Constans electrica

Constans electrica
Dimensiones	L⁻³ M⁻¹ T² Q²
Magnitudo	8.8541878188(14)×10⁻¹² C² kg⁻¹ m⁻³ s²

Constans electrica vel permittivitas vacui vel constans dielectrica vacui ( $ε 0$ notata) est constans physica quae, dimensiones oneris electrici, longitudinis et roboris inter eas nectendo, mētītur quam densus campus electricus, propter quoddam onus, oriri possit (idcirco magnitudinem permittivitatis electricae^(en)(d) vacui statuit).

Constans electrica sic definiri potest,

\varepsilon _{0}={\frac {1}{4\pi k_{e}}}={\frac {1}{\mu _{0}c^{2}}}={\frac {e^{2}}{2\alpha hc}}={\frac {e^{2}}{4\pi \alpha \hbar c}}

ubi

$k e$ est constans Coulombiana
$μ 0$ est constans magnetica^(en)(d)
$c$ est celeritas lucis in vacuo
$h$ et $ħ$ sunt constans Planckiana et constans Planckiana minuta
$α$ est constans structurae subtilis
$e$ est onus electricum elementare (i.e. onus protonis vel electronis)

Secundum unitates Systematis internationalis unitatum, constanti electricae congruunt 8.8541878188(14)×10⁻¹² C² kg⁻¹ m⁻³ s².^[1]

Exempli gratia, vis inter duo onera electrica punctiformia, secundum legem Coulombianam^(en)(d), scribi potest,

F_{\text{C}}={\frac {1}{4\pi \varepsilon _{0}}}{\frac {q_{1}q_{2}}{r^{2}}}

ubi $q 1$ et $q 2$ sunt onera electrica, $r$ est eorum mutua separatio, et $.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num,.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0 0.1em}.mw-parser-output .sfrac .den{border-top:1px solid}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}1/4𝜋ε0$ est alius modus constantem Coulombianam scribendi.

Eadem constans in aequationibus Maxwellianis quoque, quae faciem campi electrici magneticique et radiationis electromagneticae describunt, incurritur.

In arte electrica, constans $ε 0$ saepe adhibetur unitas ad permittivitatem variarum dielectricarum materierum^(en)(d) mētiendam.

Historia

Carolus de Coulomb primum intellexit vim inter duo onera electrica punctiformia ( $q 1$ et $q 2$ ), in spatio vacuo posita atque quoddam intervallum $r$ inter se distantia, hanc formam habere:

F=k_{\text{e}}{\frac {q_{1}q_{2}}{r^{2}}}

Postea, intellectum est in condicionibus geometriae sphaericae oportere factorem $4 𝜋$ in computatione includere, dehinc quandam constantem $k e'$ in locum constantis Coulombianae ( $k e$ ) substituere, et eidem aequationi sequentem formam dare:

F=k'_{\text{e}}{\frac {q_{1}q_{2}}{4\pi r^{2}}}

Haec ratio computandi appellatur “rationalizatio”, quia si novae constanti $k e'$ eadem magnitudo constantis $k e$ assignatur, onera electrica cessant esse numeri irrationales.

In locum constantis $k e'$ eius inversio $ε 0 = 1 / k e' = 1 / 4 𝜋 k e$ postremo substituta est constans primaria suo iure, quae formam hodiernam legis Coulombianae gignit:

F={\frac {q_{1}q_{2}}{4\pi r^{2}\varepsilon _{0}}}

Sic anno 1893 Oliverius Heaviside^(en)(d) in suo libro Theoria electromagnetica talem methodum descripsit:^[2]

It is not long since it was taken for granted that the common electrical units were correct. That curious and obtrusive constant $4 𝜋$ was considered by some to be a sort of blessed dispensation, without which all electrical theory would fall to pieces. I believe that this view is now nearly extinct, and that it is well recognised that the $4 𝜋$ was an unfortunate and mischievous mistake, the source of many evils.
In plain English, the common system of electrical units involves an irrationality of the same kind as would be brought into the metric system of weights and measures, were we to define the unit area to be the area, not of a square with unit side, but of a circle of unit diameter. The constant $𝜋$ would then obtrude itself into the area of a rectangle, and everywhere it should not be, and be a source of great confusion and inconvenience. So it is in the common electrical units, which are truly irrational.
Now, to make a mistake is easy and natural to man. But that is not enough. The next thing is to correct it: When a mistake has once been started, it is not necessary to go on repeating it for ever and ever with cumulative inconvenience.

Non pridem pro concesso sumebatur unitates oneris electrici esse rectas. Illa curiosa et molesta constans $4 𝜋$ a multis putabatur quasi quaedam benedicta dispensatio, sine qua quaecumque theoria electrica dilapsura erat. Credo hodie hanc opinionem esse paene exstinctam, et constantem $4 𝜋$ bene agnosci esse infelix perniciosumque mendum ac fontem multorum malorum.
Id est, commune systema unitatum electricarum quandam irrationalitatem servat eodem genere quod introduceretur si in systemate metrico ponderum et mensurarum unitatem areae definiremus haud quadrum latere unius unitatis, sed circulum diametro unius unitatis. Ideo numerus $𝜋$ se obtruderet in aream rectanguli et ubicumque esse non oporteat, atque fieret fons magnae confusionis et incommodi. Sic evenit in communibus unitatibus electricis, quae vere sunt irrationales.
Facile autem est, et naturale, homini errare. Sed postea error corrigendus est: Cum quidem semel occeptus sit error, non necesse est semper et semper, crescente incommodo, eum iterare.

―Oliverius Heaviside (1893). Theoria electromagnetica. Tom. 1.^[2]

Notae

↑ "2022 CODATA Value: vacuum electric permittivity". The NIST Reference on Constants, Units, and Uncertainty. NIST. Maio 2024
↑ ^2.0 ^2.1 Heaviside, Oliverius (1893). Theoria electromagnetica. 1. Londinii: The D. van Nostrand Company. p. xi . Alius fons eiusdem textūs: Heaviside 1893: Electromagnetic Theory volume 1 .... . Open Source Ecology Germany (wiki.opensourceecology.de)

Nexus interni

[1] "2022 CODATA Value: vacuum electric permittivity". The NIST Reference on Constants, Units, and Uncertainty. NIST. Maio 2024

[heaviside-2] 2.0 ^2.1 Heaviside, Oliverius (1893). Theoria electromagnetica. 1. Londinii: The D. van Nostrand Company. p. xi . Alius fons eiusdem textūs: Heaviside 1893: Electromagnetic Theory volume 1 .... . Open Source Ecology Germany (wiki.opensourceecology.de)

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