# 電磁場的動力學理論

## 馬克士威原本的方程式

(A) 總電流定律
$\mathbf{J}_{tot} = \mathbf{J} + \frac{\partial\mathbf{D}}{\partial t}$
(B) 磁場方程式
$\mu \mathbf{H} = \nabla \times \mathbf{A}$
(C) 安培環路定理
$\nabla \times \mathbf{H} = \mathbf{J}_{tot}$
(D) 勞侖茲力方程式
$\mathbf{E} = \mu \mathbf{v} \times \mathbf{H} - \frac{\partial\mathbf{A}}{\partial t}-\nabla \phi$
(E) 電彈性方程式
$\mathbf{E} = \frac{1}{\epsilon} \mathbf{D}$
(F) 歐姆定律
$\mathbf{E} = \frac{1}{\sigma} \mathbf{J}$
(G) 高斯定律
$\nabla \cdot \mathbf{D} = \rho$
(H) 連續方程式
$\nabla \cdot \mathbf{J} = -\frac{\partial\rho}{\partial t}$

$\mathbf{H}$磁場強度
$\mathbf{J}$傳導電流密度
$\mathbf{J}_{tot}$ 是總電流密度（包括位移電流密度），
$\mathbf{D}$電位移
$\rho$自由電荷密度，
$\mathbf{A}$磁向量勢
$\mathbf{E}$電場
$\phi$電勢
$\mu$磁導率
$\epsilon$電容率
$\sigma$電導率

——詹姆斯·馬克士威

## 馬克士威的推導

$\mathbf{B}= - \hat{x}\frac{\partial A_y}{\partial z}+ \hat{y}\frac{\partial A_x}{\partial z}$

$\mathbf{J}_{tot}= - \hat{x}\frac{\partial H_y}{\partial z}+ \hat{y}\frac{\partial H_x}{\partial z} = - \frac{1}{\mu}\left(\hat{x}\frac{\partial^2 A_x}{\partial z^2}+\hat{y}\frac{\partial^2 A_y}{\partial z^2}\right)$

$\mathbf{J}_{tot}=\frac{\partial \mathbf{D}}{\partial t}=\epsilon\frac{\partial \mathbf{E}}{\partial t}$

$\frac{\partial^2 A_x}{\partial z^2} - \mu\epsilon\frac{\partial^2 A_x}{\partial t^2}=0$
$\frac{\partial^2 A_y}{\partial z^2} - \mu\epsilon\frac{\partial^2 A_y}{\partial t^2}=0$

$\frac{\partial^2 B_x}{\partial z^2} - \mu\epsilon\frac{\partial^2 B_x}{\partial t^2}=0$
$\frac{\partial^2 B_y}{\partial z^2} - \mu\epsilon\frac{\partial^2 B_y}{\partial t^2}=0$

$\frac{\partial}{\partial z}=\frac{\partial w}{\partial z}\frac{\mathrm{d}}{\mathrm{d}w}=\frac{\mathrm{d}}{\mathrm{d}w}$
$\frac{\partial}{\partial t}=\frac{\partial w}{\partial t}\frac{\mathrm{d}}{\mathrm{d}w}= - V\frac{\mathrm{d}}{\mathrm{d}w}$

$\frac{\mathrm{d}^2 B_x}{\mathrm{d} w^2} - \mu\epsilon V^2\frac{\mathrm{d}^2 B_x}{\mathrm{d} w^2}=0$
$\frac{\mathrm{d}^2 B_y}{\mathrm{d} w^2} - \mu\epsilon V^2\frac{\mathrm{d}^2 B_y}{\mathrm{d} w^2}=0$

$V=1/\sqrt{\mu\epsilon}$

$\frac{\partial^2 E_x}{\partial z^2} - \mu\epsilon\frac{\partial^2 E_x}{\partial t^2}=0$
$\frac{\partial^2 E_y}{\partial z^2} - \mu\epsilon\frac{\partial^2 E_y}{\partial t^2}=0$
$E_z= - \frac{\partial A_z}{\partial t} - \frac{\partial \phi}{\partial z}$

## 現代推導

$\nabla \cdot \mathbf{E} = 0$(1)
$\nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}} {\partial t}$(2)
$\nabla \cdot \mathbf{B} = 0$(3)
$\nabla \times \mathbf{B} =\mu_0 \varepsilon_0 \frac{ \partial \mathbf{E}} {\partial t}$(4)

$\nabla \times(\nabla \times \mathbf{E})= - \frac{\partial } {\partial t} (\nabla \times \mathbf{B})= - \mu_0 \varepsilon_0 \frac{\partial^2 \mathbf{E} } {\partial t^2}$
$\nabla \times(\nabla \times \mathbf{B})= \mu_0 \varepsilon_0 \frac{\partial } {\partial t}(\nabla \times \mathbf{E})= - \mu_o \varepsilon_o \frac{\partial^2 \mathbf{B}}{\partial t^2}$

$\nabla \times \left( \nabla \times \mathbf{Z} \right) = \nabla \left( \nabla \cdot \mathbf{Z} \right) - \nabla^2 \mathbf{Z}$

$\left(\nabla^2 - \frac{ 1}{{c}^2 }\frac{\partial^2}{\partial t^2} \right) \mathbf{E}\ =\ 0$(5)
$\left(\nabla^2 - \frac{ 1}{{c}^2 }\frac{\partial^2}{\partial t^2} \right) \mathbf{B}\ =\ 0$(6)

## 參考文獻

1. ^ 馬克士威, 詹姆斯, A dynamical theory of the electromagnetic field (pdf), Philosophical Transactions of the Royal Society of London, 1865, 155: 459–512
2. ^ 馬克士威, 詹姆斯, On physical lines of force (pdf), Philosophical Magazine, 1861
3. ^ 馬克士威, 詹姆斯, A Dynamical Theory of the Electromagnetic Field, pp. 499, 1864
• Maxwell, James C.; Torrance, Thomas F., A Dynamical Theory of the Electromagnetic Field, Eugene, OR: Wipf and Stock, 1996-03, ISBN 1-57910-015-5
• Niven, W. D., The Scientific Papers of James Clerk Maxwell, Vol. 1, New York: Dover, 1952