(\mathbf {m} ,\mathbf {M} \right)}$
Permittivity,
Permeability
${\displaystyle \left(\epsilon ,\mu \right)}$ ${\displaystyle {\frac {1}{c}}\left(\epsilon ,\mu \right)}$
Electric susceptibility,
Magnetic susceptibility
${\displaystyle \left(\chi _{\text{e}},\chi _{\text{m}}\right)}$ ${\displaystyle 4\pi \left(\chi _{\text{e}},\chi _{\text{m}}\right)}$
Conductivity, Conductance, Capacitance ${\displaystyle \left(\sigma ,S,C\right)}$ ${\displaystyle {\frac {4\pi }{c}}\left(\sigma ,S,C\right)}$
Resistivity, Resistance, Inductance ${\displaystyle \left(\rho ,R,L\right)}$ ${\displaystyle {\frac {c}{4\pi }}\left(\rho ,R,L\right)}$

72.251.58.166 (talk) 19:20, 23 October 2017 (UTC)

Right, so "the flaw in your reasoning" is that you're not supposed to replace ε0 and μ0 by 1. ε0 and μ0 are the exact same dimensionful SI quantities as before, e.g. ε0=8.9e-12 F/m which is not equal to 1. It is however true as always that 1/sqrt(ε0μ0)=c.
If the section does not make it clear that ε0 does not turn into 1, then that's a pedagogy/description problem. I'm not denying that a pedagogy/description problem is an important problem, just trying to suggest that it is probably fixable. :-D
You say "There are infinitely many ways to convert from SI to Gaussian" but I disagree. Here is an example that is translated correctly with the old table but not the new:
${\displaystyle q_{electron}=-1.6\times 10^{-19}{\text{C}}}$
(the charge of an electron). Using the old table, we get the Gaussian formula:
${\displaystyle q_{electron}=-1.6\times 10^{-19}{\text{C}}/{\sqrt {4\pi \epsilon _{0}}}=-1.6\times 10^{-19}{\text{A s}}/{\sqrt {4\pi (8.9\times 10^{-12}{\text{A}}^{2}{\text{s}}^{4}{\text{kg}}^{-1}{\text{m}}^{-3})}}}$
${\displaystyle =-4.8\times 10^{-10}{\text{g}}^{1/2}{\text{cm}}^{3/2}/{\text{s}}}$
${\displaystyle =-4.8\times 10^{-10}{\text{statC}}}$
It works! That's the textbook value of the electron charge in gaussian units. Whereas with the new table, the same thing would not work. Ditto with any other physical quantity: "I'm holding a capacitor in my hand, and the electric field between its plates is E=1V/m". If you translate that formula "E=1V/m" with the old table, you get a correct description of that same electric field using Gaussian units. But not with the new table. And so on.
In this sense, the old table is the fundamental deep truth of SI vs gaussian. :-D It answers questions like "What does the word "charge" really mean? What does the term "displacement field" really mean?"
OK, now, to get from the old table to the new table (the Halliday/Resnick one that you put in), just take each SI quantity, and if its expression in SI base units has the Nth power of amperes, you multiply it by sqrt(4πε0)^N. Leaving aside examples like the ones I gave above (i.e. examples which reference physical objects), you are bound to have just as many powers of amperes on each side, and that means that the new table is going to work too in those cases. Still, I prefer the old one because it applies even to equations referencing physical objects, and (relatedly) I feel like it's more fundamental / profound. :-D --Steve (talk) 00:28, 24 October 2017 (UTC)

I believe that your derivation of the Gaussian current loop formula can be made rigorous if instead of using the SI equation c = 1/(ε0μ0)1/2, you use the speed of light rule in the table. Rewrite m = IA as (μ0/4π)1/2m = (μ0/4π)1/2IA = (μ0/4π)1/2(4πε0)1/2IA/(4πε0)1/2 = (I/(4πε0)1/2)A/(1/(ε0μ0)1/2). Now substitute according to the table to obtain m = IA/c. It should be noticed that all substitutions are done simultaneously, all possible substitutions are made, the table is used only once, and the variable c is introduced only because it is required by the table.

I took it for granted that, as in the forward direction of the table, one was supposed to substitute for each individual variable, not to substitute for groups of variables as you have done. On that basis I arrived at an equation equivalent to m = IA/(1/(ε0μ0)1/2). At this point I could not justify replacing c with 1/(ε0μ0)1/2 according to the table since I had already used the table. It seems therefore that my mistake was my initial assumption.

The SI column of the old table consists of only mechanical quantities, as is also true of the Gaussian column. This is what makes it possible to convert from Gaussian to SI; one is converting mechanical quantities into mechanical quantities. Using this fact you try to show that the table can also be used to convert units, specifically when used in reverse from SI to Gaussian. Your exposition is not altogether clear, and so I will reformulate it as follows. Begin with an SI equation q = k asserting that a variable q is equal to a specific quantity of charge k, where k is expressed in SI units. Rewrite this equation as q/(4πε0)1/2 = k/(4πε0)1/2. Now let h = k/(4πε0)1/2 expressed in terms of the SI base units m, kg, s, A. The equation now becomes q/(4πε0)1/2 = h. Both sides of this equation are mechanical quantities, and so it is possible to cancel out the occurrences of the unit A in h and express it in terms of the base units m, kg, s. Let hmks be h so expressed. The equation becomes q/(4πε0)1/2 = hmks. Now convert this equation from SI to Gaussian using the old table. Since the right member is a mechanical constant, it does not convert, and the equation becomes q = hmks. Let hcgs be hmks expressed in terms of the cgs base units cm, g, s. We now have the purely Gaussian equation q = hcgs. Finally, observe that hcgs gives the correct value of q in Gaussian units. The old table can be used, therefore, to convert not only formulas, but units as well. You go on to say that the same thing cannot be accomplished using the new table, but this is incorrect, since the equation q/(4πε0)1/2 = hmks still converts into q = hmks, because ε0 = 1 in Gaussian.

I believe, therefore, that the old table has no physical or philosophical significance. The tables are merely formal syntactic algorithms for translating from one language to another. There is nothing to choose between them other than convenience, and for that purpose the new table is undoubtedly the superior one. It has no square roots, and most of the electric variables are not converted, making it especially useful to electrical engineers. 72.251.60.35 (talk) 17:50, 24 October 2017 (UTC)

To Sbyrnes321: If you look carefully at both the old and the new tables, you will see that each group of variables and constants in the SI column of the old table converts into the same single variable under both tables. Just remember that ε0 = μ0 = 1 in Gaussian units. Indeed, the same thing must be true of any correct table that does not convert mechanical quantities, since each group in the old table represents a mechanical quantity. Hence, far from being contradictory, the two tables give identical results. 72.251.60.35 (talk) 19:00, 24 October 2017 (UTC)

I still believe that we should have a table like this, but it has to be right, and it has to be understood how to use it. In any case, it is not ε0 = μ0 = 1 in Gaussian units but instead (4πε0)=1. Gah4 (talk) 20:21, 24 October 2017 (UTC)

For example, the old table has the rule that q/(4πε0)1/2 converts into q. The new table has the rules that q is unchanged and ε converts into ε/(4π). Hence, ε0 converts into ε0/(4π). Therefore, q/(4πε0)1/2 converts into q/(4πε0/(4π))1/2 = q/(ε01/2) = q, because ε0 = 1 in Gaussian units. Hence, the two tables give identical results.

In both SI and Gaussian units, ε is defined by the equation D = εE. Hence, ε0 is defined by the equation D = ε0E in a vacuum. In Gaussian units, D = E in a vacuum, and so ε0 = 1.

Coulomb's law in SI and Gaussian units is F = q1q2/(4πε0r2) and F = q1q2/r2, respectively. This makes it appear that 4πε0 = 1 in the latter, but this is incorrect. SI is a rationalized system, but Gaussian is an unrationalized system, in which Coulomb's law has the form F = q1q2/(ε0r2). Therefore, ε0 = 1 in Gaussian units. 72.251.58.208 (talk) 11:13, 26 October 2017 (UTC)

72.251: Thank you for your continuing patient discussion.

I could be wrong, but I think I'm saying something simpler than you realize. Here's the reverse table as I have always thought of it. This is the old table from the article with the columns switched (and the factors correspondingly inverted), plus two more entries near the top to avoid all confusion:

TABLE D (the old one in the text, directly reversed, plus some excessive detail for purposes of this discussion)

Name SI units Gaussian units
Speed of light ${\displaystyle c}$ ${\displaystyle c}$
Vacuum permittivity ${\displaystyle \epsilon _{0}}$ ${\displaystyle 8.9\times 10^{-12}{\text{F/m}}}$
Vacuum permeability ${\displaystyle \mu _{0}}$ ${\displaystyle 1.3\times 10^{-6}{\text{N/A}}^{2}}$
Electric field, Electric potential ${\displaystyle \left(\mathbf {E} ,\varphi \right)}$ ${\displaystyle {\frac {1}{\sqrt {4\pi \epsilon _{0}}}}\left(\mathbf {E} ,\varphi \right)}$
Electric displacement field ${\displaystyle \mathbf {D} }$ ${\displaystyle {\sqrt {\frac {\epsilon _{0}}{4\pi }}}\mathbf {D} }$
Charge, Charge density, Current,
Current density, Polarization density,
Electric dipole moment
${\displaystyle \left(q,\rho ,I,\mathbf {J} ,\mathbf {P} ,\mathbf {p} \right)}$ ${\displaystyle {\sqrt {4\pi \epsilon _{0}}}\left(q,\rho ,I,\mathbf {J} ,\mathbf {P} ,\mathbf {p} \right)}$
Magnetic B field, Magnetic flux,
Magnetic vector potential
${\displaystyle \left(\mathbf {B} ,\Phi _{\text{m}},\mathbf {A} \right)}$ ${\displaystyle {\sqrt {\frac {\mu _{0}}{4\pi }}}\left(\mathbf {B} ,\Phi _{\text{m}},\mathbf {A} \right)}$
Magnetic H field ${\displaystyle \mathbf {H} }$ ${\displaystyle {\frac {1}{\sqrt {4\pi \mu _{0}}}}\mathbf {H} }$
Magnetic moment, Magnetization ${\displaystyle \left(\mathbf {m} ,\mathbf {M} \right)}$ ${\displaystyle {\sqrt {\frac {4\pi }{\mu _{0}}}}\left(\mathbf {m} ,\mathbf {M} \right)}$
Relative Permittivity,
Relative Permeability
${\displaystyle \left(\epsilon _{r},\mu _{r}\right)}$ ${\displaystyle \left(\epsilon ,\mu \right)}$
Absolute Permittivity,
Absolute Permeability
${\displaystyle \left(\epsilon ,\mu \right)=\left(\epsilon _{r}\epsilon _{0},\mu _{r}\mu _{0}\right)}$ ${\displaystyle \left(\epsilon \epsilon _{0},\mu \mu _{0}\right)}$
Electric susceptibility,
Magnetic susceptibility
${\displaystyle \left(\chi _{\text{e}},\chi _{\text{m}}\right)}$ ${\displaystyle 4\pi \left(\chi _{\text{e}},\chi _{\text{m}}\right)}$
Conductivity, Conductance, Capacitance ${\displaystyle \left(\sigma ,S,C\right)}$ ${\displaystyle 4\pi \epsilon _{0}\left(\sigma ,S,C\right)}$
Resistivity, Resistance, Inductance ${\displaystyle \left(\rho ,R,L\right)}$ ${\displaystyle {\frac {1}{4\pi \epsilon _{0}}}\left(\rho ,R,L\right)}$

Then the rule is Take any SI formula, and without thinking, substitute any quantity in the left column with the corresponding entry in the right column. That's your Gaussian formula. But do not throw out or replace any of the ε0 & μ0s!! I put the 2nd and 3rd row of the table to drive home this point as clearly as possible. I could have put the numerical values in every other row too but I didn't want to be pedantic and distracting. Again, I am treating ε0 & μ0 as specific dimensionful parameters with universal meanings of 8.9e-12F/m and 1.3e-6N/A^2 respectively.

Now I want to go through the two examples

Example 1: m=IA

m=IA --> substitute using table --> the Gaussian formula is

(sqrt(4π/μ0)m = sqrt(4πε0)IA
simplify --> m = IA * sqrt(ε0μ0)

So this is the Gaussian formula. It doesn't immediately look like a Gaussian formula yet, because it still has those alien symbols ε0 and μ0. But remember, they mean nothing more or less than 8.9e-12F/m and 1.3e-6N/A^2. OK, so sqrt(ε0μ0) is some quantity I can calculate directly, and it just happens to be exactly 1/(299792458m/s). So the answer is m=IA/c. The mindless table substitution gave the right answer.

Example 2: ${\displaystyle q_{electron}=-1.6\times 10^{-19}{\text{C}}}$

This is just another formula and I can apply the exact same mindless table substitution as before. That gives:

${\displaystyle q_{electron}{\sqrt {4\pi \epsilon _{0}}}=-1.6\times 10^{-19}{\text{C}}}$
${\displaystyle q_{electron}=-1.6\times 10^{-19}{\text{C}}/{\sqrt {4\pi \epsilon _{0}}}=-4.8\times 10^{-10}{\text{g}}^{1/2}{\text{cm}}^{3/2}/{\text{s}}}$

It works! And this is the test that I'm saying the other tables do not pass. You can always get the right answer if you think carefully and discuss etc. as above. But I don't see how you can take Example 2, do a mindless table substitution, and get the gaussian-units charge of the electron using the new table. It only works with the old table / Table D (which are equivalent), I think.

Do you agree?

I am not arguing right now that the old table / Table D is the best choice all things considered (including pedagogy etc.), just that this is one advantage that it has. PS: I am finding this conversation very helpful and feel like we're making progress, thanks again for your time. :-D --Steve (talk) 02:32, 25 October 2017 (UTC)

I have found a subtle flaw in your table and have emended it as follows. The following note should be attached to it: "Either Final substitution A or Final substitution B (not both) must be done after all the other rules have been applied."

TABLE D2

Name SI units Gaussian units
Final substitution A ${\displaystyle \epsilon _{0}}$ ${\displaystyle {\frac {1}{\mu _{0}c^{2}}}}$
Final substitution B ${\displaystyle \mu _{0}}$ ${\displaystyle {\frac {1}{\epsilon _{0}c^{2}}}}$
Speed of light ${\displaystyle c}$ ${\displaystyle c}$
Electric field, Electric potential ${\displaystyle \left(\mathbf {E} ,\varphi \right)}$ ${\displaystyle {\frac {1}{\sqrt {4\pi \epsilon _{0}}}}\left(\mathbf {E} ,\varphi \right)}$
Electric displacement field ${\displaystyle \mathbf {D} }$ ${\displaystyle {\sqrt {\frac {\epsilon _{0}}{4\pi }}}\mathbf {D} }$
Charge, Charge density, Current,
Current density, Polarization density,
Electric dipole moment
${\displaystyle \left(q,\rho ,I,\mathbf {J} ,\mathbf {P} ,\mathbf {p} \right)}$ ${\displaystyle {\sqrt {4\pi \epsilon _{0}}}\left(q,\rho ,I,\mathbf {J} ,\mathbf {P} ,\mathbf {p} \right)}$
Magnetic B field, Magnetic flux,
Magnetic vector potential
${\displaystyle \left(\mathbf {B} ,\Phi _{\text{m}},\mathbf {A} \right)}$ ${\displaystyle {\sqrt {\frac {\mu _{0}}{4\pi }}}\left(\mathbf {B} ,\Phi _{\text{m}},\mathbf {A} \right)}$
Magnetic H field ${\displaystyle \mathbf {H} }$ ${\displaystyle {\frac {1}{\sqrt {4\pi \mu _{0}}}}\mathbf {H} }$
Magnetic moment, Magnetization ${\displaystyle \left(\mathbf {m} ,\mathbf {M} \right)}$ ${\displaystyle {\sqrt {\frac {4\pi }{\mu _{0}}}}\left(\mathbf {m} ,\mathbf {M} \right)}$
Relative permittivity,
Relative permeability
${\displaystyle \left(\epsilon _{r},\mu _{r}\right)}$ ${\displaystyle \left(\epsilon ,\mu \right)}$
Vacuum permittivity,
Vacuum permeability
${\displaystyle \left(\epsilon _{0},\mu _{0}\right)}$ ${\displaystyle \left(\epsilon _{0},\mu _{0}\right)}$
Absolute permittivity,
Absolute permeability
${\displaystyle \left(\epsilon ,\mu \right)}$ ${\displaystyle \left(\epsilon _{0}\epsilon ,\mu _{0}\mu \right)}$
Electric susceptibility,
Magnetic susceptibility
${\displaystyle \left(\chi _{\text{e}},\chi _{\text{m}}\right)}$ ${\displaystyle 4\pi \left(\chi _{\text{e}},\chi _{\text{m}}\right)}$
Conductivity, Conductance, Capacitance ${\displaystyle \left(\sigma ,S,C\right)}$ ${\displaystyle 4\pi \epsilon _{0}\left(\sigma ,S,C\right)}$
Resistivity, Resistance, Inductance ${\displaystyle \left(\rho ,R,L\right)}$ ${\displaystyle {\frac {1}{4\pi \epsilon _{0}}}\left(\rho ,R,L\right)}$

72.251.57.89 (talk) 02:53, 26 October 2017 (UTC)

One more thing not well described here. In Gaussian units, ${\displaystyle \epsilon }$ is the relative permittivity, also known as the dielectric constant. In SI, it is supposed to be absolute permittivity, that is, ${\displaystyle \epsilon _{0}}$ times the relative permittivity, but often enough, even with SI, it is used for relative permittivity. It is often much easier to work with relative permittivity. Gah4 (talk) 04:27, 26 October 2017 (UTC)
I agree with everything stated here and endorse those suggestions, if I understand them right. But I also have a few more ideas that I think will help.
• First, would it help to write all the conversion factors in terms of ε0 and c, and just leave μ0 out? For example, instead of writing ${\displaystyle {\sqrt {4\pi /\mu _{0}}}}$, we can write ${\displaystyle c{\sqrt {4\pi \epsilon _{0}}}}$. It should reduce the probability that there are ε0μ0 combinations hanging around at the end. But oh, actually it's not guaranteed to work, because there might be μ0s in the original formula. Never mind.
• Second, we can potentially write out an example in the text, drawing attention to anything tricky.
• Third, I suggest writing ${\displaystyle \epsilon _{0}^{\text{SI}},\mu _{0}^{\text{SI}}}$ when it's potentially ambiguous.
• Third, there is a second table that we haven't talked about here but it could actually be very helpful: The "electromagnetic unit names" is actually closely related to "general rules to translate a formula", but the article is a missed opportunity to spell out the relationship and thereby clarify both. Here's what I mean:

TABLE E

Common electromagnetism units in SI vs Gaussian
2.998 is shorthand for exactly 2.99792458 (see speed of light)
Quantity Symbol SI unit Gaussian unit Conversion factor
electric charge q C Fr = cm3/2g1/2s−1 ${\displaystyle {\frac {q_{\text{G}}}{q_{\text{SI}}}}={\frac {1}{\sqrt {4\pi \epsilon _{0}^{\text{SI}}}}}={\frac {2.998\times 10^{9}{\text{ Fr}}}{1{\text{ C}}}}}$
electric current I A Fr/s = cm3/2g1/2s−2 ${\displaystyle {\frac {I_{\text{G}}}{I_{\text{SI}}}}={\frac {1}{\sqrt {4\pi \epsilon _{0}^{\text{SI}}}}}={\frac {2.998\times 10^{9}{\text{ Fr/s}}}{1{\text{ A}}}}}$
electric potential
voltage
φ
V
V statV = cm1/2g1/2s−1 ${\displaystyle {\frac {V_{\text{G}}}{V_{\text{SI}}}}={\sqrt {4\pi \epsilon _{0}^{\text{SI}}}}={\frac {1{\text{ statV}}}{2.998\times 10^{2}{\text{ V}}}}}$
... ... ... ... ...
(I haven't bothered to fill out the rest of the table yet.)
(Conveniently, one can check this table for consistency using a computer algebra system. I plan to do that (and did do it just now for these three rows) and will post the source code.)
If I finish expanding Table E along these lines, I think that's already technically enough by itself to translate formulas. For example, if ${\displaystyle F={\frac {q_{\text{SI}}^{2}}{4\pi \epsilon _{0}^{\text{SI}}r^{2}}}}$, and Table E says that ${\displaystyle {\frac {q_{\text{G}}}{q_{\text{SI}}}}={\frac {1}{\sqrt {4\pi \epsilon _{0}^{\text{SI}}}}}}$, well then it follows immediately that ${\displaystyle F={\frac {q_{\text{G}}^{2}}{r^{2}}}}$. I'm not sure whether actually deleting the how-to-translate-a-formula table(s) is a good idea, maybe we should have both, but either way, I think the readers will have higher probability of understanding where the translation table comes from, how it works, and how to use it.
I'm happy to hear any feedback, and in the next few days I just want to try this ... we can always change it back. :-D --Steve (talk) 02:13, 28 October 2017 (UTC)

I suggest that you try converting the following SI equations to Gaussian using Table D2 to see how it works. You should simplify the equation before making the final substitutions. (1) m = IA, (2) D = εE, (3) D = ε0E, (4) εμv2 = 1, (5) ε0μ0c2 = 1.

It is very awkward to use the numerical values that appear in Table D; it is much easer to calculate using symbols. That is the purpose of the final substitution rules of Table D2. They are merely reformulations of the speed of light rule of the old table.

It is possible, as you suggest, to make the final substitutions directly in Table D2 itself, thereby obtaining two new tables. But it will be necessary to have both, since one or the other will be easier to use in a particular case. Which will be the easier one, however, will not always be obvious. On the other hand, if one makes the initial substitutions of Table D2 and simplifies the result, it will usually be clear which of the final substitution rules is easier to use. It is best, therefore, to leave Table D as is and give the user a choice between two rules rather than two tables.

Any ε0 or μ0 in the given SI equation is replaced by its Gaussian value one and disappears. Any such in the resulting equation is new and is introduced by a conversion rule.

Any ε0 or μ0 in the table or an equation is SI; it is unnecessary to use a superscript to emphasize this. 72.251.62.243 (talk) 02:53, 29 October 2017 (UTC)

I edited it here. I personally think it's better than before. I don't think it's perfect. I want to re-read your comments above, there is probably more good advice that I haven't done yet.
I see your point about superscript SI, but I feel like this very discussion indicates that there is a possibility of confusion, so I have to assume that some of the readers will be confused, and want to head off that confusion. It is a bit of an inconsistent mix right now. I'm not sure if it's making things better or worse. :-P
This is not a topic I've tried explaining to people in person before, so I don't know if the explanations are clear or helpful or pedagogical. Please do tell me if specific things are confusing (or wrong), and how. Thanks for your continued patience as always :-D --Steve (talk) 20:09, 29 October 2017 (UTC)

I suggest that you try converting the following Gaussian equations to SI using both the old table and Table 2 to see how they work and determine which one is easier to use. (1) cm = IA, (2) B = εH, (3) B = H, (4) εμv2 = c2. Table 2 does not give the natural translation of the equation w = (E2 + H2)/(8π), which is w = (ε0E2 + μ0H2)/2. 72.251.62.115 (talk) 07:15, 30 October 2017 (UTC)

The old table results in equations in which c is absent. The new table results in equations in which either ε0 or μ0 is absent, and in which c often appears in positions that are unnatural in SI. The old table is symmetrical; the new table is asymmetrical. The old table has generally one variable in each coefficient; the new sometimes has two. The old table requires one step; the new requires three steps. The old table is clearer, simpler, and easier to use than the new one. The final substitution rules were designed to be used in the SI to Gaussian Table D2 to get rid of the ε0's and μ0's and replace them with c's; they are unnecessary in a Gaussian to SI table, in which it is better to get rid of the c's. For these reasons I propose that the old table be restored and Table 1 be edited to be consistent with it. 72.251.60.211 (talk) 03:11, 1 November 2017 (UTC)

After thinking about it and trying a few examples as you suggest, I agree! I put the μ0's back in both tables. Thanks for that! :-D --Steve (talk) 16:45, 4 November 2017 (UTC)

What you have now is an attempt to combine the old table and Table D2 into a single table, instead of having separate tables for Gaussian-to-SI and SI-to-Gaussian conversions. It cannot be said to be wholly successful. A conversion table should give a univocal translation, but giving the user license to use the final substitution rules in a Gaussian to SI conversion will result in several possible translations that are not mathematically equivalent.

More serious is the failure of the table to address the "subtle flaw" that I alluded to before. When used in reverse from SI to Gaussian the table converts, e.g., ε to ε0ε, and, as a special case, it converts ε0 to ε0ε0. But one of the two ε0's here has its Gaussian value one, so that the result should be written as ε0. This is why I added the explicit "vacuum" rules to Table D2 and placed them immediately above the more general "absolute" rules.

For these reasons I propose that the old table be restored in its original form (without the final substitution rules and with the original speed of light rule) and Table D2 (unedited) be added to the text to complement it. 72.251.59.84 (talk) 02:40, 6 November 2017 (UTC)

Oh, sorry I didn't understand what you were saying earlier. But that makes sense! I just followed your suggestion, I hope I got it right this time!! --Steve (talk) 00:08, 7 November 2017 (UTC)

A note should be attached to Table 1 stating that the SI ε0 and μ0 satisfy the constraint ε0μ0 = 1/c2.

The name of the unit of M in the table in the section "Dimensionally equivalent units" should be changed to dyn/Mx, consistent with Table 1 and other articles. The name Mx/cm2 suggests an obsolete concept of magnetic dipole moment. The unit of M in SI is N/Wb not Wb/m2. 72.251.56.236 (talk) 22:59, 9 November 2017 (UTC)

If dyn = 1 g⋅cm/s² and Mx = 1 cm−5/2⋅g1/2⋅s−1 (from Maxwell (unit) top right box), then dyn/Mx does not have the same units cm−1/2 g1/2 s−1 as everything else in the "dimensionally equivalent units" table. Or am I making a mistake? Likewise, the Magnetization article says the SI unit of M is A/m, which is not the same as N/Wb. Where is the mistake? Thanks in advance, --Steve (talk) 01:33, 10 November 2017 (UTC)

I would prefer it if the note to Table 1 were displayed in regular-size type. The note is not merely informational. The equation in the note will be needed if the table is to be used for conversions.

The expression for Mx that you quote is a mistake. See Gauss (unit) or the table I mentioned. Magnetic moment is torque per unit B, and B is flux density. Hence, magnetic moment is force-volume per unit flux, and M is force per unit flux. Since Mx and Wb are units of flux, the rest follows. (See also Weber (unit).) 72.251.62.211 (talk) 06:11, 11 November 2017 (UTC)

If the expression Mx = 1 cm−5/2⋅g1/2⋅s−1 from Maxwell (unit) is wrong, then we better fix it! But I'm still not sure.
First of all, I have the following three claims which are all consistent with each other, therefore either they are all correct or at least two are incorrect. So which of these are you disagreeing with and how would you change them? (Sorry if you already answered this above.)
(A) Mx = 1 cm−5/2⋅g1/2⋅s−1 (from Maxwell (unit) box)
(B) G = 1 cm−1/2⋅g1/2⋅s−1 (from the "dimensionally equivalent units" section of this article, and also Gauss (unit))
(C) Mx = G/cm² (stated in both Gauss (unit) and Maxwell (unit))
Second of all, I agree that magnetic moment is torque per B. But B is an areal density of flux, and M is a volume density of magnetic moment, and also torque is dimensionally equivalent to erg not dyn. So I end up with Gaussian units of erg/Mx/cm for M, which is indeed equivalent to cm−1/2⋅g1/2⋅s−1 as expected IF the (A,B,C) statements above are all correct. Which part of that do you disagree with? Thanks, --Steve (talk) 17:53, 11 November 2017 (UTC)
(A) Mx = cm3/2⋅g1/2⋅s−1
(B) G = cm−1/2⋅g1/2⋅s−1
(C) Mx = G⋅cm²
(D) erg/Mx/cm = dyn/Mx
72.251.63.11 (talk) 05:31, 12 November 2017 (UTC)
Thank you, I agree on all counts. Please forgive me for making various sloppy mistakes in my previous post. I just edited both this article and Maxwell (unit). Thanks again for your diligence and patience! --Steve (talk) 23:57, 12 November 2017 (UTC)

## Table

A recent edit added a table, and another removed it. I believe that the table is useful, though it should also be consistent with the rest of the article. Instead of reverting the removal, I am asking here, for thoughts both ways. Gah4 (talk) 02:59, 23 October 2017 (UTC)

## Python code to check the correctness of the unit-relationship table

I used the code below as a way to double-check the unit-relationship table (called "Table 1" as of this writing):

# Using Python 3.6
from math import sqrt, isclose, pi as π
# https://pypi.python.org/pypi/numericalunits
from numericalunits import (reset_units, cm, g, s, C, A, ε0,
μ0, V, m, T, erg, Wb, Ω, F, H)
reset_units()
# Electric charge
Fr = cm**(3/2) * g**(1/2) * s**-1
assert isclose( 1/sqrt(4*π*ε0),
(2.99792458e9*Fr) / (1*C))
# Electric current
assert isclose( 1/sqrt(4*π*ε0),
(2.99792458e9*Fr/s) / (1*A))
# Electric potential / voltage
statV = cm**(1/2) * g**(1/2) * s**-1
assert isclose( sqrt(4*π*ε0),
(1*statV) / (2.99792458e2*V))
# Electric field
assert isclose( sqrt(4*π*ε0),
(1*statV/cm) / (2.99792458e4*V/m))
# Electric displacement field
assert isclose( sqrt(4*π/ε0),
(4*π*2.99792458e5*Fr/cm**2) / (1*C/m**2))
# Magnetic B field
# Note, numericalunits has a gauss (G), but it's defined for
# use in SI formulas, so we need to define it ourselves
G = cm**(-1/2) * g**(1/2) * s**-1
assert isclose( sqrt(4*π/μ0),
(1e4*G) / (1*T))
# Magnetic H field
Oe = G
assert isclose( sqrt(4*π*μ0),
(4*π * 1e-3 * Oe) / (1*A/m))
# Magnetic dipole moment
assert isclose( sqrt(μ0/(4*π)),
(1e3 * erg/G) / (1*A*m**2))
# Magnetic flux
assert isclose( sqrt(4*π/μ0),
(1e8*G*cm**2) / (1*Wb))
# Resistance
assert isclose( 4*π*ε0,
(1*s/cm) / (1e11*2.99792458**2*Ω))
# Resistivity
assert isclose( 4*π*ε0,
(1*s) / (1e9*2.99792458**2*Ω*m))
# Capacitance
assert isclose( 1/(4*π*ε0),
(1e11 * 2.99792458**2 *cm) / F)
# Inductance
assert isclose( 4*π*ε0,
(1 * s**2/cm) / (2.99792458**2 * 1e11 * H))


--Steve (talk) 19:58, 29 October 2017 (UTC)

## References

1. ^ Jackson, John David (1962). Classical Electrodynamics (PDF). New York: John Wiley & Sons. Appendix, Table 3, p. 619.
2. ^ Resnick, Robert; Halliday, David (1966). Physics, Part I. New York: John Wiley & Sons. Appendix L.

## state ε0 = μ0 = 1 in beginning

Hi, physics student here. Gaussian units basically have two aspects: (1) CGS is used, and (2) ε0 = μ0 = 1. It would be more clear if this was stated in the very beginning (like first or second paragraph at the top). The first two paragraphs right now are very ambiguous, stating that Gaussian units are a type of CGS system.... That's not so informative. Or what do yall think? -- Blue.painting (talk) 23:00, 4 March 2019 (UTC)

I find that when you tell someone "ε0 = μ0 = 1", they learn "OK great, I can take an SI electromagnetism formula and plug in ε0 = μ0 = 1, and then I'll wind up with a Gaussian formula", which of course is not correct. I would say instead "Gaussian units does not contain anything analogous to ε0 or μ0 in SI". Of course I could make the same statement of all the other CGS electromagnetism units (ESU, EMU, Lorentz–Heaviside).
So can you clarify what you're talking about when you say ε0 = μ0 = 1?
Can you think of a simpler way to describe Gaussian units in the first paragraph or two that does not (ambiguously) also describe ESU & EMU & Lorentz-Heaviside units equally well? --Steve (talk) 01:22, 5 March 2019 (UTC)
Above I note that ε0=1/(4π). I haven't checked this recently, though, but that sounds like not 1. I think it is Heaviside-Lorentz units where it is 1. Gah4 (talk) 05:34, 5 March 2019 (UTC)
Hi Steve and Gah4, thank you for the responses, and sorry for getting back late. I thought about what yall said, and have been working more on making sure I understand Gaussian vs Lorentz-Heaviside vs SI better, before proceeding further with this. I'll think about this more and then get back to yall again... Thanks. -- Blue.painting (talk) 23:24, 12 March 2019 (UTC)
You might find it interesting to look at this book. (Or, I believe, the second edition, but not the third edition.) This book teaches E&M as a relativistic effect. (Consider the force on a charge moving near a wire, parallel to the wire at the same speed as the moving charges in the wire.) This shows the connection between E and B in a way that you don't see in most books. And the older editions are available for even more reasonable prices than the still plenty reasonable price for the new ones. I suppose you could do even better and also get the new one, and compare them. In any case, the derivation of the units is more obvious when you see it this way. Gah4 (talk) 00:14, 13 March 2019 (UTC)
Thanks Gah4, I'll take a look for sure. -- Blue.painting (talk) 01:30, 16 March 2019 (UTC)