# 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

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

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

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

## 註釋

1. ^ 在帶電粒子的拉格朗日力學裡，正則動量${\displaystyle \mathbf {p} }$
${\displaystyle \mathbf {p} =m\mathbf {v} +q\mathbf {A} }$
其中，${\displaystyle m}$ 是質量，${\displaystyle \mathbf {v} }$ 是速度，${\displaystyle q}$ 是電荷。
在某些狀況下，守恆的不是${\displaystyle m\mathbf {v} }$ ，而是${\displaystyle \mathbf {p} }$ ，因此，${\displaystyle m\mathbf {v} }$ ${\displaystyle q\mathbf {A} }$ 兩個項共同貢獻出正確的守恆動量。
在量子力學裡，帶電粒子的薛丁格方程使用的基礎物理量是磁向量勢，而不是磁場。[3]

## 參考文獻

1. ^ 黑維塞, 奧利弗, On the self-induction of wires, Philosophy Magazine: 173, [1886], ... E and H, which have physical significance in really defining the state of the medium anywhere, which A and P do not.
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], magnitudes which serve for calculation only
3. Semon, Mark; Taylor, John, Thoughts on the magnetic vector potential, Am. J. Phys., 1996, 64 (11): pp. 1361–1369, doi:10.1119/1.18400, vector potential can be seen as a stored momentum per unit charge in much the same way that electric potential is the stored energy per unit charge
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 (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, 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 φ(the scalar potential)
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. 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. ^ 術語在線. 全国科学技术名词审定委员会. electrontonic
13. ^ Whittaker 1951，第272-273页
14. ^ 馬克士威, 詹姆斯, 8, (编) Nivin, William, The scientific papers of James Clerk Maxwell 1, New York: Doer Publications, 1890

## 参考书目

• Duffin, W.J. Electricity and Magnetism, Fourth Edition. McGraw-Hill. 1990. ISBN 007707209X.
• Jackson, John Davd, Classical Electrodynamics 3rd, John-Wiley, 1999, ISBN 047130932X
• Kraus, John D., Electromagnetics 3rd, McGraw-Hill, 1984, ISBN 0070354235
• Ulaby, Fawwaz. Fundamentals of Applied Electromagnetics, Fifth Edition. Pearson Prentice Hall. 2007: 226–228. ISBN 0-13-241326-4.