Electromagnetic unit

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Because I cannot express the physical quantity to be related to electromagnetism only in length, mass, time when I am going to incorporate an electromagnetic unit (でんじきのたんい) in a unit system, it seems to be necessary to add physics quantity according to the other to a unit system. In fact, in international unit system (SI) becoming the international standard of the unit system, I am in that way now. However, there are really other methods.

Equation system of the electromagnetism

The equation of the electromagnetism is written in the different form variously by a unit system. Therefore I show a quantity equation to be established without depending on a unit system in the form including the coefficient and show it how it changes in each unit system.

At first the Lorentz force to give electromagnetic power

${\boldsymbol {f}}=q({\boldsymbol {E}}+\gamma ^{-1}{\boldsymbol {v}}\times {\boldsymbol {B}})$

となる.

Next the Maxwell equations

$\nabla \cdot {\boldsymbol {D}}=\alpha \rho$

$\gamma \nabla \times {\boldsymbol {H}}-{\frac {\partial {\boldsymbol {D}}}{\partial t}}=\alpha {\boldsymbol {j}}$

$\nabla \cdot {\boldsymbol {B}}=\alpha \rho _{\text{m}}$

$\gamma \nabla \times {\boldsymbol {E}}+{\frac {\partial {\boldsymbol {B}}}{\partial t}}=\alpha {\boldsymbol {j}}_{\text{m}}$

となる. I write it in form including 磁荷密度 $ρ m$ to discuss it later about 磁荷.

Finally the constitution equation to connect $E$ , $H$ and $B$ with $D$

${\boldsymbol {D}}=\epsilon _{0}{\boldsymbol {E}}+\alpha {\boldsymbol {P}}$

${\boldsymbol {H}}={\boldsymbol {B}}/\mu _{0}-\alpha {\boldsymbol {M}}$

となる.

Equations consecutive from Maxwell equations

$\nabla \cdot {\boldsymbol {j}}+{\frac {\partial \rho }{\partial t}}=0$

But, I am led. In addition, the speed of light in the vacuum

$c^{2}={\frac {\gamma ^{2}}{\mu _{0}\epsilon _{0}}}$

It is expressed と. It is Coulomb's law in an electrostatic field, the static magnetic field

${\boldsymbol {E}}={\frac {\alpha }{4\pi \epsilon _{0}}}{\frac {q{\boldsymbol {r}}}{r^{3}}}$

${\boldsymbol {H}}={\frac {\alpha }{4\pi \mu _{0}}}{\frac {q_{\text{m}}{\boldsymbol {r}}}{r^{3}}}$

But, I am led. It is a law of ビオ Savart for the stationary electric current

${\boldsymbol {B}}={\frac {\alpha \mu _{0}}{4\pi \gamma }}\int {\frac {I\,d{\boldsymbol {l}}\times {\boldsymbol {r}}}{r^{3}}}$

But, I am led.

Classification of the unit system

The Maxwell equations include two coefficient $α, γ$ . With an existence physics and chemistry coefficient, a unit system is $α = 1$ at the time of Yuri system, and coefficient $α$ is $α = 4 π$ in the me on physical science. Coefficient $γ$ is the symmetrization fixed number, a coefficient called the connection factor, and an electrical place and a magnetic place become the same dimension if I choose you with $γ = c$ . ε ₀ included in the constitution equation $, μ 0$ are called the electric fixed number, the magnetic fixed number each. These are connected with speed of light and are not independent. As for the unit system of the electromagnetism, pro-4 yuan by giving these one a new unit.

Unit system	Dynamics unit	Existence physics and chemistry	Symmetrization	Dimension	$α$	$γ$	$ε 0$	$μ 0$	Base unit
CGS electromagnetic unit system (CGS-emu)	CGS unit system	Non-Yuri	Asymmetry	3	$4 π$	1	$1/ c 2$	1	cm, g, s
CGS electrostatic unit system (CGS-esu)		Non-Yuri	Asymmetry	3	$4 π$	1	1	$1/ c 2$	cm, g, s
CGS gauss unit system		Non-Yuri	Symmetry	3	$4 π$	$c$	1	1	cm, g, s
Heavy side unit system		Yuri	Symmetry	3	1	$c$	1	1	cm, g, s
Generalization electromagnetic unit system		Non-Yuri	Asymmetry	4	$4 π$	1	$1/ μ 0 c 2$	1 dyn/Bi²	cm, g, s, Bi
Generalization electrostatic unit system		Non-Yuri	Asymmetry	4	$4 π$	1	1 Fr²/erg·m	$1/ ε 0 c 2$	cm, g, s, Fr
MKSA system of units	MKS system	Yuri	Asymmetry	4	1	1	$1/ μ 0 c 2$	4 π *10^-7 H/m	m, kg, s, A
MKSC unit system									m, kg, s, C
MKSΩ unit system									m, kg, s,Ω
MKSP unit system		Yuri	Symmetry	3	1	$c$	1	1	m, kg, s
Practical use unit system		Non-Yuri	Asymmetry	3	$4 π$	1	$1/ c 2$	1	10⁹cm, 10^-11 g, s

In addition, how to put these coefficients is not necessity and can push forward a story equally even if how to put is different. I imitated Yoichi Okabe here (cf. outside link).

Dynamics unit

It is the difference that you assume a base unit of the dynamic quantity MKS system, or you assume CGS unit system. The form of the expression does not change without influencing only the size of the unit.

Existence physics and chemistry

Existence physics and chemistry coefficient $α$ takes (rationalized system) $α = 1$ of Yuri line and takes $α = 4 π$ in the me on physical science (non-rationalized system). It is quantity of no dimension and influences a dimension of the electromagnetic quantity neither.

Because Coulomb's law and a law of ビオ Savart were known earlier than Maxwell equation historically, it was the me on physical science that these laws coefficient $α /4π$ disappeared in the early unit system. Existence physics and chemistry (rationalizarion) to remove coefficient $4 π$ which appeared in the Maxwell equation was proposed by the Maxwell equation that was a basic expression of relations having been established by the next. It is an origin called "existence physics and chemistry" to delete $4 π$ which is an irrational number. $4 π$ does not completely disappear in the Yuri system, and $4 π$ appears to the coefficient of the law of Coulomb's law and ビオ Savart who did not appear in the me on physical science.

I think that quantity of electricity to have of the point charge is equal to all electric flux to start in the neighborhood in the Yuri system physically and think that it is equal to electric flux to start in 1 sr in the me on physical science.

Symmetrization

The form of the expression is symmetric, but, as for the law of the electromagnetism, a dimension may not accord with electrical quantity and magnetic quantity about electricity and magnetism. A dimension of electrical quantity and the magnetic quantity agrees to give a dimension of the speed in symmetrization coefficient $γ$ . The fixed number with a dimension of the speed in the electromagnetism is communication speed of the electromagnetic wave in the vacuum namely speed of light $c$ . I assume it $γ = c$ in the symmetric unit system to synchronize a dimension of electrical quantity and the magnetic quantity.

Because it is attached correspondence, in the special theory of relativity, electrical quantity and magnetic quantity improve a theoretical prospect. (but $c =1$ and a unit system to do are often chosen when I treat the special theory of relativity.)

Dimension

A base unit is the difference of three or four. I bring about an electromagnetic unit without adding a new base unit to a unit system of the dynamics in the case of 3 yuan system.

For example, in the CGS electrostatic unit system, ε ₀ is removed by Coulomb's law by putting it with ε ₀ = 1 (no dimension), and it is in F = Qq/r² (I substituted α = 4 π), and unit esu = dyn^1/2cm of the electric charge is arrived at if I substitute F = 1 dyn, r = 1cm, Q = q = 1 esu for this. This is a derived unit not a base unit as I am identified as dyn from cm because it is assembled. There are various unit systems which fixed number you put in what kind of value, and not only the size of the unit but also the dimension is different.

It is, and the quantity that α, γ, ε ₀, all of μ ₀ can calculate from the mathematics fixed number such as 4 π and other physics fixed number such as 1/c² in electromagnetic unit system of CGS origin, electrostatic unit system, the gauss unit system can remove it from an equation. These are 3 yuan system.

μ ₀ = 4 π *10^-7 H/m is the independent physical fixed number and, for them, cannot remove the MKSA system of units. Therefore the electromagnetic unit has the dimension that is not assembled from a dynamics unit. Because the extra physical fixed number is one, flexibility is one, and those dimensions should add one base unit. I still less put it in a number because I can remove ε ₀ with ε ₀ = 1/μ₀c².

The 3 yuan system is convenient for the theoretical handling and is liked for theory science and a numerical value experiment. However, because flexibility is low, the size of the unit is easy to become the non-daily size and is inconvenient for experimental science and engineering. A unit of all electromagnetism is fixed uniformly if I decide α, γ, ε ₀, μ ₀ because three base units have been already decided as a dynamics unit system by the particularly electromagnetic unit.

In contrast, because there is much flexibility in the 4 yuan system, I can adjust the size of the unit and can bring it close to daily size. For example, 10^-2 = 1/100 should double a value of μ ₀ to double the unit of electric current 10 because the dimension of μ ₀ is L2 M T^-2 I^-2, and the index of current (I) is -2. It is such a result that it coordinated that μ ₀ of the MKSA system of units is a mysterious value.

Each unit system

I adopt an electric current as an electromagnetism-related base unit in international unit system (SI) and define it by power to act between the objects which the electric current flows in. The electric charge becomes the derived unit defined as an electric current as the product of the time in international unit system (SI). There were various changes before it became such a form.

When a study on electromagnetism was begun, and the unit had begun to be made, the unit system used widely was CGS unit system. The early electromagnetic unit was built on CGS unit system.

Main unit system

CGS electromagnetic unit system

CGS electromagnetic unit system (CGS-emu) is the me on physical science that is asymmetry of 3 yuan. I was made with a built electromagnetic unit system first by Weber.

The Weber did the size of the electric current to give power of 2 dyn per 1cm to at distance 1cm with the unit of the electric current so that a law of ビオ Savart included no coefficient. Equal to putting this with αμ0/4π=μ0 = 1 (because is non-Yuri form using α = 4 π).

μ ₀ has a dimension of power / electric current ², but, as for the dimension of the electric current, it is in power ^1/2 by assuming this quantity of no dimension, and the unit of the electric current is assembled with dyn^1/2. I call this an electromagnetic unit (electromagnetic unit) and abbreviate it to emu.

Furthermore, I put it with γ = 1 and led the unit of the electric charge to lose a current consecutive laws coefficient. I applied this to Coulomb's law, and power to work between objects with a unit electric charge put every 1cm did it with αμ0/4π=μ0 = 1/c² so that it was in 1 dyn.

CGS electrostatic unit system

CGS electrostatic unit system (CGS-esu) is the me on physical science that is asymmetry of 3 yuan. I reversed CGS electromagnetic unit system about electricity and magnetism theoretically. Maxwell suggested it.

That is and Maxwell did it with αε0/4π=ε0 = 1 so that Coulomb's law included no coefficient I did the size of the electric charge that I gave power of 1 dyn to with the unit of the electric charge. The unit of electric charge is assembled with dyn^1/2cm and calls this an electrostatic unit (electrostatic unit) and abbreviates it to esu.

I led the unit of electric current for γ = 1 with a current consecutive laws coefficient like CGS electromagnetic unit system. I applied this to a law of ビオ Savart, and power to work between the unit electric currents which flowed every 1cm did it with αε0/4π=ε0 = 1/c² so that it was in 1 dyn.

CGS gauss unit system

CGS gauss unit system (gauss unit system) is symmetric me on physical science of 3 yuan. I use electrostatic unit system for an electromagnetic unit system, a unit about the electricity to a unit about the magnetism. Hz proposed it with Helmholtz.

As a result, it becomes ε ₀ = μ ₀ = 1. In addition, $\gamma =c{\sqrt {\varepsilon _{0}\mu _{0}}}$ It becomes により, γ = c.

Because the equation of an electric field and the magnetic field is symmetrized, and there is the good point that a theoretical prospect has good, this unit system may be used in theoretical physics or astronomy even now.

Heavy side Lorenz unit system

The heavy side Lorenz unit system (heavy side unit system) is symmetric Yuri system of 3 yuan. A heavy side proposed it in 1883 and, in the CGS unit system that Hendrick Lorenz reorganized, made a gauss unit system Yuri

The heavy side assumed that there was a previous unit system as α = 4 π tacitly α = 1 and prevented 4 π from appearing in the Maxwell equation by putting 4 π in the denominator of the expression of relations to express electric magnetic charge and relations with the dynamics quantity and called this with existence physics and chemistry. The Maxwell equations came to be described in a simple form by existence physics and chemistry, but in the case of the conversion with the unit system conventional as the compensation ${\sqrt {4\pi}}$ But, I appeared in large quantities. For an experiment scientist and the engineer whom the conversion of the unit was necessary for frequently, it was not a practical unit system. However, I may be rarely used because the size of the unit is not important to a theorist.

Practical use unit system

Practical use unit system (practical units) or BA unit system (British Association units) is the size that 10 冪倍 does an electromagnetic unit while assuming an electromagnetic unit system the cause, and is practical and the unit system that I did. How to put fixed number is the same as an electromagnetic unit system.

Ampere (A), most of electromagnetic units that are used now including bolt (V) are practical units formerly (with the history older partly).

The practical use unit system does not have a credit in dynamics with only an electromagnetic unit, but will do 10⁹cm, 10^-11 g, a second with a base unit if I count it backward from a theory (it was the calculation top, and such a unit was not used to the last).

MKSA system of units

MKSA system of units is the Yuri system who is asymmetry of 4 yuan. In addition, unlike a past unit system, I expanded the MKS system.

Because the base unit of the previous CGS unit system was too small, by industrial development, a shift to MKS system came to be carried out as a more practical unit system. To this, the need that the electromagnetic unit shifted to a thing on the basis of the MKS system came out. In the electrical engineering, a practical unit spread out.

Giovanni ジョルジ proposed 4 yuan system to assume the practical unit ampere of the electric current another base unit. By this, epsilon ₀ that was not conscious of for the physical fixed number and μ ₀ came to have meaningful quantity for the physical fixed number until now.

Furthermore, I changed the dynamics unit to the MKS system at the same time and adopted existence physics and chemistry. By these three changes, it is ε ₀ = 10⁷/4π(c·s/m⁾2 F/m (c·It was in thing) which s/m fell below velocity of light in unit m/s, and did it in the numerical value of no dimension, μ ₀ = 4 π *10^-7 H/m. It is existence physics and chemistry like a heavy side unit system ${\sqrt {4\pi}}$ But, in order to avoid an evil to appear in large quantities, 4 π was included in ε ₀ and μ ₀ (they do not have this solution in the 3 yuan system).

International unit system (SI) adopts MKSA system of units about the electromagnetism.

Minor unit system

Generalization CGS unit system

It is the unit system that revised CGS electromagnetic unit system of 3 yuan, electrostatic unit system in 4 yuan system such as the MKSA system of units formally. As transient measures in case of the shifts to MKSA system of units, an international sign unit of association of international pure applied physics (IUPAP) predicate committee (SUN Committee) introduced it in 1961.

I give unit of electric current emu a name ビオ (Bi), and the generalization CGS electromagnetic unit system assumes it a base unit. これにより、μ₀ は無次元量の1ではなく、次元を持つ 1 dyn/Bi² となる。

一般化CGS静電単位系は、電荷の単位としての esu をフランクリン (Fr) と呼び基本単位とする。これにより、ε₀ は無次元量の1ではなく、次元を持つ 1 Fr²/erg·cm となる。ただし、この基本単位の名称フランクリンは、（一般化電磁単位系のビオと異なり）従来からあった名称である。

ただしこれらの変更では単位の大きさは変わらず、基本単位からの組み立てのみが変わる。

MKSC単位系・MKSΩ単位系

MKSA単位系と同様の4元系だが、第4の基本単位としてアンペア (A) の代わりにクーロン (C) やオーム (Ω) を使った単位系である。実用上はMKSA単位系とまったく同じで、単位の定義のしかたが違うだけである。

MKSP単位系

MKSP単位系は、ヘヴィサイド単位系のような、3元の対称な有理系である。ただし、力学単位系としてMKS単位系を採用している。鈴木範人・小塩高文による。

ヘヴィサイド単位系と同様に理論計算が簡便で、しかしヘヴィサイド単位系と異なり力学単位はSIと同じで比較的相性がいいので、数値実験に使われることがある。

単位名称

CGS電磁単位系・静電単位系・ガウス単位系は、3元系なので理論上は力学単位から全ての単位を組み立てられるが、電磁単位系での電流の単位が dyn^1/2 になるなど指数に半整数が表れる問題があるので、そのような表現はされなかった。

電磁単位系の電流の単位は電磁単位 (emu)、静電単位系の電荷の単位は静電単位 (esu) と呼ばれた。これらはガウス単位系でも使うことができる。また、3元系の特徴としていくつかの物理量の次元が同じになり、たとえば磁束も emu で表せた。

MKSA単位系の元となった実用単位は、当初より単位名称と共に考案された。電圧のボルト (V)、電流のアンペア (A)、電荷のクーロン (C) などがそうである。ただし電気抵抗のオーム (Ω) は、実用単位以前から存在した単位と名称である。

実用単位との比較の問題から、実用単位の名称に接頭辞アブ (ab; absoluteの略) を付けて、元となった電磁単位を表すようになった。たとえば、電磁単位系の電流の単位 (emu) はアブアンペア (abA) となる。静電単位系でもこれに倣ってスタット（stat; staticの略）をつけて表すこともある。たとえば、静電単位系の電荷の単位 (esu) はスタットクーロン (statC) となる。

いくつかのガウス単位系の単位には、固有の名称が与えられた。

フランクリン以外は磁気系の単位、つまり、電磁単位系と共通の単位である。

一般化電磁単位系では、電流の単位を新しくビオ (Bi) と名づけた（これはガウス単位系でない唯一の単位名称となった）。

MKSA単位系（および国際単位系）では、実用単位の名称がそのまま使われる。

換算

各単位系を相互に変換するには、簡単な計算で求められる係数を乗算すればよい。なお、CGS単位系の基本単位となる物理量や、国際単位系(SI)の基本単位となる電流（アンペア）は、その定義通りの実験が困難であるため、より高い精度の別の実験から間接的に求められている。

CGSガウス単位系の単位を1とした場合、各単位系の単位の換算は以下のようになる。ただし、c は光速そのものではなく、光速を cm/s で表した場合の数値 c = 2.99792458×10¹⁰（単位なし）とする。

	Electric current	磁束	Electric charge	電圧
CGS electromagnetic unit system	1	1	1/c	c
CGS electrostatic unit system	c	1/c	1	1
CGS gauss unit system	1	1	1	1
Heavy side unit system	${\sqrt {4\pi}}$	$1/{\sqrt {4\pi }}$	${\sqrt {4\pi}}$	$1/{\sqrt {4\pi }}$
MKSA system of units	10	10^-8	10/c	10^-8c

MKSA/SI	物理量		emu	esu/gauss	MKSA/SI	Physical quantity		emu/gauss	esu
アンペア (A)	電流	I	10⁻¹ Bi	10⁻¹c	~~ボルト (V)~~	-	-	-	-
ボルト (V)	起電力・電位	V	10⁸	10⁸/c	アンペア (A)	起磁力・磁位	F_m	10⁻¹×4π Gb	10⁻¹×4πc
オーム (Ω)	電気抵抗	R	10⁹	10⁹/c²	~~ジーメンス (S)~~	-	-	-	-
クーロン (C)	電荷	Q	10⁻¹	10⁻¹c Fr	ウェーバ (Wb)	磁荷	Q_m	10⁸/4π	10⁸/4πc
クーロン (C)	電束	ψ	10⁻¹×4π	10⁻¹×4πc	ウェーバ (Wb)	磁束	Φ	10⁸ Mx	10⁸/c
ファラド (F)	静電容量	C	10⁻⁹	10⁻⁹c²	ヘンリー (H)	インダクタンス	L	10⁹	10⁹/c²
V/m	電場	E	10⁶	10⁶/c	A/m	磁場	H	10⁻³×4π Oe	10⁻³×4πc
V/m	-	-	-	-	A/m	磁化	M	10⁻³	10⁻³/c
C/m²	電束密度	D	10⁻⁵×4π	10⁻⁵×4πc	テスラ (T)	磁束密度	B	10⁴ G	10⁴/c
C/m²	電気分極	P	10⁻⁵	10⁻⁵×c	テスラ (T)	磁気分極	P_m	10⁴/4π	10⁴/4πc
F/m	誘電率	ε	10⁻¹¹×4π	10⁻¹¹×4πc²	H/m	透磁率	μ	10⁷/4π	10⁷/4πc²
表の見方

外部リンク

単位系について岡部洋一
電磁気学における単位系山﨑勝義

「https://ja.wikipedia.org/w/index.php?title=電磁気の単位&oldid=59330039」から取得

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2017년 3월 28일 화요일