給予在源位置r ′ {\displaystyle \mathbf {r} '} r {\displaystyle \mathbf {r} } 在真空 內，電場E {\displaystyle \mathbf {E} } B {\displaystyle \mathbf {B} }
E ( r , t ) = 1 4 π ϵ 0 ∫ V ′ [ ρ ( r ′ , t r ) r − r ′ | r − r ′ | 3 + ρ ˙ ( r ′ , t r ) c r − r ′ | r − r ′ | 2 − J ˙ ( r ′ , t r ) c 2 | r − r ′ | ] d 3 r ′ {\displaystyle \mathbf {E} (\mathbf {r} ,\,t)={\frac {1}{4\pi \epsilon _{0}}}\int _{{\mathcal {V}}'}\left[\rho (\mathbf {r} ',\,t_{r}){\frac {\mathbf {r} -\mathbf {r} '}{|\mathbf {r} -\mathbf {r} '|^{3}}}+{\frac {{\dot {\rho }}(\mathbf {r} ',\,t_{r})}{c}}{\frac {\mathbf {r} -\mathbf {r} '}{|\mathbf {r} -\mathbf {r} '|^{2}}}-{\frac {{\dot {\mathbf {J} }}(\mathbf {r} ',\,t_{r})}{c^{2}|\mathbf {r} -\mathbf {r} '|}}\right]d^{3}\mathbf {r} '} B ( r , t ) = μ 0 4 π ∫ V ′ [ J ( r ′ , t r ) | r − r ′ | 3 + J ˙ ( r ′ , t r ) c | r − r ′ | 2 ] × ( r − r ′ ) d 3 r ′ {\displaystyle \mathbf {B} (\mathbf {r} ,t)={\frac {\mu _{0}}{4\pi }}\int _{{\mathcal {V}}'}\left[{\frac {\mathbf {J} (\mathbf {r} ',\,t_{r})}{|\mathbf {r} -\mathbf {r} '|^{3}}}+{\frac {{\dot {\mathbf {J} }}(\mathbf {r} ',\,t_{r})}{c|\mathbf {r} -\mathbf {r} '|^{2}}}\right]\times (\mathbf {r} -\mathbf {r} ')\ d^{3}\mathbf {r} '} 其中，r {\displaystyle \mathbf {r} } r ′ {\displaystyle \mathbf {r} '} t {\displaystyle t} 時間 ，t r {\displaystyle t_{r}} 推遲時間 ，ϵ 0 {\displaystyle \epsilon _{0}} 電常數 ，μ 0 {\displaystyle \mu _{0}} 磁常數 ，ρ {\displaystyle \rho } 電荷密度 ，ρ ˙ = d e f ∂ ρ ∂ t {\displaystyle {\dot {\rho }}\ {\stackrel {def}{=}}\ {\frac {\partial \rho }{\partial t}}} 偏導數 ，J {\displaystyle \mathbf {J} } 電流密度 ，J ˙ = d e f ∂ J ∂ t {\displaystyle {\dot {\mathbf {J} }}\ {\stackrel {def}{=}}\ {\frac {\partial \mathbf {J} }{\partial t}}} 偏導數 ，V ′ {\displaystyle {\mathcal {V}}'} d 3 r ′ {\displaystyle d^{3}\mathbf {r} '}
給予電荷密度分佈ρ ( r ′ , t ) {\displaystyle \rho (\mathbf {r} ',\,t)} J ( r ′ , t ) {\displaystyle \mathbf {J} (\mathbf {r} ',\,t)} Φ ( r , t ) {\displaystyle \Phi (\mathbf {r} ,\,t)} A ( r , t ) {\displaystyle \mathbf {A} (\mathbf {r} ,\,t)} 推遲勢 ）
Φ ( r , t ) = d e f 1 4 π ϵ 0 ∫ V ′ ρ ( r ′ , t r ) | r − r ′ | d 3 r ′ {\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} '} A ( r , t ) = d e f μ 0 4 π ∫ V ′ J ( r ′ , t r ) | r − r ′ | d 3 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t r {\displaystyle t_{r}} t {\displaystyle t} 光波 傳播的時間：
t r = d e f t − | r − r ′ | c {\displaystyle t_{r}\ {\stackrel {def}{=}}\ t-{\frac {|\mathbf {r} -\mathbf {r} '|}{c}}} 其中，c {\displaystyle c} 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t r {\displaystyle t_{r}}
推遲勢與電場E {\displaystyle \mathbf {E} } B {\displaystyle \mathbf {B} }
E = − ∇ Φ − ∂ A ∂ t {\displaystyle \mathbf {E} =-\nabla \Phi -{\frac {\partial \mathbf {A} }{\partial t}}} B = ∇ × A {\displaystyle \mathbf {B} =\nabla \times \mathbf {A} } 設定R {\displaystyle {\boldsymbol {\mathfrak {R}}}}
R = r − r ′ {\displaystyle {\boldsymbol {\mathfrak {R}}}=\mathbf {r} -\mathbf {r} '} 場位置r {\displaystyle \mathbf {r} } r ′ {\displaystyle \mathbf {r} '} t {\displaystyle t} 自變數 。分離向量R {\displaystyle {\boldsymbol {\mathfrak {R}}}} R {\displaystyle {\mathfrak {R}}} 應變數 ，跟場位置r {\displaystyle \mathbf {r} } r ′ {\displaystyle \mathbf {r} '} t r = t − R / c {\displaystyle t_{r}=t-{\mathfrak {R}}/c} t {\displaystyle t} R {\displaystyle {\mathfrak {R}}}
推遲純量勢Φ ( r , t ) {\displaystyle \Phi (\mathbf {r} ,\,t)} 梯度 是
∇ Φ ( r , t ) = 1 4 π ϵ 0 ∫ V ′ ∇ ( ρ ( r ′ , t r ) R ) d 3 r ′ = 1 4 π ϵ 0 ∫ V ′ [ ∇ ρ ( r ′ , t r ) R + ρ ( r ′ , t r ) ∇ ( 1 R ) ] d 3 r ′ {\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ρ ( r ′ , t r ) {\displaystyle \rho (\mathbf {r} ',\,t_{r})} 全微分 是
d ρ ( r ′ , t r ) = ∇ ′ ρ ⋅ d r ′ + ∂ ρ ∂ t r d t r = ∇ ′ ρ ⋅ d r ′ + ∂ ρ ∂ t r ( ∂ t r ∂ t d t + ∂ t r ∂ R d R ) = ∇ ′ ρ ⋅ d r ′ + ∂ ρ ∂ t r ( d t − 1 c d R ) = ∇ ′ ρ ⋅ d r ′ + ∂ ρ ∂ t r [ d t − 1 c ( ∇ R ⋅ d r + ∇ ′ R ⋅ d r ′ ) ] {\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
∂ ρ ( r ′ , t r ) ∂ t = ∂ t r ∂ t ∂ ρ ( r ′ , t r ) ∂ t r = ∂ ρ ( r ′ , t r ) ∂ t r {\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}}}} ∇ R = R ^ {\displaystyle \nabla {\mathfrak {R}}={\hat {\boldsymbol {\mathfrak {R}}}}} 所以，源電荷密度ρ ( r ′ , t r ) {\displaystyle \rho (\mathbf {r} ',\,t_{r})}
∇ ρ ( r ′ , t r ) = − 1 c ∂ ρ ( r ′ , t r ) ∂ t r ∇ R = − 1 c ∂ ρ ( r ′ , t r ) ∂ t R ^ = − ρ ˙ ( r ′ , t r ) c R ^ {\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，ρ ˙ ( r ′ , t r ) {\displaystyle {\dot {\rho }}(\mathbf {r} ',\,t_{r})} ∂ ρ ( r ′ , t r ) ∂ t {\displaystyle {\frac {\partial \rho (\mathbf {r} ',\,t_{r})}{\partial t}}}
將這公式代入，推遲純量勢Φ ( r , t ) {\displaystyle \Phi (\mathbf {r} ,\,t)}
∇ Φ ( r , t ) = 1 4 π ϵ 0 ∫ V ′ [ − ρ ˙ ( r ′ , t r ) c R ^ R − ρ ( r ′ , t r ) ( R ^ R 2 ) ] d 3 r ′ {\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A ( r , t ) {\displaystyle \mathbf {A} (\mathbf {r} ,\,t)}
∂ A ( r , t ) ∂ t = μ 0 4 π ∫ V ′ J ˙ ( r ′ , t r ) | r − r ′ | d 3 r ′ = 1 4 π ϵ 0 c 2 ∫ V ′ J ˙ ( r ′ , t r ) | r − r ′ | d 3 r ′ {\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。
![{\displaystyle \mathbf {E} (\mathbf {r} ,\,t)={\frac {1}{4\pi \epsilon _{0}}}\int _{{\mathcal {V}}'}\left[\rho (\mathbf {r} ',\,t_{r}){\frac {\mathbf {r} -\mathbf {r} '}{|\mathbf {r} -\mathbf {r} '|^{3}}}+{\frac {{\dot {\rho }}(\mathbf {r} ',\,t_{r})}{c}}{\frac {\mathbf {r} -\mathbf {r} '}{|\mathbf {r} -\mathbf {r} '|^{2}}}-{\frac {{\dot {\mathbf {J} }}(\mathbf {r} ',\,t_{r})}{c^{2}|\mathbf {r} -\mathbf {r} '|}}\right]d^{3}\mathbf {r} '}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6527650dbac9a3a460139b0f4baf8322568e36e)
![{\displaystyle \mathbf {B} (\mathbf {r} ,t)={\frac {\mu _{0}}{4\pi }}\int _{{\mathcal {V}}'}\left[{\frac {\mathbf {J} (\mathbf {r} ',\,t_{r})}{|\mathbf {r} -\mathbf {r} '|^{3}}}+{\frac {{\dot {\mathbf {J} }}(\mathbf {r} ',\,t_{r})}{c|\mathbf {r} -\mathbf {r} '|^{2}}}\right]\times (\mathbf {r} -\mathbf {r} ')\ d^{3}\mathbf {r} '}](https://wikimedia.org/api/rest_v1/media/math/render/svg/17d6a631f0553f09719d125ba570555874379771)
![{\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} '}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f57b1e0b413f1a6cbf9c412072a825920491d4c6)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c21d8218c9aa790db8eff2d5897509c6171be41b)
![{\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} '}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b736019d784e70000dfa40f82fcab856d48bb9d)
