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: ∇ 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}}} 為磁矢勢, φ {\displaystyle \varphi } 為電勢, c {\displaystyle c} 為真空光速。[1] Remove ads推導 經典電動力學中的麥克斯韋方程組如下所示 ∇ × 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}}} 的無源性可以引入磁矢勢 A {\displaystyle {\boldsymbol {A}}} ,有 B = ∇ × A {\displaystyle {\boldsymbol {B}}={\boldsymbol {\nabla }}\times {\boldsymbol {A}}} ,代入麥克斯韋方程組的第一式得 ∇ × ( E + ∂ A ∂ t ) = 0 {\displaystyle {\boldsymbol {\nabla }}\times \left({\boldsymbol {E}}+{\frac {\partial {\boldsymbol {A}}}{\partial t}}\right)=0} 。這說明矢量 E + ∂ A ∂ t {\displaystyle {\boldsymbol {E}}+{\frac {\partial {\boldsymbol {A}}}{\partial t}}} 是無旋場,可以用純量勢 φ {\displaystyle \varphi } 的負梯度描述: 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}}}} ,代入並整理得 ∇ 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}}}} 採用洛倫茨規範,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}}}} 此即達朗貝爾方程,其自由項為電流密度和電荷密度。 Remove ads參考資料Loading content...Loading related searches...Wikiwand - on Seamless Wikipedia browsing. On steroids.Remove ads