傑斐緬柯方程式

在真空內的電磁場

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

推導

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

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

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

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

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

\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}

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

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

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

$\frac{\partial \mathbf{A}(\mathbf{r},\,t)}{\partial t}=\frac{\mu_0}{4\pi}\int_{\mathcal{V}'} \frac{ \dot{\mathbf{J}}(\mathbf{r}',\,t_r)}{|\mathbf{r} - \mathbf{r}'|}\, d^3\mathbf{r}' =\frac{1}{4\pi\epsilon_0 c^2}\int_{\mathcal{V}'} \frac{ \dot{\mathbf{J}}(\mathbf{r}',\,t_r)}{|\mathbf{r} - \mathbf{r}'|}\, d^3\mathbf{r}'$

電場和磁場的因果關係

——歐雷格·傑斐緬柯，《Causality Electromagnetic Induction and Gravitation, 第 16 頁》

參考文獻

1. ^ The relation between expressions for time-dependent electromagnetic fields given by Jefimenko and by Panofsky and Phillips, American Journal of Physics, 1997, 65 (11): pp. 1074–1076
2. ^ Jefimenko, Oleg D., Electricity and magnetism: an introduction to the theory of electric and magnetic fields 2nd, Electret Scientific Co., 1989, ISBN 9780917406089
3. ^ Griffiths, David J. Introduction to Electrodynamics (3rd ed.). Prentice Hall. 1998. ISBN 0-13-805326-X.
4. ^ Oleg D. Jefimenko, Solutions of Maxwell's equations for electric and magnetic fields in arbitrary media, American Journal of Physics 60 (10) (1992), 899-902.
5. ^ Jefimenko, Oleg D., Causality Electromagnetic Induction and Gravitation 2nd, Electret Scientific, pp. 16, 2000, ISBN 0-917406-23-0