# 黎納-維謝勢

## 物理理論

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

### 表達方程式

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

$\Phi(\mathbf{r},\,t) = \frac{1}{4\pi\epsilon_0}\ \frac{q c}{\mathfrak{R} c - \boldsymbol{\mathfrak{R}}\cdot \mathbf{v}}\,\!$
$\mathbf{A}(\mathbf{r},\,t) =\frac{\mathbf{v}}{c^2}\Phi(\mathbf{r},\,t) \,\!$

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

### 推導

$\Phi(\mathbf{r},\,t)\ \stackrel{def}{=}\ \frac{1}{4\pi\epsilon_0}\int_{\mathcal{V}'} \frac{\rho(\mathbf{r}' ,\, t_r)}{\mathfrak{R}}\, d^3\mathbf{r}'\,\!$
$\mathbf{A}(\mathbf{r},\,t)\ \stackrel{def}{=}\ \frac{\mu_0}{4\pi}\int_{\mathcal{V}'} \frac{\mathbf{J}(\mathbf{r}',\,t_r)}{\mathfrak{R}}\, d^3\mathbf{r}'\,\!$

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

$\Phi(\mathbf{r},\,t)=\frac{q}{4\pi\epsilon_0}\int_{\mathcal{V}'} \frac{\delta(\mathbf{r}' - \mathbf{w}(t_r))}{\mathfrak{R}}\, d^3\mathbf{r}'\,\!$

$\Phi(\mathbf{r},\,t)=\frac{q}{4\pi\epsilon_0\mathfrak{R}}\int_{\mathcal{V}'} \delta(\mathbf{r}' - \mathbf{w}(t_r))\, d^3\mathbf{r}'\,\!$

$\mathfrak{J}=\cfrac{\partial \boldsymbol{\eta}}{\partial \mathbf{r}'} =\begin{vmatrix} \cfrac{\partial \eta_x}{\partial x'} & \cfrac{\partial \eta_x}{\partial y'} & \cfrac{\partial \eta_x}{\partial z'} \\ \cfrac{\partial \eta_y}{\partial x'} & \cfrac{\partial \eta_y}{\partial y'} & \cfrac{\partial \eta_y}{\partial z'} \\ \cfrac{\partial \eta_z}{\partial x'} & \cfrac{\partial \eta_z}{\partial y'} & \cfrac{\partial \eta_z}{\partial z'} \\ \end{vmatrix}\,\!$

$\cfrac{\partial \eta_x}{\partial x'}=1 - \cfrac{\partial w_x}{\partial x'}=1 - \cfrac{\partial w_x}{\partial t_r}\ \cfrac{\partial t_r}{\partial x'}=1 - v_x\cfrac{\partial t_r}{\partial x'}\,\!$
$\cfrac{\partial \eta_y}{\partial x'}=\cfrac{\partial w_y}{\partial x'}=\cfrac{\partial w_y}{\partial t_r}\ \cfrac{\partial t_r}{\partial x'}=v_y\cfrac{\partial t_r}{\partial x'}\,\!$

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

$\Phi(\mathbf{r},\,t)=\frac{q}{4\pi\epsilon_0\mathfrak{R}}\int_{\mathcal{V}'} \delta(\boldsymbol{\eta})\cfrac{\partial \mathbf{r}'}{\partial \boldsymbol{\eta}}\, d^3\boldsymbol{\eta} =\frac{q}{4\pi\epsilon_0\mathfrak{R}}\int_{\mathcal{V}'} \cfrac{\delta(\boldsymbol{\eta})}{\mathfrak{J}}\, d^3\boldsymbol{\eta} =\frac{q}{4\pi\epsilon_0\mathfrak{R}}\int_{\mathcal{V}'} \cfrac{\delta(\boldsymbol{\eta})}{1 - \hat{\boldsymbol{\mathfrak{R}}}\cdot\mathbf{v}/c}\, d^3\boldsymbol{\eta} \,\!$

$\Phi(\mathbf{r},\,t)= \frac{1}{4\pi\epsilon_0}\ \frac{q c}{\mathfrak{R} c - \boldsymbol{\mathfrak{R}}\cdot \mathbf{v}}\,\!$

#### 物理意義

$\Phi(\mathbf{r},\,t)= \frac{1}{4\pi\epsilon_0}\ \frac{q}{\mathfrak{R}}\,\!$

### 移動中的帶電粒子的電磁場

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

$\mathbf{E}(\mathbf{r},\,t)= \frac{q}{4\pi\epsilon_0}\ \cfrac{\mathfrak{R}}{(\boldsymbol{\mathfrak{R}}\cdot \mathbf{u})^3} [(c^2 - v^2)\mathbf{u}+\boldsymbol{\mathfrak{R}}\times(\mathbf{u}\times\mathbf{a})]\,\!$
$\mathbf{B}(\mathbf{r},\,t)= \frac{1}{c}\hat{\boldsymbol{\mathfrak{R}}}\times\mathbf{E}(\mathbf{r},\,t)\,\!$ ;

$\mathbf{E} = \frac{q}{4\pi\epsilon_0 }\ \frac{\hat{\boldsymbol{\mathfrak{R}}}}{\mathfrak{R}^2}\,\!$

## 參考文獻

1. ^ Marc Jouguet, La vie et l'oeuvre scientifique de Alfred-Marie Liénard, Exposé fait en séance mensuelle de la Société française des Electriciens, le 4 décembre, 1958
2. ^ Mulligan, Joseph F., Emil Wiechert (1861–1928): Esteemed seismologist, forgotten physicist, American Journal of Physics, March, 69 (3): pp. 277–287
3. ^ Ribarič, Marijan; Šušteršič, Luka, Expansion in terms of time-dependent, moving charges and currents, SIAM Journal on Applied Mathematics, June, 55 (3): pp. 593–624, doi:10.1137/S0036139992241972
4. ^ Griffiths, David; Heald, Mark, Time-Dependent Generalization of the Biot-Savart and Coulomb laws, American Journal of Physics, Feb., 59 (2): pp. 111–117
5. ^ Griffiths, David J. Introduction to Electrodynamics (3rd ed.). Prentice Hall. 1998: pp. 435–440. ISBN 0-13-805326-X.