达朗贝尔方程

在经典电动力学中，将描述电磁波的势所满足的一个微分方程组称作达朗贝尔方程（英文：d'Alembert equation）。达朗贝尔方程是一个非齐次（英语：Homogeneity and heterogeneity）的波动方程。^[1]

形式

达朗贝尔方程的形式如下：

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

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

其中${\displaystyle {\boldsymbol {A}}}$ 为磁矢势，${\displaystyle \varphi }$ 为电势，${\displaystyle c}$ 为真空光速。^[1]

推导

经典电动力学中的麦克斯韦方程组如下所示

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

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

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

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

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

由${\displaystyle {\boldsymbol {B}}}$ 的无源性可以引入磁矢势${\displaystyle {\boldsymbol {A}}}$ ，有${\displaystyle {\boldsymbol {B}}={\boldsymbol {\nabla }}\times {\boldsymbol {A}}}$ ，代入麦克斯韦方程组的第一式得${\displaystyle {\boldsymbol {\nabla }}\times \left({\boldsymbol {E}}+{\frac {\partial {\boldsymbol {A}}}{\partial t}}\right)=0}$ 。这说明矢量${\displaystyle {\boldsymbol {E}}+{\frac {\partial {\boldsymbol {A}}}{\partial t}}}$ 是无旋场，可以用标量势${\displaystyle \varphi }$ 的负梯度描述：

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

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${\displaystyle \mu _{0}\varepsilon _{0}={\frac {1}{c^{2}}}}$ ，代入并整理得

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

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

采用洛伦茨规范，即${\displaystyle {\boldsymbol {\nabla }}\cdot {\boldsymbol {A}}+{\frac {1}{c^{2}}}{\frac {\partial \varphi }{\partial t}}=0}$ ，可得

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

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

此即达朗贝尔方程，其自由项为电流密度和电荷密度。

参考资料

1. 郭硕鸿. 《电动力学（第三版）》. 北京: 高等教育出版社. 2008. ISBN 978-7-04-023924-9.