# 非齐次的电磁波方程

## 国际单位

${\displaystyle \nabla ^{2}\varphi +{{\partial } \over \partial t}\left(\nabla \cdot \mathbf {A} \right)=-{\rho \over \varepsilon _{0}}}$
${\displaystyle \nabla ^{2}\mathbf {A} -{1 \over c^{2}}{\partial ^{2}\mathbf {A} \over \partial t^{2}}-\nabla \left({1 \over c^{2}}{{\partial \varphi } \over {\partial t}}+\nabla \cdot \mathbf {A} \right)=-\mu _{0}\mathbf {J} }$

${\displaystyle \mathbf {E} =-\nabla \varphi -{\partial \mathbf {A} \over \partial t}}$

${\displaystyle \mathbf {B} =\nabla \times \mathbf {A} }$ .

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

${\displaystyle \nabla ^{2}\varphi -{1 \over c^{2}}{\partial ^{2}\varphi \over \partial t^{2}}=-{\rho \over \varepsilon _{0}}}$
${\displaystyle \nabla ^{2}\mathbf {A} -{1 \over c^{2}}{\partial ^{2}\mathbf {A} \over \partial t^{2}}=-\mu _{0}\mathbf {J} }$  .

## 厘米-克-秒单位和洛伦兹-赫维赛德单位

${\displaystyle \nabla ^{2}\varphi -{1 \over c^{2}}{\partial ^{2}\varphi \over \partial t^{2}}=-{4\pi \rho }}$
${\displaystyle \nabla ^{2}\mathbf {A} -{1 \over c^{2}}{\partial ^{2}\mathbf {A} \over \partial t^{2}}=-{4\pi \over c}\mathbf {J} }$

${\displaystyle \mathbf {E} =-\nabla \varphi -{1 \over c}{\partial \mathbf {A} \over \partial t}}$
${\displaystyle \mathbf {B} =\nabla \times \mathbf {A} }$

${\displaystyle {1 \over c}{{\partial \varphi } \over {\partial t}}+\nabla \cdot \mathbf {A} =0}$ .

${\displaystyle \rho \rightarrow {\rho \over {4\pi }}}$
${\displaystyle \mathbf {J} \rightarrow {1 \over {4\pi }}\mathbf {J} }$ .

## 非齐次波方程的协变形式

${\displaystyle \Box A^{\mu }\ {\stackrel {\mathrm {def} }{=}}\ \partial _{\beta }\partial ^{\beta }A^{\mu }\ {\stackrel {\mathrm {def} }{=}}\ {A^{\mu ,\beta }}_{\beta }=-\mu _{0}J^{\mu }}$  （国际单位制）
${\displaystyle \Box A^{\mu }\ {\stackrel {\mathrm {def} }{=}}\ \partial _{\beta }\partial ^{\beta }A^{\mu }\ {\stackrel {\mathrm {def} }{=}}\ {A^{\mu ,\beta }}_{\beta }=-{\frac {4\pi }{c}}J^{\mu }}$

（厘米-克-秒制）

${\displaystyle J^{\mu }=\left(c\rho ,\mathbf {J} \right)}$ ,
${\displaystyle {\partial \over {\partial x^{a}}}\ {\stackrel {\mathrm {def} }{=}}\ \partial _{a}\ {\stackrel {\mathrm {def} }{=}}\ {}_{,a}\ {\stackrel {\mathrm {def} }{=}}\ (\partial /\partial ct,\nabla )}$

${\displaystyle A^{\mu }=(\varphi ,\mathbf {A} c)}$  （国际单位制）
${\displaystyle A^{\mu }=(\varphi ,\mathbf {A} )}$  （厘米-克-秒制）

${\displaystyle \partial _{\mu }A^{\mu }=0}$ .

${\displaystyle \Box =\partial _{\beta }\partial ^{\beta }=\nabla ^{2}-{1 \over c^{2}}{\frac {\partial ^{2}}{\partial t^{2}}}}$ 达朗贝尔算符

## 弯曲时空

${\displaystyle -{A^{\alpha ;\beta }}_{\beta }+{R^{\alpha }}_{\beta }A^{\beta }=\mu _{0}J^{\alpha }}$

${\displaystyle {R^{\alpha }}_{\beta }}$

${\displaystyle {A^{\mu }}_{;\mu }=0}$

## 非齐次电磁波方程的解

${\displaystyle \varphi (\mathbf {r} ,t)=\int {{\delta \left(t'+{{\left|\mathbf {r} -\mathbf {r} '\right|} \over c}-t\right)} \over {\left|\mathbf {r} -\mathbf {r} '\right|}}\rho (\mathbf {r} ',t')d^{3}r'dt'}$

${\displaystyle \mathbf {A} (\mathbf {r} ,t)=\int {{\delta \left(t'+{{\left|\mathbf {r} -\mathbf {r} '\right|} \over c}-t\right)} \over {\left|\mathbf {r} -\mathbf {r} '\right|}}{\mathbf {J} (\mathbf {r} ',t') \over c}d^{3}r'dt'}$

${\displaystyle {\delta \left(t'+{{\left|\mathbf {r} -\mathbf {r} '\right|} \over c}-t\right)}}$

${\displaystyle \rho \rightarrow {\rho \over {4\pi \varepsilon _{0}}}}$
${\displaystyle \mathbf {J} \rightarrow {\mu _{0} \over {4\pi }}\mathbf {J} }$ .

${\displaystyle \rho \rightarrow {\rho \over {4\pi }}}$
${\displaystyle \mathbf {J} \rightarrow {1 \over {4\pi }}\mathbf {J} }$ .

${\displaystyle \varphi (\mathbf {r} ,t)=\int {{\delta \left(t'-{{\left|\mathbf {r} -\mathbf {r} '\right|} \over c}-t\right)} \over {\left|\mathbf {r} -\mathbf {r} '\right|}}\rho (\mathbf {r} ',t')d^{3}r'dt'}$

${\displaystyle \mathbf {A} (\mathbf {r} ,t)=\int {{\delta \left(t'-{{\left|\mathbf {r} -\mathbf {r} '\right|} \over c}-t\right)} \over {\left|\mathbf {r} -\mathbf {r} '\right|}}{\mathbf {J} (\mathbf {r} ',t') \over c}d^{3}r'dt'}$ .

## 参考文献

### 电磁学

#### 期刊论文

• James Clerk Maxwell, "A Dynamical Theory of the Electromagnetic Field", Philosophical Transactions of the Royal Society of London 155, 459-512 (1865). (This article accompanied a December 8, 1864 presentation by Maxwell to the Royal Society.)

#### 本科水平教科书

• Griffiths, David J. Introduction to Electrodynamics (3rd ed.). Prentice Hall. 1998. ISBN 0-13-805326-X.
• Tipler, Paul. Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics (5th ed.). W. H. Freeman. 2004. ISBN 0-7167-0810-8.
• Edward M. Purcell, Electricity and Magnetism (McGraw-Hill, New York, 1985).
• Hermann A. Haus and James R. Melcher, Electromagnetic Fields and Energy (Prentice-Hall, 1989) ISBN 0-13-249020-X
• Banesh Hoffman, Relativity and Its Roots (Freeman, New York, 1983).
• David H. Staelin, Ann W. Morgenthaler, and Jin Au Kong, Electromagnetic Waves (Prentice-Hall, 1994) ISBN 0-13-225871-4
• Charles F. Stevens, The Six Core Theories of Modern Physics, (MIT Press, 1995) ISBN 0-262-69188-4.

### 矢量微积分

• H. M. Schey, Div Grad Curl and all that: An informal text on vector calculus, 4th edition (W. W. Norton & Company, 2005) ISBN 0-393-92516-1.