# 杰斐缅柯方程

## 在真空内的电磁场

${\displaystyle \mathbf {E} (\mathbf {r} ,\,t)={\frac {1}{4\pi \epsilon _{0))}\int _((\mathcal {V))'}\left[\rho (\mathbf {r} ',\,t_{r}){\frac {\mathbf {r} -\mathbf {r} '}{|\mathbf {r} -\mathbf {r} '|^{3))}+{\frac ((\dot {\rho ))(\mathbf {r} ',\,t_{r})}{c)){\frac {\mathbf {r} -\mathbf {r} '}{|\mathbf {r} -\mathbf {r} '|^{2))}-{\frac ((\dot {\mathbf {J} ))(\mathbf {r} ',\,t_{r})}{c^{2}|\mathbf {r} -\mathbf {r} '|))\right]d^{3}\mathbf {r} '}$
${\displaystyle \mathbf {B} (\mathbf {r} ,t)={\frac {\mu _{0)){4\pi ))\int _((\mathcal {V))'}\left[{\frac {\mathbf {J} (\mathbf {r} ',\,t_{r})}{|\mathbf {r} -\mathbf {r} '|^{3))}+{\frac ((\dot {\mathbf {J} ))(\mathbf {r} ',\,t_{r})}{c|\mathbf {r} -\mathbf {r} '|^{2))}\right]\times (\mathbf {r} -\mathbf {r} ')\ d^{3}\mathbf {r} '}$ ;

### 推导

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

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

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

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

${\displaystyle \nabla \Phi (\mathbf {r} ,\,t)={\frac {1}{4\pi \epsilon _{0))}\int _((\mathcal {V))'}\nabla \left({\frac {\rho (\mathbf {r} ',\,t_{r})}{\mathfrak {R))}\right)\,d^{3}\mathbf {r} '={\frac {1}{4\pi \epsilon _{0))}\int _((\mathcal {V))'}\left[{\frac {\nabla \rho (\mathbf {r} ',\,t_{r})}{\mathfrak {R))}+\rho (\mathbf {r} ',\,t_{r})\nabla \left({\frac {1}{\mathfrak {R))}\right)\right]\,d^{3}\mathbf {r} '}$

{\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))}

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

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

${\displaystyle \nabla \Phi (\mathbf {r} ,\,t)={\frac {1}{4\pi \epsilon _{0))}\int _((\mathcal {V))'}\left[-{\frac ((\dot {\rho ))(\mathbf {r} ',\,t_{r})}{c)){\frac {\hat {\boldsymbol {\mathfrak {R)))){\mathfrak {R))}-\rho (\mathbf {r} ',\,t_{r})\left({\frac {\hat {\boldsymbol {\mathfrak {R))))((\mathfrak {R))^{2))}\right)\right]\,d^{3}\mathbf {r} '}$

${\displaystyle {\frac {\partial \mathbf {A} (\mathbf {r} ,\,t)}{\partial t))={\frac {\mu _{0)){4\pi ))\int _((\mathcal {V))'}{\frac ((\dot {\mathbf {J} ))(\mathbf {r} ',\,t_{r})}{|\mathbf {r} -\mathbf {r} '|))\,d^{3}\mathbf {r} '={\frac {1}{4\pi \epsilon _{0}c^{2))}\int _((\mathcal {V))'}{\frac ((\dot {\mathbf {J} ))(\mathbf {r} ',\,t_{r})}{|\mathbf {r} -\mathbf {r} '|))\,d^{3}\mathbf {r} '}$

## 参考文献

1. ^ McDonald, Kirk T., 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 Authors list列表缺少|last1= (帮助)
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