磁矢势

${\displaystyle \mathbf {B} =\nabla \times \mathbf {A} }$

定義與公式

${\displaystyle \nabla \cdot \mathbf {B} =0}$

${\displaystyle \mathbf {B} =\nabla \times \mathbf {A} }$(1)

${\displaystyle \nabla \times \mathbf {B} =\mu _{0}\mathbf {J} }$

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

${\displaystyle \nabla ^{2}\mathbf {A} =-\mu _{0}\mathbf {J} }$

${\displaystyle \mathbf {A} (\mathbf {r} )=\ {\frac {\mu _{0}}{4\pi }}\iiint _{\mathbb {V} '}{\frac {\mathbf {J} (\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}\,d^{3}\mathbf {r} '}$

${\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}=-{\frac {\partial }{\partial t}}(\nabla \times \mathbf {A} )}$

${\displaystyle \nabla \times \left(\mathbf {E} +{\frac {\partial \mathbf {A} }{\partial t}}\right)=0}$

${\displaystyle \mathbf {E} +{\frac {\partial \mathbf {A} }{\partial t}}=-\nabla \phi }$

${\displaystyle \mathbf {E} =-\nabla \phi -{\frac {\partial \mathbf {A} }{\partial t}}}$(2)

規範設定

${\displaystyle \mathbf {B} =\nabla \times \mathbf {A} =\nabla \times (\mathbf {A} +\nabla \lambda )}$

採用庫侖規範的馬克士威方程組

${\displaystyle \nabla \cdot {\textbf {E}}=\rho /\epsilon _{0}}$

${\displaystyle \nabla \cdot {\textbf {E}}=-\nabla ^{2}\phi -{\frac {\partial (\nabla \cdot \mathbf {A} )}{\partial t}}=-\nabla ^{2}\phi }$

${\displaystyle \nabla \times \mathbf {B} =\mu _{0}\mathbf {J} +\mu _{0}\epsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}}$

${\displaystyle \nabla \times (\nabla \times \mathbf {A} )=\mu _{0}\mathbf {J} -\mu _{0}\epsilon _{0}\left[\nabla \left({\frac {\partial \phi }{\partial t}}\right)+{\frac {\partial ^{2}\mathbf {A} }{\partial t^{2}}}\right]}$

${\displaystyle \nabla ^{2}\phi =-\rho /\epsilon _{0}}$
${\displaystyle \nabla ^{2}{\textbf {A}}-\mu _{0}\epsilon _{0}{\frac {\partial ^{2}{\textbf {A}}}{\partial t^{2}}}=-\mu _{0}{\textbf {J}}+\mu _{0}\epsilon _{0}\nabla \left({\frac {\partial \phi }{\partial t}}\right)}$

採用勞侖次規範的馬克士威方程組

${\displaystyle \nabla \cdot {\textbf {A}}+\mu _{0}\epsilon _{0}{\frac {\partial \phi }{\partial t}}=0}$

${\displaystyle \nabla ^{2}\phi -\mu _{0}\epsilon _{0}{\frac {\partial ^{2}\phi }{\partial t^{2}}}=-\rho /\epsilon _{0}}$
${\displaystyle \nabla ^{2}{\textbf {A}}-\mu _{0}\epsilon _{0}{\frac {\partial ^{2}{\textbf {A}}}{\partial t^{2}}}=-\mu _{0}{\textbf {J}}}$

${\displaystyle \Box \phi =-\rho /\epsilon _{0}}$
${\displaystyle \Box {\textbf {A}}=-\mu _{0}{\textbf {J}}}$

電磁四維勢

• 第一、電磁四維勢乃是一個四維向量。使用標準四維向量變換規則，假若知道在某慣性參考系的電磁四維勢，很容易就可以計算出在其它慣性參考系的數值。
• 第二、经典电磁学的內容可以更簡要、更便利地以電磁四維勢表達，特別是當採用勞侖次規範時。
• 第三、電磁四維勢在量子電動力學裏佔有重要的角色。

${\displaystyle A^{\alpha }\ {\stackrel {def}{=}}\ (\phi /c,\,\mathbf {A} )}$

${\displaystyle \partial _{\alpha }A^{\alpha }=0}$

${\displaystyle \Box A^{\alpha }=-\mu _{0}J^{\alpha }}$

${\displaystyle P^{\alpha }=\left({\frac {E}{c}},\,\mathbf {p} \right)}$

從源分佈計算位勢

${\displaystyle \phi (\mathbf {r} )\ {\stackrel {def}{=}}\ {\frac {1}{4\pi \epsilon _{0}}}\int _{\mathbb {V} '}{\frac {\rho (\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}\,d^{3}\mathbf {r} '}$
${\displaystyle \mathbf {A} (\mathbf {r} )\ {\stackrel {def}{=}}\ {\frac {\mu _{0}}{4\pi }}\int _{\mathbb {V} '}{\frac {\mathbf {J} (\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}\,d^{3}\mathbf {r} '}$

${\displaystyle t_{r}\ {\stackrel {def}{=}}\ t-{\frac {|\mathbf {r} -\mathbf {r} '|}{c}}}$

${\displaystyle \phi (\mathbf {r} ,\,t)\ {\stackrel {def}{=}}\ {\frac {1}{4\pi \epsilon _{0}}}\int _{\mathbb {V} '}{\frac {\rho (\mathbf {r} ',\,t_{r})}{|\mathbf {r} -\mathbf {r} '|}}\,d^{3}\mathbf {r} '}$
${\displaystyle \mathbf {A} (\mathbf {r} ,\,t)\ {\stackrel {def}{=}}\ {\frac {\mu _{0}}{4\pi }}\int _{\mathbb {V} '}{\frac {\mathbf {J} (\mathbf {r} ',\,t_{r})}{|\mathbf {r} -\mathbf {r} '|}}\,d^{3}\mathbf {r} '}$

磁向量勢場線圖

${\displaystyle \nabla \cdot \mathbf {B} =0}$
${\displaystyle \nabla \times \mathbf {B} =\mu _{0}\mathbf {J} }$

${\displaystyle \nabla \cdot \mathbf {A} =0}$
${\displaystyle \nabla \times \mathbf {A} =\mathbf {B} }$

歷史

${\displaystyle \mathbf {B} =\nabla \times \mathbf {A} }$

${\displaystyle \mathbf {E} =-{\dot {\mathbf {A} }}}$

${\displaystyle \nabla \times \mathbf {E} =-{\dot {\mathbf {B} }}}$

參考文獻

1. ^ 黑維塞, 奧利弗, On the self-induction of wires, Philosophy Magazine: 118–137, [1886]
2. ^ 赫兹, 海因里希, Electric waves: being researches on the propagation of electric action with finite velocity through space, Macmillan: pp.196, [1893]
3. ^ 馬克士威, 詹姆斯. 電磁場的動力學理論 (pdf). Philosophical Transactions of the Royal Society of London. 1865, 155: 459–512. doi:10.1098/rstl.1865.0008.
4. ^ Aharonov, Y; Bohm, D, Significance of electromagnetic potentials in quantum theory, Physical Review, 1959, 115: 485–491, doi:10.1103/PhysRev.115.485
5. ^ Chambers, R. G., Shift of an Electron Interference Pattern by Enclosed Magnetic Flux, Physical Review Letters, 1960, 5 (1): pp. 3–5, doi:10.1103/PhysRevLett.5.3
6. ^ 費曼, 理查; 雷頓, 羅伯; 山德士, 馬修, 15, 費曼物理學講義 II 電磁與物質(2)－－ 介電質、磁與感應定律, 台灣: 天下文化書: pp. 162–175, 2006, ISBN 978-986-216-231-6, knowledge of the classical electromagnetic field acting locally on a particle is not sufficient to predict its quantum-mechanical behavior. and Is the vector potential a "real" field? ... a real field is a mathematical device for avoiding the idea of action at a distance. .... for a long time it was believed that A was not a "real" field. .... there are phenomena involving quantum mechanics which show that in fact A is a "real" field in the sense that we have defined it. ... E and B are slowly disappearing from the modern expression of physical laws; they are being replaced by A (the vector potential) and ${\displaystyle \varphi }$(the scalar potential) 参数|quote=值左起第650位存在delete character (帮助)
7. ^ Konopinski, E. J., What the electromagnetic vector potential describes, American Jounal of Physics, 1978, 46 (5): pp. 499–502
8. ^ Yang, ChenNing. The conceptual origins of Maxwell’s equations and gauge theory. Physics Today. 2014, 67 (11): 45–51. doi:10.1063/PT.3.2585.
9. ^ Griffiths, David J. Introduction to Electrodynamics (3rd ed.). Prentice Hall. 1998: pp. 422–428. ISBN 0-13-805326-X.
10. ^ 費曼, 理查; 雷頓, 羅伯; 山德士, 馬修, 費曼物理學講義 II (2) 介電質、磁與感應定律, 台灣: 天下文化書: pp. 167, 2008, ISBN 9789862162316
11. ^ 11.0 11.1 11.2 11.3 Yang, ChenNing. The conceptual origins of Maxwell’s equations and gauge theory. Physics Today. 2014, 67 (11): 45–51. doi:10.1063/PT.3.2585.
12. ^ Whittaker 1951，第272-273页
13. ^ 馬克士威, 詹姆斯, 8, (编) Nivin, William, The scientific papers of James Clerk Maxwell 1, New York: Doer Publications, 1890
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