達朗貝爾方程

在經典電動力學中，將描述電磁波的勢所滿足的一個微分方程組稱作達朗貝爾方程（英文：d'Alembert equation）。達朗貝爾方程以數學家讓·勒朗·達朗貝爾的名字命名，他於 1747 年將其作為振動弦問題的解決方案推導出來。

達朗貝爾方程是一個非齊次（英語：Homogeneity and heterogeneity）的波動方程。^[1]

形式

達朗貝爾方程的形式如下：

${\displaystyle {\boldsymbol {\nabla }}^{2}{\boldsymbol {A}}-{\frac {1}{c^{2}}}{\frac {\partial ^{2}{\boldsymbol {A}}}{\partial t^{2}}}=-\mu _{0}{\boldsymbol {J}}}$

${\displaystyle {\boldsymbol {\nabla }}^{2}\varphi -{\frac {1}{c^{2}}}{\frac {\partial ^{2}\varphi }{\partial t^{2}}}=-{\frac {\rho }{\varepsilon _{0}}}}$

其中${\displaystyle {\boldsymbol {A}}}$為磁矢勢，${\displaystyle \varphi }$為電勢，${\displaystyle c}$為真空光速。^[1]

推導

經典電動力學中的麥克斯韋方程組如下所示

${\displaystyle {\boldsymbol {\nabla }}\times {\boldsymbol {E}}=-{\frac {\partial {\boldsymbol {B}}}{\partial t}}}$

${\displaystyle {\boldsymbol {\nabla }}\times {\boldsymbol {H}}={\frac {\partial {\boldsymbol {D}}}{\partial t}}+{\boldsymbol {J}}}$

${\displaystyle {\boldsymbol {\nabla }}\cdot {\boldsymbol {D}}=\rho }$

${\displaystyle {\boldsymbol {\nabla }}\cdot {\boldsymbol {B}}=0}$

且有${\displaystyle {\boldsymbol {D}}=\varepsilon _{0}{\boldsymbol {E}},{\boldsymbol {B}}=\mu _{0}{\boldsymbol {H}}}$。

由${\displaystyle {\boldsymbol {B}}}$的無源性可以引入磁矢勢${\displaystyle {\boldsymbol {A}}}$，有${\displaystyle {\boldsymbol {B}}={\boldsymbol {\nabla }}\times {\boldsymbol {A}}}$，代入麥克斯韋方程組的第一式得${\displaystyle {\boldsymbol {\nabla }}\times \left({\boldsymbol {E}}+{\frac {\partial {\boldsymbol {A}}}{\partial t}}\right)=0}$。這說明矢量${\displaystyle {\boldsymbol {E}}+{\frac {\partial {\boldsymbol {A}}}{\partial t}}}$是無旋場，可以用標量勢${\displaystyle \varphi }$的負梯度描述：

${\displaystyle {\boldsymbol {E}}+{\frac {\partial {\boldsymbol {A}}}{\partial t}}=-{\boldsymbol {\nabla }}\varphi }$

也即${\displaystyle {\boldsymbol {E}}=-{\boldsymbol {\nabla }}\varphi -{\frac {\partial {\boldsymbol {A}}}{\partial t}}}$。

因此

${\displaystyle {\boldsymbol {\nabla }}\times \left({\boldsymbol {\nabla }}\times {\boldsymbol {A}}\right)=\mu _{0}{\boldsymbol {J}}-\mu _{0}\varepsilon _{0}{\frac {\partial }{\partial t}}\left({\boldsymbol {\nabla }}\varphi \right)-\mu _{0}\varepsilon _{0}{\frac {\partial ^{2}{\boldsymbol {A}}}{\partial t^{2}}}}$

${\displaystyle -{\boldsymbol {\nabla }}^{2}\varphi -{\frac {\partial }{\partial t}}\left({\boldsymbol {\nabla }}\cdot {\boldsymbol {A}}\right)={\frac {\rho }{\varepsilon _{0}}}}$

而${\displaystyle \mu _{0}\varepsilon _{0}={\frac {1}{c^{2}}}}$，代入並整理得

${\displaystyle {\boldsymbol {\nabla }}^{2}{\boldsymbol {A}}-{\frac {1}{c^{2}}}{\frac {\partial ^{2}{\boldsymbol {A}}}{\partial t^{2}}}-{\boldsymbol {\nabla }}\left({\boldsymbol {\nabla }}\cdot {\boldsymbol {A}}+{\frac {1}{c^{2}}}{\frac {\partial \varphi }{\partial t}}\right)=-\mu _{0}{\boldsymbol {J}}}$

${\displaystyle {\boldsymbol {\nabla }}^{2}\varphi +{\frac {\partial }{\partial t}}\left({\boldsymbol {\nabla }}\cdot {\boldsymbol {A}}\right)=-{\frac {\rho }{\varepsilon _{0}}}}$

採用洛倫茨規範，即${\displaystyle {\boldsymbol {\nabla }}\cdot {\boldsymbol {A}}+{\frac {1}{c^{2}}}{\frac {\partial \varphi }{\partial t}}=0}$，可得

${\displaystyle {\boldsymbol {\nabla }}^{2}{\boldsymbol {A}}-{\frac {1}{c^{2}}}{\frac {\partial ^{2}{\boldsymbol {A}}}{\partial t^{2}}}=-\mu _{0}{\boldsymbol {J}}}$

${\displaystyle {\boldsymbol {\nabla }}^{2}\varphi -{\frac {1}{c^{2}}}{\frac {\partial ^{2}\varphi }{\partial t^{2}}}=-{\frac {\rho }{\varepsilon _{0}}}}$

此即達朗貝爾方程，其自由項為電流密度和電荷密度。

參考資料

1. 郭碩鴻. 《電動力學（第三版）》. 北京: 高等教育出版社. 2008. ISBN 978-7-04-023924-9.