# 真空电容率

${\displaystyle \epsilon _{0}\approx 8.854\ 187\ 817\dots \ \times 10^{-12}}$ 法拉

${\displaystyle \epsilon _{0}\ {\stackrel {def}{=}}\ {\frac {1}{\mu _{0}{c_{0}}^{2}}}}$

${\displaystyle \epsilon _{0}\approx 8.854\ 187\ 817\ldots \times 10^{-12}}$ 安培24公斤-1-3（或者法拉／米）。

${\displaystyle \mathbf {D} \ {\stackrel {def}{=}}\ \epsilon _{0}\mathbf {E} +\mathbf {P} \,\!}$

## 歷史背景

### 單位理想化

${\displaystyle F={\frac {k_{\mathrm {e} }Q^{2}}{r^{2}}}\,\!}$

${\displaystyle F={\frac {{q_{s}}^{2}}{r^{2}}}\,\!}$

${\displaystyle F=\;k'_{\mathrm {e} }{q'_{s}}^{2}/4\pi r^{2}\,\!}$

${\displaystyle F=q^{2}/4\pi \epsilon _{0}r^{2}\,\!}$

${\displaystyle q_{s}=q/{\sqrt {4\pi \epsilon _{0}}}\,\!}$

### ε0數值的設定

${\displaystyle \nabla \cdot \mathbf {E} =0\,\!}$
${\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}\,\!}$
${\displaystyle \nabla \cdot \mathbf {B} =0\,\!}$
${\displaystyle \nabla \times \mathbf {B} =\epsilon _{0}\mu _{0}{\frac {\partial \mathbf {E} }{\partial t}}\,\!}$  ;

${\displaystyle \nabla \times (\nabla \times \mathbf {B} )=\epsilon _{0}\mu _{0}{\frac {\partial (\nabla \times \mathbf {E} )}{\partial t}}\,\!}$

${\displaystyle \nabla \times (\nabla \times \mathbf {B} )=-\epsilon _{0}\mu _{0}{\frac {\partial ^{2}\mathbf {B} }{\partial t^{2}}}\,\!}$

${\displaystyle \nabla \times (\nabla \times \mathbf {B} )=\nabla (\nabla \cdot \mathbf {B} )-\nabla ^{2}\mathbf {B} \,\!}$

${\displaystyle \nabla \times (\nabla \times \mathbf {B} )=-\nabla ^{2}\mathbf {B} \,\!}$

${\displaystyle \nabla ^{2}\mathbf {B} -\epsilon _{0}\mu _{0}{\frac {\partial ^{2}\mathbf {B} }{\partial t^{2}}}=0\,\!}$

${\displaystyle \nabla ^{2}\mathbf {E} -\epsilon _{0}\mu _{0}{\frac {\partial ^{2}\mathbf {E} }{\partial t^{2}}}=0}$

${\displaystyle c_{0}=1/{\sqrt {\epsilon _{0}\mu _{0}}}\,\!}$

## 註釋

1. ^ 取第二個馬克士威方程式（法拉第方程式）的旋度，并將第四個馬克士威方程式${\displaystyle \nabla \times \mathbf {B} =\epsilon _{0}\mu _{0}{\frac {\partial \mathbf {E} }{\partial t}}}$ 代入，則可得到
{\displaystyle {\begin{aligned}\nabla \times (\nabla \times \mathbf {E} )&=-{\frac {\partial (\nabla \times \mathbf {B} )}{\partial t}}\\&=-\epsilon _{0}\mu _{0}{\frac {\partial ^{2}\mathbf {E} }{\partial t^{2}}}\\\end{aligned}}}
應用一個向量恆等式，再代入第一个馬克士威方程式${\displaystyle \nabla \cdot \mathbf {E} =0}$ ，即得
{\displaystyle {\begin{aligned}\nabla \times (\nabla \times \mathbf {E} )&=\nabla (\nabla \cdot \mathbf {E} )-\nabla ^{2}\mathbf {E} \\&=-\nabla ^{2}\mathbf {E} \\\end{aligned}}}
這樣，就可以得到光波的电场波動方程式
${\displaystyle \nabla ^{2}\mathbf {E} -\epsilon _{0}\mu _{0}{\frac {\partial ^{2}\mathbf {E} }{\partial t^{2}}}=0}$

## 參考文獻

1. CODATA. Electric constant. 2006 CODATA recommended values. NIST. [2007-08-08]. （原始内容存档于2007-04-23）.
2. ^ 引述自 NIST（國家標準與技術學院）：現行的慣例是按照ISO 31的建議，用 ${\displaystyle c_{0}}$  來標記在真空的光速。原本的1983年建議書主張採用 ${\displaystyle c}$  來做此用途。
3. ^ NIST對於公尺的定義 (html). NIST. [2009-06-01]. （原始内容存档于2011-08-22）.
4. ^ NIST對於安培的定義 (html). NIST. [2009-06-01]. （原始内容存档于2017-04-25）.
5. ^ CODATA報告 (pdf). NIST. [2009-06-01]. （原始内容存档 (PDF)于2018-06-12）.
6. ^ Cardarelli, François. Encyclopaedia of Scientific Units, Weights and Measures: Their SI Equivalences and Origins 2nd. Springer. 2004. ISBN 9781852336820.
7. ^ John David Jackson. Classical electrodynamics Third. New York: Wiley. 1999: Appendix on units and dimensions; pp. 775 et seq.。. ISBN 047130932X.怎樣選擇獨立單位的敘述
8. ^ 物理術語部分真空指出，近似真空和自由空間的一個主要分歧源點，是來自於無法達到0氣壓。但是，還有其它非理想性的可能源點。參閱，例如，Di Piazza, Antonino; K. Hatsagortsyan & C. Keitel, Light diffraction by a strong standing electromagnetic wave, Phys.Rev.Lett., 2006, 97: 083603 [2016-02-21], （原始内容存档于2021-05-20）Gies, Holger; J. Jaeckel & A. Ringwald, Polarized light propagating in a magnetic field as a probe for millicharged fermions, Phys. Rev. Letts., 2006, 97: 140402 [2009-06-01], （原始内容存档于2021-05-20）
9. ^ Astrid Lambrecht (Hartmut Figger, Dieter Meschede, Claus Zimmermann Eds.). Observing mechanical dissipation in the quantum vacuum: an experimental challenge；在物理書 Laser physics at the limits. Berlin/New York: Springer. 2002: 197. ISBN 3540424180.
10. ^ Walter Dittrich & Gies H. Probing the quantum vacuum: perturbative effective action approach. Berlin: Springer. 2000. ISBN 3540674284.
11. ^ 對於這類修正，CIPM RECOMMENDATION 1 (CI-2002)p. 195页面存档备份，存于互联网档案馆）的建議是：
♦ …在每一個案例裏，為了要處理真實發生的事件，像繞射、地心引力，或不完美的真空等等，任何必要的修正都必須仔細執行。
除此以外，
♦ …科學家認為公尺是單位固有長度proper length）。公尺的定義，只適用於一個足夠小的區域內，這樣，可以忽略重力場的不均勻性。
CIPM是國際重量和度量會議International Committee for Weights and Measures）的首字母縮略字。