本条目中，向量 與标量 分別用粗體 與斜體 顯示。例如，位置向量通常用 r {\displaystyle \mathbf {r} \,\!} r {\displaystyle r\,\!} 檢驗變數或場變數 的標記的後面沒有單撇號「 ′ {\displaystyle '\,\!} ′ {\displaystyle '\,\!} 在電磁學 裏，給予含時電荷密度 分佈和電流密度 分佈，可以使用傑斐緬柯方程式 （Jefimenko equation）來計算電場 和磁場 。這方程式因其發現者物理學家歐雷格·傑斐緬柯 [ 1] 馬克士威方程組 對於這些電荷密度分佈和電流密度分佈的解答[ 2] [ 3]
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。
對於任意介質，將前面所述電場和磁場的方程式加以延伸[ 4] ρ {\displaystyle \rho } J {\displaystyle \mathbf {J} } 電極化強度 P {\displaystyle \mathbf {P} } 磁化强度 M {\displaystyle \mathbf {M} } E {\displaystyle \mathbf {E} } 電位移 D {\displaystyle \mathbf {D} } 磁感應強度 B {\displaystyle \mathbf {B} } 磁場強度 H {\displaystyle \mathbf {H} }
很多物理學家藉著馬克士威方程組來詮釋為甚麼含時電場與含時磁場會互相生成。這詮釋常常會被納入電磁波形成的理論。但是，傑斐緬柯方程式顯示出，實際上並不是這樣[ 5]
馬克士威方程組和其解答，都沒有指明電場和磁場之間的因果關係。因此總結，電磁場是一個
對偶 實體，是由含時電荷分佈和含時電流分佈共同同時產生的電場和磁場。
— 歐雷格·傑斐緬柯， Causality Electromagnetic Induction and Gravitation，第16頁 ^ McDonald, Kirk T., The relation between expressions for time-dependent electromagnetic fields given by Jefimenko and by Panofsky and Phillips, American Journal of Physics, 1997, 65 (11): pp. 1074–1076 ^ Jefimenko, Oleg D., Electricity and magnetism: an introduction to the theory of electric and magnetic fields 2nd, Electret Scientific Co., 1989, ISBN 9780917406089 ^ Griffiths, David J. Introduction to Electrodynamics (3rd ed.). Prentice Hall. 1998. ISBN 0-13-805326-X ^ Oleg D. Jefimenko, Solutions of Maxwell's equations for electric and magnetic fields in arbitrary media , American Journal of Physics 60(10) (1992), 899-902. ^ Jefimenko, Oleg D., Causality Electromagnetic Induction and Gravitation 2nd, Electret Scientific: pp. 16, 2000, ISBN 0-917406-23-0 
![{\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)
