# 李纳-维谢势

## 物理理论

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

### 表达方程

${\displaystyle {\boldsymbol {\mathfrak {R))}=\mathbf {r} -\mathbf {r} '=\mathbf {r} -\mathbf {w} (t)\,\!}$

${\displaystyle \Phi (\mathbf {r} ,\,t)={\frac {1}{4\pi \epsilon _{0))}\ {\frac {qc}((\mathfrak {R))c-{\boldsymbol {\mathfrak {R))}\cdot \mathbf {v} ))\,\!}$
${\displaystyle \mathbf {A} (\mathbf {r} ,\,t)={\frac {\mathbf {v} }{c^{2))}\Phi (\mathbf {r} ,\,t)\,\!}$

${\displaystyle \mathbf {r} '=\mathbf {w} (t_{r})\,\!}$
${\displaystyle \mathbf {v} =\mathbf {v} (t_{r})\,\!}$

### 推导

${\displaystyle \Phi (\mathbf {r} ,\,t)\ {\stackrel {def}{=))\ {\frac {1}{4\pi \epsilon _{0))}\int _((\mathcal {V))'}{\frac {\rho (\mathbf {r} ',\,t_{r})}{\mathfrak {R))}\,d^{3}\mathbf {r} '\,\!}$
${\displaystyle \mathbf {A} (\mathbf {r} ,\,t)\ {\stackrel {def}{=))\ {\frac {\mu _{0)){4\pi ))\int _((\mathcal {V))'}{\frac {\mathbf {J} (\mathbf {r} ',\,t_{r})}{\mathfrak {R))}\,d^{3}\mathbf {r} '\,\!}$

${\displaystyle \rho (\mathbf {r} ,\,t)=q\delta (\mathbf {r} -\mathbf {w} (t))\,\!}$

${\displaystyle \Phi (\mathbf {r} ,\,t)={\frac {q}{4\pi \epsilon _{0))}\int _((\mathcal {V))'}{\frac {\delta (\mathbf {r} '-\mathbf {w} (t_{r}))}{\mathfrak {R))}\,d^{3}\mathbf {r} '\,\!}$

${\displaystyle \Phi (\mathbf {r} ,\,t)={\frac {q}{4\pi \epsilon _{0}{\mathfrak {R))))\int _((\mathcal {V))'}\delta (\mathbf {r} '-\mathbf {w} (t_{r}))\,d^{3}\mathbf {r} '\,\!}$

${\displaystyle {\mathfrak {J))={\cfrac {\partial {\boldsymbol {\eta ))}{\partial \mathbf {r} '))={\begin{vmatrix}{\cfrac {\partial \eta _{x)){\partial x'))&{\cfrac {\partial \eta _{x)){\partial y'))&{\cfrac {\partial \eta _{x)){\partial z'))\\{\cfrac {\partial \eta _{y)){\partial x'))&{\cfrac {\partial \eta _{y)){\partial y'))&{\cfrac {\partial \eta _{y)){\partial z'))\\{\cfrac {\partial \eta _{z)){\partial x'))&{\cfrac {\partial \eta _{z)){\partial y'))&{\cfrac {\partial \eta _{z)){\partial z'))\\\end{vmatrix))\,\!}$

${\displaystyle {\cfrac {\partial \eta _{x)){\partial x'))=1-{\cfrac {\partial w_{x)){\partial x'))=1-{\cfrac {\partial w_{x)){\partial t_{r))}\ {\cfrac {\partial t_{r)){\partial x'))=1-v_{x}{\cfrac {\partial t_{r)){\partial x'))\,\!}$
${\displaystyle {\cfrac {\partial \eta _{y)){\partial x'))={\cfrac {\partial w_{y)){\partial x'))={\cfrac {\partial w_{y)){\partial t_{r))}\ {\cfrac {\partial t_{r)){\partial x'))=v_{y}{\cfrac {\partial t_{r)){\partial x'))\,\!}$

${\displaystyle {\mathfrak {J))=1-\mathbf {v} \cdot \nabla 't_{r}=1-{\hat {\boldsymbol {\mathfrak {R))))\cdot \mathbf {v} /c\,\!}$

${\displaystyle \Phi (\mathbf {r} ,\,t)={\frac {q}{4\pi \epsilon _{0}{\mathfrak {R))))\int _((\mathcal {V))'}\delta ({\boldsymbol {\eta ))){\cfrac {\partial \mathbf {r} '}{\partial {\boldsymbol {\eta ))))\,d^{3}{\boldsymbol {\eta ))={\frac {q}{4\pi \epsilon _{0}{\mathfrak {R))))\int _((\mathcal {V))'}{\cfrac {\delta ({\boldsymbol {\eta )))}{\mathfrak {J))}\,d^{3}{\boldsymbol {\eta ))={\frac {q}{4\pi \epsilon _{0}{\mathfrak {R))))\int _((\mathcal {V))'}{\cfrac {\delta ({\boldsymbol {\eta )))}{1-{\hat {\boldsymbol {\mathfrak {R))))\cdot \mathbf {v} /c))\,d^{3}{\boldsymbol {\eta ))\,\!}$

${\displaystyle \Phi (\mathbf {r} ,\,t)={\frac {1}{4\pi \epsilon _{0))}\ {\frac {qc}((\mathfrak {R))c-{\boldsymbol {\mathfrak {R))}\cdot \mathbf {v} ))\,\!}$

### 相对论性导引

${\displaystyle \phi '={\frac {q}{4\pi \epsilon _{0}{\mathfrak {R))'))}$
${\displaystyle A'=0}$

${\displaystyle \phi =\gamma (\phi '-c\beta A')}$
${\displaystyle A=\gamma (-A'+\beta \phi '/c)}$

${\displaystyle \phi ={\frac {\gamma q}{4\pi \epsilon _{0}{\mathfrak {R))'))}$
${\displaystyle {\boldsymbol {A))={\frac {\gamma q{\boldsymbol {\beta ))}{4\pi \epsilon _{0}{\mathfrak {R))'c))}$

${\displaystyle {\mathfrak {R))'}$${\displaystyle {\mathfrak {R))}$的变换关系也由洛仑兹变换给出：

${\displaystyle {\mathfrak {R))'=c\Delta t'=c\gamma (\Delta t-{\boldsymbol {\beta ))\cdot {\boldsymbol {\mathfrak {R))}/c)=\gamma ({\mathfrak {R))-{\boldsymbol {\beta ))\cdot {\boldsymbol {\mathfrak {R))})}$

${\displaystyle {\mathfrak {R))'}$的表达式代入即得到李纳-维谢势。

### 物理意义

${\displaystyle \Phi (\mathbf {r} ,\,t)={\frac {1}{4\pi \epsilon _{0))}\ {\frac {q}{\mathfrak {R))}\,\!}$

### 移动中的带电粒子的电磁场

${\displaystyle \mathbf {E} =-\nabla \Phi -{\dfrac {\partial \mathbf {A} }{\partial t))\,\!}$
${\displaystyle \mathbf {B} =\nabla \times \mathbf {A} \,\!}$

${\displaystyle \mathbf {E} (\mathbf {r} ,\,t)={\frac {q}{4\pi \epsilon _{0))}\ {\cfrac {\mathfrak {R)){({\boldsymbol {\mathfrak {R))}\cdot \mathbf {u} )^{3))}[(c^{2}-v^{2})\mathbf {u} +{\boldsymbol {\mathfrak {R))}\times (\mathbf {u} \times \mathbf {a} )]\,\!}$
${\displaystyle \mathbf {B} (\mathbf {r} ,\,t)={\frac {1}{c)){\hat {\boldsymbol {\mathfrak {R))))\times \mathbf {E} (\mathbf {r} ,\,t)\,\!}$ ;

${\displaystyle \mathbf {E} ={\frac {q}{4\pi \epsilon _{0))}\ {\frac {\hat {\boldsymbol {\mathfrak {R))))((\mathfrak {R))^{2))}\,\!}$

## 参考文献

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5. 俞允强. 《电动力学简明教程》. 北京大学出版社. 1999: p298.
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