给予在源位置
r
′
{\displaystyle \mathbf {r} '}
的电流或电荷分布,计算在场位置
r
{\displaystyle \mathbf {r} }
产生的电势或磁向量势。
在真空 内,电场
E
{\displaystyle \mathbf {E} }
和磁场
B
{\displaystyle \mathbf {B} }
可以用杰斐缅柯方程式表达为:
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} }
是场位置,
r
′
{\displaystyle \mathbf {r} '}
是源位置,
t
{\displaystyle t}
是现在时间 ,
t
r
{\displaystyle t_{r}}
是推迟时间 ,
ϵ
0
{\displaystyle \epsilon _{0}}
是电常数 ,
μ
0
{\displaystyle \mu _{0}}
是磁常数 ,
ρ
{\displaystyle \rho }
是电荷密度 ,
ρ
˙
=
d
e
f
∂
ρ
∂
t
{\displaystyle {\dot {\rho }}\ {\stackrel {def}{=}}\ {\frac {\partial \rho }{\partial t}}}
定义为电荷密度对于时间的偏导数 ,
J
{\displaystyle \mathbf {J} }
是电流密度 ,
J
˙
=
d
e
f
∂
J
∂
t
{\displaystyle {\dot {\mathbf {J} }}\ {\stackrel {def}{=}}\ {\frac {\partial \mathbf {J} }{\partial t}}}
定义为电流密度对于时间的偏导数 ,
V
′
{\displaystyle {\mathcal {V}}'}
是体积分的空间,
d
3
r
′
{\displaystyle d^{3}\mathbf {r} '}
是微小体元素。
给予电荷密度分布
ρ
(
r
′
,
t
)
{\displaystyle \rho (\mathbf {r} ',\,t)}
和电流密度分布
J
(
r
′
,
t
)
{\displaystyle \mathbf {J} (\mathbf {r} ',\,t)}
,推迟纯量势
Φ
(
r
,
t
)
{\displaystyle \Phi (\mathbf {r} ,\,t)}
和推迟向量势
A
(
r
,
t
)
{\displaystyle \mathbf {A} (\mathbf {r} ,\,t)}
分别用方程式定义为(参阅推迟势 )
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} '}
。
推迟时间
t
r
{\displaystyle t_{r}}
定义为现在时间
t
{\displaystyle t}
减去光波 传播的时间:
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}
是光速 。
在这两个非静态的推迟势方程式内,源电荷密度和源电流密度都跟推迟时间
t
r
{\displaystyle t_{r}}
有关,而不是跟时间无关。
推迟势与电场
E
{\displaystyle \mathbf {E} }
、磁场
B
{\displaystyle \mathbf {B} }
的关系分别为
E
=
−
∇
Φ
−
∂
A
∂
t
{\displaystyle \mathbf {E} =-\nabla \Phi -{\frac {\partial \mathbf {A} }{\partial t}}}
、
B
=
∇
×
A
{\displaystyle \mathbf {B} =\nabla \times \mathbf {A} }
。
设定
R
{\displaystyle {\boldsymbol {\mathfrak {R}}}}
为从源位置到场位置的分离向量:
R
=
r
−
r
′
{\displaystyle {\boldsymbol {\mathfrak {R}}}=\mathbf {r} -\mathbf {r} '}
。
场位置
r
{\displaystyle \mathbf {r} }
、源位置
r
′
{\displaystyle \mathbf {r} '}
和时间
t
{\displaystyle t}
都是自变数 。分离向量
R
{\displaystyle {\boldsymbol {\mathfrak {R}}}}
和其大小
R
{\displaystyle {\mathfrak {R}}}
都是应变数 ,跟场位置
r
{\displaystyle \mathbf {r} }
、源位置
r
′
{\displaystyle \mathbf {r} '}
有关。推迟时间
t
r
=
t
−
R
/
c
{\displaystyle t_{r}=t-{\mathfrak {R}}/c}
也是应变数,跟时间
t
{\displaystyle t}
、分离距离
R
{\displaystyle {\mathfrak {R}}}
有关。
推迟纯量势
Φ
(
r
,
t
)
{\displaystyle \Phi (\mathbf {r} ,\,t)}
的梯度 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} '}
。
源电荷密度
ρ
(
r
′
,
t
r
)
{\displaystyle \rho (\mathbf {r} ',\,t_{r})}
的全微分 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,源电荷密度
ρ
(
r
′
,
t
r
)
{\displaystyle \rho (\mathbf {r} ',\,t_{r})}
的梯度是
∇
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})}
定义为
∂
ρ
(
r
′
,
t
r
)
∂
t
{\displaystyle {\frac {\partial \rho (\mathbf {r} ',\,t_{r})}{\partial t}}}
。
将这公式代入,推迟纯量势
Φ
(
r
,
t
)
{\displaystyle \Phi (\mathbf {r} ,\,t)}
的梯度是
∇
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} '}
。
推迟向量势
A
(
r
,
t
)
{\displaystyle \mathbf {A} (\mathbf {r} ,\,t)}
对于时间的偏导数为:
∂
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} '}
。
综合前面这两个公式,可以得到电场的杰斐缅柯方程式。同样方法,可以得到磁场的杰斐缅柯方程式。