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

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

## 馬克士威的推導

${\displaystyle \mathbf {B} =-{\hat {x}}{\frac {\partial A_{y}}{\partial z}}+{\hat {y}}{\frac {\partial A_{x}}{\partial z}}}$

${\displaystyle \mathbf {J} _{tot}=-{\hat {x}}{\frac {\partial H_{y}}{\partial z}}+{\hat {y}}{\frac {\partial H_{x}}{\partial z}}=-{\frac {1}{\mu }}\left({\hat {x}}{\frac {\partial ^{2}A_{x}}{\partial z^{2}}}+{\hat {y}}{\frac {\partial ^{2}A_{y}}{\partial z^{2}}}\right)}$

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

${\displaystyle {\frac {\partial ^{2}A_{x}}{\partial z^{2}}}-\mu \epsilon {\frac {\partial ^{2}A_{x}}{\partial t^{2}}}=0}$
${\displaystyle {\frac {\partial ^{2}A_{y}}{\partial z^{2}}}-\mu \epsilon {\frac {\partial ^{2}A_{y}}{\partial t^{2}}}=0}$

${\displaystyle {\frac {\partial ^{2}B_{x}}{\partial z^{2}}}-\mu \epsilon {\frac {\partial ^{2}B_{x}}{\partial t^{2}}}=0}$
${\displaystyle {\frac {\partial ^{2}B_{y}}{\partial z^{2}}}-\mu \epsilon {\frac {\partial ^{2}B_{y}}{\partial t^{2}}}=0}$

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

${\displaystyle {\frac {\mathrm {d} ^{2}B_{x}}{\mathrm {d} w^{2}}}-\mu \epsilon V^{2}{\frac {\mathrm {d} ^{2}B_{x}}{\mathrm {d} w^{2}}}=0}$
${\displaystyle {\frac {\mathrm {d} ^{2}B_{y}}{\mathrm {d} w^{2}}}-\mu \epsilon V^{2}{\frac {\mathrm {d} ^{2}B_{y}}{\mathrm {d} w^{2}}}=0}$

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

${\displaystyle {\frac {\partial ^{2}E_{x}}{\partial z^{2}}}-\mu \epsilon {\frac {\partial ^{2}E_{x}}{\partial t^{2}}}=0}$
${\displaystyle {\frac {\partial ^{2}E_{y}}{\partial z^{2}}}-\mu \epsilon {\frac {\partial ^{2}E_{y}}{\partial t^{2}}}=0}$
${\displaystyle E_{z}=-{\frac {\partial A_{z}}{\partial t}}-{\frac {\partial \phi }{\partial z}}}$

## 現代推導

${\displaystyle \nabla \cdot \mathbf {E} =0}$(1)
${\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}$(2)
${\displaystyle \nabla \cdot \mathbf {B} =0}$(3)
${\displaystyle \nabla \times \mathbf {B} =\mu _{0}\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}}$(4)

${\displaystyle \nabla \times (\nabla \times \mathbf {E} )=-{\frac {\partial }{\partial t}}(\nabla \times \mathbf {B} )=-\mu _{0}\varepsilon _{0}{\frac {\partial ^{2}\mathbf {E} }{\partial t^{2}}}}$
${\displaystyle \nabla \times (\nabla \times \mathbf {B} )=\mu _{0}\varepsilon _{0}{\frac {\partial }{\partial t}}(\nabla \times \mathbf {E} )=-\mu _{o}\varepsilon _{o}{\frac {\partial ^{2}\mathbf {B} }{\partial t^{2}}}}$

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

${\displaystyle \left(\nabla ^{2}-{\frac {1}{{c}^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\right)\mathbf {E} \ =\ 0}$(5)
${\displaystyle \left(\nabla ^{2}-{\frac {1}{{c}^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\right)\mathbf {B} \ =\ 0}$(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. ^ 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.
4. ^ 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, March 1996, ISBN 1-57910-015-5
• Niven, W. D., The Scientific Papers of James Clerk Maxwell Vol. 1, New York: Dover, 1952