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Values of parameters in different systems of units

The table of parameters in the section "Various extensions of the CGS system to electromagnetism" has incorrect entries in the columns for ε0, μ0, λ, and λ′ that contradict the text of the article and other articles. According to the text ESU and Gaussian are not rationalized, and so, according to the text, λ = λ′ = 4π in these systems. Whence it follows from the formula for λ in the table that ε0 = 1 in these systems.

The table assumes that ε0μ0 = 1/c2 in all systems, but this is incorrect, the correct formula being ε0μ0 = 1/αL2/c2. Hence, μ0 = 1/c2 in ESU, and μ0 = 1 in Gaussian and Lorentz–Heaviside.

The same conclusions can be reached in another way. According to the text, D = ε0E and B = μ0H in free space in all systems. But according to the articles Gaussian units and Lorentz–Heaviside units D = E and B = H in free space in Gaussian and Lorentz–Heaviside. Therefore, ε0 = μ0 = 1 in these systems. The formulas for D and B in these articles also plainly show that λ and λ′ are 4π in Gaussian and 1 in Lorentz–Heaviside.

Accordingly, the quadruple (ε0, μ0, λ, λ′) should be changed to (1, 1/c2, 4π, 4π) for ESU, to (1, 1, 4π, 4π) for Gaussian, and to (1, 1, 1, 1) for Lorentz–Heaviside. 72.251.58.64 (talk) 02:05, 14 November 2017 (UTC)[reply]

This is quite a long series of posts, so I'll start by noting that the values for ε0 and μ0 are in complete agreement with Table 2 of the Appendix on Units and Dimensions in the Jackson reference, while (with the conversions given below the table) the various k constants agree with Table 1. In Wikipedia, the references are paramount. Contrary to the statement by the IP editor, the column for λ does agree with those for kC and ε0 through the formula λ = 4 π kCε0. RockMagnetist(talk) 16:42, 15 December 2017 (UTC)[reply]
I have belatedly realized that the table is correct because the IP editor changed it. Thank you! RockMagnetist(talk) 17:44, 15 December 2017 (UTC)[reply]

The following pertains to the section "Alternate derivations of CGS units in electromagnetism."

Theorem. λ = 4πε0kC and λ′ = 4παB/(μ0αL).

Corollary. If λ = λ′, then c2 = 1/(ε0μ0αL2).

Proof. Begin with the equations in SI. Let λ, λ′, σ, τ be independent variables. Define β = λστ and β′ = λ′στ. Perform the following multiplications:

B by σ/β′ and D by τ/β
E by σ and H by τ
M by β′/σ and Q, ρ, I, J, P by λτ/β
μ by λ′σ2/β′2 and ε by λτ22

Notice that the cross products P × E and M × B are invariant under this transformation since they represent torque densities. After eliminating σ and τ from the resulting equations, we obtain the most general system subject to the usual constraints. It may be seen that λ and λ′ have the meanings given them in the text. That the formulas in the theorem are valid in the general system can be verified directly. For λ this is obvious by inspection; for λ′ use the fact that λ/β = λ′/β′.

The corollary follows from the theorem and the formulas kC/kA = c2 and kA = αLαB given in the text. 72.251.58.81 (talk) 02:28, 12 December 2017 (UTC)[reply]

The general system has six parameters, λ, λ′, β, β′, ε0, μ0, subject to two constraints, λ/β = λ′/β′ and c2 = ββ′/(ε0μ0). It therefore has four degrees of freedom in the choice of units. It may be thought of as having seven base units, including the three mechanical units. The units of P and M, however, are directly derived from those of E and B, respectively.

In terms of these parameters the constants defined in the text have the following values:

αL = 1/β′
kC = λ/(4πε0)
αB = λμ0/(4πβ)
kA = λμ0/(4πββ′)

72.251.58.233 (talk) 04:22, 13 December 2017 (UTC)[reply]

The text states that 4πε0kC is a dimensionless quantity, but since ε0 is arbitrary, including its unit, this is not necessarily so. In general both ε0 and μ0 may be selected at will, but if it is required that λ = λ′, then the only limitation is that indicated in the corollary. It would be very convenient to assign a unit to λ and λ′ (if they are equal) in order to facilitate conversion from one system to another. The unit that seems most appropriate is that of a solid angle. The difference between unrationalized and rationalized systems would be that the former use the steradian as the unit of solid angle whereas the latter use the sphere. 72.251.62.29 (talk) 03:02, 14 December 2017 (UTC)[reply]

The proof of the theorem refers to the "usual constraints." These are conditions and equations that are invariant under the transformation in the proof. They include the equation of continuity, the formulas D = εE and B = μH, and the definitions of electric and magnetic moments as torques per units of E and B, respectively. There are, however, systems that violate the constraints. The most notorious offenders are variants of the Gaussian system. One such measures charge in ESU and current in EMU; this system violates the equation of continuity. The standard Gaussian system was carefully constructed to satisfy the constraints. For example, the magnetic moment of a small current loop is defined as m = IA/c. The c is inserted here to ensure that M is measured in EMU, as are B and H. If it is omitted, then M is measured in ESU and λ′ = 4π/c. If electric and magnetic dipole moments are defined by p = 4πQd and m = 4πIA/c, respectively, then λ = λ′ = 1. This in no way, however, makes the system rationalized. The way to rationalize the standard Gaussian system (without changing the units of E, B, P, M) is to choose ε0 = 1/(4π) and μ0 = 4π.

Rationalization, as the word is ordinarily understood, requires that ε0 and μ0 be so chosen that kC = 1/(4πε0) and αB = μ0αL/(4π). 72.251.59.120 (talk) 03:39, 15 December 2017 (UTC)[reply]

Can anyone summarize the issue? Also, it would help if you make a wiki user. MaoGo (talk) 13:32, 15 December 2017 (UTC)[reply]

The IP editor was correct about the table, and I would encourage them to add the other material to the article. However, they should be aware of the core Wikipedia policy of verifiability and provide citations of a reliable source for anything they add. RockMagnetist(talk) 17:55, 15 December 2017 (UTC)[reply]
This all seems reasonable and I would encourage 72.251 (who was also very helpful at Talk:Gaussian units last month) to feel welcome and encouraged to edit the text yourself.
My only gripe about the table is that I wish the ε0 and μ0 columns were named something else, ideally meaningless symbols with no prior associations, like "k_2" and "k_3" for example. For example, the statement "ε0=1 in Gaussian units" is I think prone to being misunderstood ... for example I worry that a reader will see that and then feel entitled to replace ε0 with 1 in translating random formulas (like Coulomb's law) from SI to Gaussian. That statement is really supposed to be "ε0=1 in Gaussian units, where ε0 is by definition the ratio D/E in free space". That statement is correct, but it would still be equally correct if we used a different symbol besides ε0. Just my opinion, and I don't think it's a huge deal. --Steve (talk) 20:24, 15 December 2017 (UTC)[reply]

I am the IP editor. There are three problems with the text as currently written. (1) It implies that λ and λ′ may be chosen independently of ε0 and μ0; I hold that this is only possible if P and M have unusual units. (2) It states that 4πε0kC is a dimensionless quantity; I hold that this is not necessarily so. (3) It states that rationalization depends upon the values of λ and λ′; I hold that, if the formulas of the theorem do not hold, then it depends, rather, on the values of ε0 and μ0.

I cannot cite any sources, since I do not have access to any books that discuss these issues in sufficient detail. But the statements of the text ought themselves to be verifiable if they are to stand. The only reference in the text that seems to be relevant is to Jackson, whose discussion of the subject is wholly inadequate. The text seems to draw unwarranted inferences from what he does say (or doesn't say). (1) Jackson says nothing about the relationships between λ and λ′, on the one hand, and ε0 and μ0, on the other; the text infers that there are no necessary relationships. (2) Jackson says, "λ and λ′ are chosen as pure numbers"; the text infers that they must be so chosen. (3) Jackson says, "λ = λ′ = 1 in rationalized systems"; the text infers that this is the definition of "rationalization," and it calls λ and λ′ "rationalization constants."

I propose that the text "The factors … be 'rationalized'" be replaced with the following:

The units of P and M are usually so chosen that the factors λ and λ′ are equal to the "rationalization constants" and , respectively. If the rationalization constants are equal, then . If they are equal to one, then the system is said to be "rationalized"

Zophar (talk) 04:44, 17 December 2017 (UTC)[reply]

Here is a direct proof that rationalization means that what I call the rationalization constants are equal to one. Rationalization means that Maxwell's equations in material media for static fields have the form

∇·D = ρ and ∇×E = 0
∇·B = 0 and ∇×H = αLJ

In free space these become

∇·E = ρ/ε0 and ∇×E = 0
∇·B = 0 and ∇×B = μ0αLJ

From these we may derive Coulomb's law and the Biot-Savart law using the usual mathematical arguments. The results are

kC = 1/(4πε0) and αB = μ0αL/(4π)

whence it follows that the rationalization constants are equal to one. Zophar (talk) 04:58, 18 December 2017 (UTC)[reply]

emfd

Gavo atoms (talk) 07:08, 28 February 2020 (UTC)can you help me to have question and answers of EMFD unit[reply]

As others have noted, the talk page is for discussing the article, or improvements to the article. If there is an EMFD unit that should be included, then we can discuss that. (As far as I know, there is no such unit, at least in the CGS contexts.) I try to be a little flexible, and give people the benefit of the doubt, that the question might have some use for improving the article. But deleting this posts doesn't even allow discussion of the relevance to the article. (Though I agree, that I suspect that there isn't any.) Gah4 (talk) 01:44, 29 February 2020 (UTC)[reply]
What is an EMFD unit? Dondervogel 2 (talk) 09:29, 29 February 2020 (UTC)[reply]