# 倫敦方程

## 數學表述

$\frac{\partial \mathbf{j}_s}{\partial t} = \frac{n_s e^2}{m}\mathbf{E}, \qquad \mathbf{\nabla}\times\mathbf{j}_s =-\frac{n_s e^2}{mc}\mathbf{B} \,$

$\mathbf{j}_s =-\frac{n_se^2}{mc}\mathbf{A}\,$

## 倫敦穿透深度

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

$\nabla^2 \mathbf{B} = \frac{1}{\lambda^2}\mathbf{B}, \qquad \lambda \equiv \sqrt{\frac{m c^2}{4 \pi n_s e^2}}\,$

$B_z(x) = B_0 e^{-x / \lambda}\,$

## 倫敦方程的基本原理

### 最初的論述

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

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

$\frac{\partial}{\partial t}\left( \nabla \times \mathbf{j}_s + \frac{n_s e^2}{m c} \mathbf{B} \right) = 0\,$

### 正則動量論述

$\mathbf{j}_s = n_s e \mathbf{v}$

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

$\mathbf{j}_s =-\frac{n_se_s^2}{mc}\mathbf{A}$

## 註釋及參考資料

1. ^ London, F.; H. London. The Electromagnetic Equations of the Supraconductor. Proc. Roy. Soc. (London). 1935-03, 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. ^ 6.0 6.1 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. ^ 10.0 10.1 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.