# 推遲勢

## 理論概念

$\Phi(\mathbf{r})\ \stackrel{def}{=}\ \frac{1}{4\pi\epsilon_0}\int_{\mathbb{V}'} \frac{\rho(\mathbf{r}')}{|\mathbf{r} - \mathbf{r}'|}\, d^3\mathbf{r}'$
$\mathbf{A}(\mathbf{r})\ \stackrel{def}{=}\ \frac{\mu_0}{4\pi}\int_{\mathbb{V}'} \frac{\mathbf{J}(\mathbf{r}')}{|\mathbf{r} - \mathbf{r}'|}\, d^3\mathbf{r}'$

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

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

## 非齊次的電磁波方程式

$\nabla^2\Phi(\mathbf{r},\,t) - {1 \over c^2} {\partial^2 \Phi(\mathbf{r},\,t) \over \partial t^2} = - {\rho(\mathbf{r},\,t) \over \epsilon_0}$
$\nabla^2 \mathbf{A}(\mathbf{r},\,t) - {1 \over c^2} {\partial^2 \mathbf{A}(\mathbf{r},\,t) \over \partial t^2} = - \mu_0 \mathbf{J}(\mathbf{r},\,t)$

$\boldsymbol{\mathfrak{R}}=\mathbf{r} - \mathbf{r}'$

$\nabla\Phi(\mathbf{r},\,t)= \frac{1}{4\pi\epsilon_0}\int_{\mathbb{V}'} \nabla\left(\frac{\rho(\mathbf{r}' ,\, t_r)}{\mathfrak{R}}\right)\, d^3\mathbf{r}'= \frac{1}{4\pi\epsilon_0}\int_{\mathbb{V}'} \left[\frac{\nabla\rho(\mathbf{r}' ,\, t_r)}{\mathfrak{R}}+\rho(\mathbf{r}' ,\, t_r)\nabla\left(\frac{1}{\mathfrak{R}}\right)\right]\, d^3\mathbf{r}'$

\begin{align} 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{align}

$\frac{\partial\rho(\mathbf{r}' ,\, t)}{\partial t}=\frac{\partial t_r}{\partial t}\ \frac{\partial\rho(\mathbf{r}' ,\, t_r)}{\partial t_r}=\frac{\partial\rho(\mathbf{r}' ,\, t_r)}{\partial t_r}$
$\nabla \mathfrak{R}=\hat{\boldsymbol{\mathfrak{R}}}$

$\nabla\rho(\mathbf{r}' ,\, t_r)= - \frac{1}{c}\ \frac{\partial\rho(\mathbf{r}' ,\, t_r)}{\partial t_r}\nabla \mathfrak{R}= - \frac{1}{c}\ \frac{\partial\rho(\mathbf{r}' ,\, t_r)}{\partial t}\hat{\boldsymbol{\mathfrak{R}}}= - \frac{\dot{\rho}(\mathbf{r}' ,\, t_r)}{c}\hat{\boldsymbol{\mathfrak{R}}}$

$\nabla\Phi(\mathbf{r},\,t)= \frac{1}{4\pi\epsilon_0}\int_{\mathbb{V}'} \left[ - \frac{\dot{\rho}(\mathbf{r}' ,\, t_r)}{c}\frac{\hat{\boldsymbol{\mathfrak{R}}}}{\mathfrak{R}} - \rho(\mathbf{r}' ,\, t_r) \left(\frac{\hat{\boldsymbol{\mathfrak{R}}}}{\mathfrak{R}^2}\right)\right]\, d^3\mathbf{r}'$

\begin{align} \nabla^2\Phi(\mathbf{r},\,t) & = \frac{1}{4\pi\epsilon_0}\int_{\mathbb{V}'} \left[ - \frac{\nabla\dot{\rho}(\mathbf{r}' ,\, t_r)}{c}\cdot\frac{\hat{\boldsymbol{\mathfrak{R}}}}{\mathfrak{R}} - \frac{\dot{\rho}(\mathbf{r}' ,\, t_r)}{c}\nabla\cdot\left(\frac{\hat{\boldsymbol{\mathfrak{R}}}}{\mathfrak{R}}\right) - [\nabla\rho(\mathbf{r}' ,\, t_r)] \cdot\left(\frac{\hat{\boldsymbol{\mathfrak{R}}}}{\mathfrak{R}^2}\right) - \rho(\mathbf{r}' ,\, t_r) \nabla\cdot\left(\frac{\hat{\boldsymbol{\mathfrak{R}}}}{\mathfrak{R}^2}\right)\right] \, d^3\mathbf{r} \\ & = \frac{1}{4\pi\epsilon_0}\int_{\mathbb{V}'} \left[ - \frac{\ddot{\rho}(\mathbf{r}' ,\, t_r)}{c^2 \mathfrak{R}} - \frac{\dot{\rho}(\mathbf{r}' ,\, t_r)}{c \mathfrak{R}^2} + \frac{\dot{\rho}(\mathbf{r}' ,\, t_r)}{c \mathfrak{R}^2} - 4\pi\rho(\mathbf{r}' ,\, t_r) \delta^3(\boldsymbol{\mathfrak{R}})\right]\, d^3\mathbf{r}' \\ & = - \frac{1}{c^2}\frac{\partial^2}{\partial t^2}\left[\frac{1}{4\pi\epsilon_0}\int_{\mathbb{V}'} \frac{\rho(\mathbf{r}' ,\, t_r)}{\mathfrak{R}}\, d^3\mathbf{r}' \right] - \frac{\rho(\mathbf{r},\, t)}{\epsilon_0} \\ \end{align}

$\nabla^2\Phi(\mathbf{r},\,t)+ \frac{1}{c^2}\frac{\partial^2 \Phi(\mathbf{r},\,t)}{\partial t^2}= - \frac{\rho(\mathbf{r},\, t)}{\epsilon_0}$

## 勞侖次規範條件

$\nabla \cdot \mathbf{A} +{1 \over c^2} {{\partial \Phi } \over {\partial t}}= 0$

## 廣義的含時電磁場

$\mathbf{E}= - \nabla\Phi - \frac{\partial\mathbf{A}}{\partial t}$
$\mathbf{B}=\nabla\times\mathbf{A}$

$\mathbf{E}(\mathbf{r},\,t) = \frac{1}{4\pi\epsilon_0}\int_{\mathbb{V}'} \left[\rho(\mathbf{r}',\,t_r)\frac{\mathbf{r} - \mathbf{r}'}{|\mathbf{r} - \mathbf{r}'|^3} +\frac{\dot{\rho}(\mathbf{r}',\,t_r)}{c}\frac{\mathbf{r} - \mathbf{r}'}{ |\mathbf{r} - \mathbf{r}'|^2} - \frac{\dot{\mathbf{J}}(\mathbf{r}',\,t_r)}{c^2 |\mathbf{r} - \mathbf{r}'|}\right] d^3\mathbf{r}'$
$\mathbf{B}(\mathbf{r},t) = \frac{\mu_0}{4\pi}\int_{\mathbb{V}'} \left[\frac{\mathbf{J}(\mathbf{r}',\,t_r)}{|\mathbf{r} - \mathbf{r}'|^3} +\frac{\dot{\mathbf{J}}(\mathbf{r}',\,t_r)}{c |\mathbf{r} - \mathbf{r}'|^2}\right]\times(\mathbf{r} - \mathbf{r}')\ d^3\mathbf{r}'$

## 超前勢

$t_a\ \stackrel{def}{=}\ t + \frac{|\mathbf{r} - \mathbf{r}'|}{c}$

$\Phi_a(\mathbf{r},\,t)= \frac{1}{4\pi\epsilon_0}\int_{\mathbb{V}'} \frac{\rho(\mathbf{r}' ,\, t_a)}{|\mathbf{r} - \mathbf{r}'|}\, d^3\mathbf{r}'$
$\mathbf{A}_a(\mathbf{r},\,t)= \frac{\mu_0}{4\pi}\int_{\mathbb{V}'} \frac{\mathbf{J}(\mathbf{r}',\,t_a)}{|\mathbf{r} - \mathbf{r}'|}\, d^3\mathbf{r}'$

## 參考文獻

1. ^ 1.0 1.1 Griffiths, David J. Introduction to Electrodynamics (3rd ed.). Prentice Hall. 1998: pp. 422–428. ISBN 0-13-805326-X.