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。
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 
![{\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} '}](http://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} '}](http://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} '}](http://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}}}](http://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} '}](http://wikimedia.org/api/rest_v1/media/math/render/svg/1b736019d784e70000dfa40f82fcab856d48bb9d)
