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）。达朗贝尔方程以数学家让·勒朗·达朗贝尔的名字命名，他于 1747 年将其作为振动弦问题的解决方案推导出来。 达朗贝尔方程是一个非齐次（英语：Homogeneity and heterogeneity）的波动方程。 达朗贝尔方程的形式如下： ∇ 2 A − 1 c 2 ∂ 2 A ∂ t 2 = − μ 0 J {\displaystyle {\boldsymbol {\nabla }}^{2}{\boldsymbol {A}}-{\frac {1}{c^{2}}}{\frac {\partial ^{2}{\boldsymbol {A}}}{\partial t^{2}}}=-\mu _{0}{\boldsymbol {J}}} ∇ 2 φ − 1 c 2 ∂ 2 φ ∂ t 2 = − ρ ε 0 {\displaystyle {\boldsymbol {\nabla }}^{2}\varphi -{\frac {1}{c^{2}}}{\frac {\partial ^{2}\varphi }{\partial t^{2}}}=-{\frac {\rho }{\varepsilon _{0}}}} 其中 A {\displaystyle {\boldsymbol {A}}} 为磁矢势， φ {\displaystyle \varphi } 为电势， c {\displaystyle c} 为真空光速。 经典电动力学中的麦克斯韦方程组如下所示 ∇ × E = − ∂ B ∂ t {\displaystyle {\boldsymbol {\nabla }}\times {\boldsymbol {E}}=-{\frac {\partial {\boldsymbol {B}}}{\partial t}}} ∇ × H = ∂ D ∂ t + J {\displaystyle {\boldsymbol {\nabla }}\times {\boldsymbol {H}}={\frac {\partial {\boldsymbol {D}}}{\partial t}}+{\boldsymbol {J}}} ∇ ⋅ D = ρ {\displaystyle {\boldsymbol {\nabla }}\cdot {\boldsymbol {D}}=\rho } ∇ ⋅ B = 0 {\displaystyle {\boldsymbol {\nabla }}\cdot {\boldsymbol {B}}=0} 且有 D = ε 0 E , B = μ 0 H {\displaystyle {\boldsymbol {D}}=\varepsilon _{0}{\boldsymbol {E}},{\boldsymbol {B}}=\mu _{0}{\boldsymbol {H}}} 。 由 B {\displaystyle {\boldsymbol {B}}} 的无源性可以引入磁矢势 A {\displaystyle {\boldsymbol {A}}} ，有 B = ∇ × A {\displaystyle {\boldsymbol {B}}={\boldsymbol {\nabla }}\times {\boldsymbol {A}}} ，代入麦克斯韦方程组的第一式得 ∇ × ( E + ∂ A ∂ t ) = 0 {\displaystyle {\boldsymbol {\nabla }}\times \left({\boldsymbol {E}}+{\frac {\partial {\boldsymbol {A}}}{\partial t}}\right)=0} 。这说明矢量 E + ∂ A ∂ t {\displaystyle {\boldsymbol {E}}+{\frac {\partial {\boldsymbol {A}}}{\partial t}}} 是无旋场，可以用标量势 φ {\displaystyle \varphi } 的负梯度描述： E + ∂ A ∂ t = − ∇ φ {\displaystyle {\boldsymbol {E}}+{\frac {\partial {\boldsymbol {A}}}{\partial t}}=-{\boldsymbol {\nabla }}\varphi } 也即 E = − ∇ φ − ∂ A ∂ t {\displaystyle {\boldsymbol {E}}=-{\boldsymbol {\nabla }}\varphi -{\frac {\partial {\boldsymbol {A}}}{\partial t}}} 。 因此 ∇ × ( ∇ × A ) = μ 0 J − μ 0 ε 0 ∂ ∂ t ( ∇ φ ) − μ 0 ε 0 ∂ 2 A ∂ t 2 {\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}}}} − ∇ 2 φ − ∂ ∂ t ( ∇ ⋅ A ) = ρ ε 0 {\displaystyle -{\boldsymbol {\nabla }}^{2}\varphi -{\frac {\partial }{\partial t}}\left({\boldsymbol {\nabla }}\cdot {\boldsymbol {A}}\right)={\frac {\rho }{\varepsilon _{0}}}} 而 μ 0 ε 0 = 1 c 2 {\displaystyle \mu _{0}\varepsilon _{0}={\frac {1}{c^{2}}}} ，代入并整理得 ∇ 2 A − 1 c 2 ∂ 2 A ∂ t 2 − ∇ ( ∇ ⋅ A + 1 c 2 ∂ φ ∂ t ) = − μ 0 J {\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}}} ∇ 2 φ + ∂ ∂ t ( ∇ ⋅ A ) = − ρ ε 0 {\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 ∇ ⋅ A + 1 c 2 ∂ φ ∂ t = 0 {\displaystyle {\boldsymbol {\nabla }}\cdot {\boldsymbol {A}}+{\frac {1}{c^{2}}}{\frac {\partial \varphi }{\partial t}}=0} ，可得 ∇ 2 A − 1 c 2 ∂ 2 A ∂ t 2 = − μ 0 J {\displaystyle {\boldsymbol {\nabla }}^{2}{\boldsymbol {A}}-{\frac {1}{c^{2}}}{\frac {\partial ^{2}{\boldsymbol {A}}}{\partial t^{2}}}=-\mu _{0}{\boldsymbol {J}}} ∇ 2 φ − 1 c 2 ∂ 2 φ ∂ t 2 = − ρ ε 0 {\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。 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.