# 倫敦方程

## 數學表述

${\displaystyle {\frac {\partial \mathbf {j} _{s}}{\partial t}}={\frac {n_{s}e^{2}}{m}}\mathbf {E} ,\qquad \mathbf {\nabla } \times \mathbf {j} _{s}=-{\frac {n_{s}e^{2}}{mc}}\mathbf {B} \,}$

${\displaystyle \mathbf {j} _{s}=-{\frac {n_{s}e^{2}}{mc}}\mathbf {A} \,}$

## 倫敦穿透深度

${\displaystyle \nabla \times \mathbf {B} ={\frac {4\pi \mathbf {j} }{c}}}$

${\displaystyle \nabla ^{2}\mathbf {B} ={\frac {1}{\lambda ^{2}}}\mathbf {B} ,\qquad \lambda \equiv {\sqrt {\frac {mc^{2}}{4\pi n_{s}e^{2}}}}\,}$

${\displaystyle B_{z}(x)=B_{0}e^{-x/\lambda }\,}$

## 倫敦方程的基本原理

### 最初的論述

${\displaystyle \mathbf {F} =e\mathbf {E} +{\frac {e}{c}}\mathbf {v} \times \mathbf {B} }$

${\displaystyle \nabla \times \mathbf {E} =-{\frac {1}{c}}{\frac {\partial \mathbf {B} }{\partial t}}}$

${\displaystyle {\frac {\partial }{\partial t}}\left(\nabla \times \mathbf {j} _{s}+{\frac {n_{s}e^{2}}{mc}}\mathbf {B} \right)=0\,}$

### 正則動量論述

${\displaystyle \mathbf {j} _{s}=n_{s}e\mathbf {v} }$

${\displaystyle \mathbf {v} ={\frac {1}{m}}\left(\mathbf {p} -{\frac {e}{c}}\mathbf {A} \right)}$

${\displaystyle \mathbf {j} _{s}=-{\frac {n_{s}e_{s}^{2}}{mc}}\mathbf {A} }$

## 註釋及參考資料

1. ^ London, F.; H. London. The Electromagnetic Equations of the Supraconductor. Proc. Roy. Soc. (London). March 1935, A149 (866): 71. ISSN 0080-4630.
2. ^ Michael Tinkham. Introduction to Superconductivity. McGraw-Hill. 1996. ISBN 0-07-064878-6.
3. ^ Neil W. Ashcroft; N. David Mermin. Solid State Physics. Saunders College. 1976: 738. ISBN 0-03-083993-9.
4. ^ Charles Kittel. Introduction to Solid State Physics. 1999. ISBN 0-47-141526-X.
5. ^ Meissner, W.; R. Ochsenfeld. Ein neuer Effekt bei Eintritt der Supraleitfähigkeit. Naturwissenschaften. 1933, 21 (44): 787. Bibcode:1933NW.....21..787M. doi:10.1007/BF01504252.
6. James F. Annett. Superconductivity, Superfluids and Condensates. Oxford. 2004: 58. ISBN 0-19-850756-9.
7. ^ John David Jackson. Classical Electrodynamics. John Wiley & Sons. 1999: 604. ISBN 0-19-850756-9.
8. ^ Michael Tinkham. Introduction to Superconductivity. McGraw-Hill. 1996: 6. ISBN 0-07-064878-6.
9. ^ （因為假設了電場只會隨着時間緩慢地變動，而且位移電流項已經受到1/c這個因子的壓抑，因此可以視位移為零。）
10. Michael Tinkham. Introduction to Superconductivity. McGraw-Hill. 1996: 5. ISBN 0-07-064878-6.
11. ^ John David Jackson. Classical Electrodynamics. John Wiley & Sons. 1999: 603–604. ISBN 0-19-850756-9.
12. ^ Michael Tinkham. Introduction to Superconductivity. McGraw-Hill. 1996: 5–6. ISBN 0-07-064878-6.
13. ^ L. D. Landau and E. M. Lifshitz. Quantum Mechanics- Non-relativistic Theory. Butterworth-Heinemann. 1977: 455–458. ISBN 0-7506-3539-8.