傑斐緬柯方程式

給予電荷密度分佈${\displaystyle \rho (\mathbf {r} ',\,t)}$和電流密度分佈${\displaystyle \mathbf {J} (\mathbf {r} ',\,t)}$，推遲純量勢${\displaystyle \Phi (\mathbf {r} ,\,t)}$和推遲向量勢${\displaystyle \mathbf {A} (\mathbf {r} ,\,t)}$分別用方程式定義為（參閱推遲勢）

${\displaystyle \Phi (\mathbf {r} ,\,t)\ {\stackrel {def}{=}}\ {\frac {1}{4\pi \epsilon _{0}}}\int _{{\mathcal {V}}'}{\frac {\rho (\mathbf {r} ',\,t_{r})}{|\mathbf {r} -\mathbf {r} '|}}\,d^{3}\mathbf {r} '}$、

${\displaystyle \mathbf {A} (\mathbf {r} ,\,t)\ {\stackrel {def}{=}}\ {\frac {\mu _{0}}{4\pi }}\int _{{\mathcal {V}}'}{\frac {\mathbf {J} (\mathbf {r} ',\,t_{r})}{|\mathbf {r} -\mathbf {r} '|}}\,d^{3}\mathbf {r} '}$。

推遲時間${\displaystyle t_{r}}$定義為現在時間${\displaystyle t}$減去光波傳播的時間：

${\displaystyle t_{r}\ {\stackrel {def}{=}}\ t-{\frac {|\mathbf {r} -\mathbf {r} '|}{c}}}$；

其中，${\displaystyle c}$是光速。

在這兩個非靜態的推遲勢方程式內，源電荷密度和源電流密度都跟推遲時間${\displaystyle t_{r}}$有關，而不是跟時間無關。

推遲勢與電場${\displaystyle \mathbf {E} }$、磁場${\displaystyle \mathbf {B} }$的關係分別為

${\displaystyle \mathbf {E} =-\nabla \Phi -{\frac {\partial \mathbf {A} }{\partial t}}}$、

${\displaystyle \mathbf {B} =\nabla \times \mathbf {A} }$。

設定${\displaystyle {\boldsymbol {\mathfrak {R}}}}$為從源位置到場位置的分離向量：

${\displaystyle {\boldsymbol {\mathfrak {R}}}=\mathbf {r} -\mathbf {r} '}$。

場位置${\displaystyle \mathbf {r} }$、源位置${\displaystyle \mathbf {r} '}$和時間${\displaystyle t}$都是自變數。分離向量${\displaystyle {\boldsymbol {\mathfrak {R}}}}$和其大小${\displaystyle {\mathfrak {R}}}$都是應變數，跟場位置${\displaystyle \mathbf {r} }$、源位置${\displaystyle \mathbf {r} '}$有關。推遲時間${\displaystyle t_{r}=t-{\mathfrak {R}}/c}$也是應變數，跟時間${\displaystyle t}$、分離距離${\displaystyle {\mathfrak {R}}}$有關。

推遲純量勢${\displaystyle \Phi (\mathbf {r} ,\,t)}$的梯度是

${\displaystyle \nabla \Phi (\mathbf {r} ,\,t)={\frac {1}{4\pi \epsilon _{0}}}\int _{{\mathcal {V}}'}\nabla \left({\frac {\rho (\mathbf {r} ',\,t_{r})}{\mathfrak {R}}}\right)\,d^{3}\mathbf {r} '={\frac {1}{4\pi \epsilon _{0}}}\int _{{\mathcal {V}}'}\left[{\frac {\nabla \rho (\mathbf {r} ',\,t_{r})}{\mathfrak {R}}}+\rho (\mathbf {r} ',\,t_{r})\nabla \left({\frac {1}{\mathfrak {R}}}\right)\right]\,d^{3}\mathbf {r} '}$。

源電荷密度${\displaystyle \rho (\mathbf {r} ',\,t_{r})}$的全微分是

${\displaystyle {\begin{aligned}d\rho (\mathbf {r} ',\,t_{r})&=\nabla '\rho \cdot d\mathbf {r} '+{\frac {\partial \rho }{\partial t_{r}}}dt_{r}\\&=\nabla '\rho \cdot d\mathbf {r} '+{\frac {\partial \rho }{\partial t_{r}}}\left({\frac {\partial t_{r}}{\partial t}}dt+{\frac {\partial t_{r}}{\partial {\mathfrak {R}}}}d{\mathfrak {R}}\right)\\&=\nabla '\rho \cdot d\mathbf {r} '+{\frac {\partial \rho }{\partial t_{r}}}\left(dt-{\frac {1}{c}}d{\mathfrak {R}}\right)\\&=\nabla '\rho \cdot d\mathbf {r} '+{\frac {\partial \rho }{\partial t_{r}}}\left[dt-{\frac {1}{c}}(\nabla {\mathfrak {R}}\cdot d\mathbf {r} +\nabla '{\mathfrak {R}}\cdot d\mathbf {r} ')\right]\\\end{aligned}}}$ 。

注意到

${\displaystyle {\frac {\partial \rho (\mathbf {r} ',\,t_{r})}{\partial t}}={\frac {\partial t_{r}}{\partial t}}\ {\frac {\partial \rho (\mathbf {r} ',\,t_{r})}{\partial t_{r}}}={\frac {\partial \rho (\mathbf {r} ',\,t_{r})}{\partial t_{r}}}}$、

${\displaystyle \nabla {\mathfrak {R}}={\hat {\boldsymbol {\mathfrak {R}}}}}$。

所以，源電荷密度${\displaystyle \rho (\mathbf {r} ',\,t_{r})}$的梯度是

${\displaystyle \nabla \rho (\mathbf {r} ',\,t_{r})=-{\frac {1}{c}}\ {\frac {\partial \rho (\mathbf {r} ',\,t_{r})}{\partial t_{r}}}\nabla {\mathfrak {R}}=-{\frac {1}{c}}\ {\frac {\partial \rho (\mathbf {r} ',\,t_{r})}{\partial t}}{\hat {\boldsymbol {\mathfrak {R}}}}=-{\frac {{\dot {\rho }}(\mathbf {r} ',\,t_{r})}{c}}{\hat {\boldsymbol {\mathfrak {R}}}}}$；

其中，${\displaystyle {\dot {\rho }}(\mathbf {r} ',\,t_{r})}$定義為${\displaystyle {\frac {\partial \rho (\mathbf {r} ',\,t_{r})}{\partial t}}}$。

將這公式代入，推遲純量勢${\displaystyle \Phi (\mathbf {r} ,\,t)}$的梯度是

${\displaystyle \nabla \Phi (\mathbf {r} ,\,t)={\frac {1}{4\pi \epsilon _{0}}}\int _{{\mathcal {V}}'}\left[-{\frac {{\dot {\rho }}(\mathbf {r} ',\,t_{r})}{c}}{\frac {\hat {\boldsymbol {\mathfrak {R}}}}{\mathfrak {R}}}-\rho (\mathbf {r} ',\,t_{r})\left({\frac {\hat {\boldsymbol {\mathfrak {R}}}}{{\mathfrak {R}}^{2}}}\right)\right]\,d^{3}\mathbf {r} '}$。

推遲向量勢${\displaystyle \mathbf {A} (\mathbf {r} ,\,t)}$對於時間的偏導數為：

${\displaystyle {\frac {\partial \mathbf {A} (\mathbf {r} ,\,t)}{\partial t}}={\frac {\mu _{0}}{4\pi }}\int _{{\mathcal {V}}'}{\frac {{\dot {\mathbf {J} }}(\mathbf {r} ',\,t_{r})}{|\mathbf {r} -\mathbf {r} '|}}\,d^{3}\mathbf {r} '={\frac {1}{4\pi \epsilon _{0}c^{2}}}\int _{{\mathcal {V}}'}{\frac {{\dot {\mathbf {J} }}(\mathbf {r} ',\,t_{r})}{|\mathbf {r} -\mathbf {r} '|}}\,d^{3}\mathbf {r} '}$。

綜合前面這兩個公式，可以得到電場的傑斐緬柯方程式。同樣方法，可以得到磁場的傑斐緬柯方程式。