﻿ 麦克斯韦方程组在相对性原理中与洛伦兹变换的关系

# 麦克斯韦方程组在相对性原理中与洛伦兹变换的关系The Relationship between Maxwell Equations and Lorentz Transformation in the Principle of Relativity

Abstract: It is necessary for Maxwell equations to obey the principle of relativity with the help of Lorentz Transformation in two coordinate systems K and K' which are equal weight to each other. When the plane wave (including electromagnetic and mechanical wave) from coordinate systems of relative motion at a constant speed with each other spreads K to K' when the Doppler effect occurs. Accord-ing to the principle of relativity, only with the help of Lorentz transformation can make the x and t of the plane wave equation in the coordinate system K be covariated to the x' and t' of the plane wave equation in the coordinate system K'.

1. 引言

2. 洛伦兹变换下的麦克斯韦方程组

$\frac{\partial {E}_{z}}{\partial y}-\frac{\partial {E}_{y}}{\partial z}=-\frac{\partial {B}_{x}}{\partial t}$ (1)

$\frac{\partial {E}_{x}}{\partial z}-\frac{\partial {E}_{z}}{\partial x}=-\frac{\partial {B}_{y}}{\partial t}$ (2)

$\frac{\partial {E}_{y}}{\partial x}-\frac{\partial {E}_{x}}{\partial y}=-\frac{\partial {B}_{z}}{\partial t}$ (3)

$\frac{\partial {B}_{x}}{\partial x}+\frac{\partial {B}_{y}}{\partial y}+\frac{\partial {B}_{z}}{\partial z}=0$ (4)

$\frac{\partial {H}_{z}}{\partial y}-\frac{\partial {H}_{y}}{\partial z}={J}_{x}+\frac{\partial {D}_{x}}{\partial t}$ (5)

$\frac{\partial {H}_{x}}{\partial z}-\frac{\partial {H}_{z}}{\partial x}={J}_{y}+\frac{\partial {D}_{y}}{\partial t}$ (6)

$\frac{\partial {H}_{y}}{\partial x}-\frac{\partial {H}_{x}}{\partial y}={J}_{z}+\frac{\partial {D}_{z}}{\partial t}$ (7)

$\frac{\partial {D}_{x}}{\partial x}+\frac{\partial {D}_{y}}{\partial y}+\frac{\partial {D}_{z}}{\partial z}=\rho$ (8)

$\frac{\partial {{E}^{\prime }}_{z}}{\partial {y}^{\prime }}-\frac{\partial {{E}^{\prime }}_{y}}{\partial {z}^{\prime }}=-\frac{\partial {{B}^{\prime }}_{x}}{\partial {t}^{\prime }}$ (9)

$\frac{\partial {{E}^{\prime }}_{x}}{\partial {z}^{\prime }}-\frac{\partial {{E}^{\prime }}_{z}}{\partial {x}^{\prime }}=-\frac{\partial {{B}^{\prime }}_{y}}{\partial {t}^{\prime }}$ (10)

$\frac{\partial {{E}^{\prime }}_{y}}{\partial {x}^{\prime }}-\frac{\partial {{E}^{\prime }}_{x}}{\partial {y}^{\prime }}=-\frac{\partial {{B}^{\prime }}_{z}}{\partial {t}^{\prime }}$ (11)

$\frac{\partial {{B}^{\prime }}_{x}}{\partial {x}^{\prime }}+\frac{\partial {{B}^{\prime }}_{y}}{\partial {y}^{\prime }}+\frac{\partial {{B}^{\prime }}_{z}}{\partial {z}^{\prime }}=0$ (12)

$\frac{\partial {{H}^{\prime }}_{z}}{\partial {y}^{\prime }}-\frac{\partial {{H}^{\prime }}_{y}}{\partial {z}^{\prime }}={{J}^{\prime }}_{x}+\frac{\partial {{D}^{\prime }}_{x}}{\partial {t}^{\prime }}$ (13)

$\frac{\partial {{H}^{\prime }}_{x}}{\partial {z}^{\prime }}-\frac{\partial {{H}^{\prime }}_{z}}{\partial {x}^{\prime }}={{J}^{\prime }}_{y}+\frac{\partial {{D}^{\prime }}_{y}}{\partial {t}^{\prime }}$ (14)

$\frac{\partial {{H}^{\prime }}_{y}}{\partial {x}^{\prime }}-\frac{\partial {{H}^{\prime }}_{x}}{\partial {y}^{\prime }}={{J}^{\prime }}_{z}+\frac{\partial {{D}^{\prime }}_{z}}{\partial {t}^{\prime }}$ (15)

$\frac{\partial {{D}^{\prime }}_{x}}{\partial {x}^{\prime }}+\frac{\partial {{D}^{\prime }}_{y}}{\partial {y}^{\prime }}+\frac{\partial {{D}^{\prime }}_{z}}{\partial {z}^{\prime }}={\rho }^{\prime }$ (16)

${x}^{\prime }=\gamma \left(x-vt\right)$ , ${y}^{\prime }=y$ , ${z}^{\prime }=z$ , ${t}^{\prime }=\gamma \left(t-\frac{vx}{{c}^{2}}\right)$ .

$\gamma =\frac{1}{\sqrt{1-\frac{{v}^{2}}{{c}^{2}}}}$

$\frac{\partial }{\partial x}=\gamma \left(\frac{\partial }{\partial {x}^{\prime }}-\frac{v}{{c}^{2}}\frac{\partial }{\partial {t}^{\prime }}\right)$ (17)

$\frac{\partial }{\partial y}=\frac{\partial }{\partial {y}^{\prime }}$ (18)

$\frac{\partial }{\partial z}=\frac{\partial }{\partial {z}^{\prime }}$ (19)

$\frac{\partial }{\partial {t}^{\prime }}=\gamma \left(\frac{\partial }{\partial t}-\frac{\partial }{\partial {x}^{\prime }}\right)$ (20)

$x=\gamma \left({x}^{\prime }+v{t}^{\prime }\right)$ , $y={y}^{\prime }$ , $z={z}^{\prime }$ , $t=\gamma \left({t}^{\prime }+\frac{v{x}^{\prime }}{{c}^{2}}\right)$ .

$\frac{\partial }{\partial {x}^{\prime }}=\gamma \left(\frac{\partial }{\partial x}+\frac{v}{{c}^{2}}\frac{\partial }{\partial t}\right)$ (21)

$\frac{\partial }{\partial {y}^{\prime }}=\frac{\partial }{\partial y}$ (22)

$\frac{\partial }{\partial {z}^{\prime }}=\frac{\partial }{\partial z}$ (23)

$\frac{\partial }{\partial {t}^{\prime }}=\gamma \left(\frac{\partial }{\partial t}+\frac{\partial }{\partial x}\right)$ (24)

$\frac{\partial {E}_{x}}{\partial {z}^{\prime }}-\frac{\partial }{\partial {x}^{\prime }}\gamma \left({E}_{z}+v{B}_{y}\right)=-\frac{\partial }{\partial {t}^{\prime }}\gamma \left({B}_{y}+\frac{v}{{c}^{2}}{E}_{z}\right)$ (25)

${{E}^{\prime }}_{x}={E}_{x}$ ,

${{E}^{\prime }}_{z}=\gamma \left({E}_{z}+v{E}_{y}\right)$ (26)

${{E}^{\prime }}_{y}=\gamma \left({E}_{y}+\frac{v}{{c}^{2}}{E}_{z}\right)$ (27)

$\frac{\partial {{E}^{\prime }}_{x}}{\partial z}-\frac{\partial }{\partial x}\gamma \left({{E}^{\prime }}_{z}+v{{B}^{\prime }}_{y}\right)=-\frac{\partial }{\partial t}\gamma \left({{B}^{\prime }}_{y}+\frac{v}{{c}^{2}}{{E}^{\prime }}_{z}\right)$ (28)

${E}_{x}={{E}^{\prime }}_{x}$ ,

${E}_{z}=\gamma \left({{E}^{\prime }}_{z}-v{{E}^{\prime }}_{y}\right)$ (29)

${E}_{y}=\gamma \left({{E}^{\prime }}_{y}-\frac{v}{{c}^{2}}{{E}^{\prime }}_{z}\right)$ (30)

$\gamma \frac{\partial }{\partial {x}^{\prime }}\left({E}_{y}-v{B}_{z}\right)-\frac{\partial {E}_{x}}{\partial {y}^{\prime }}=-\gamma \frac{\partial }{\partial {t}^{\prime }}\left({B}_{z}+\frac{v}{{c}^{2}}{E}_{y}\right)$ (31)

${{E}^{\prime }}_{x}={E}_{x}$ ,

${{E}^{\prime }}_{y}=\gamma \left({E}_{y}-v{B}_{z}\right)$ (32)

${{B}^{\prime }}_{z}=\gamma \left({B}_{z}-\frac{v}{{c}^{2}}{E}_{y}\right)$ (33)

$\gamma \frac{\partial }{\partial x}\left({{E}^{\prime }}_{y}+v{{B}^{\prime }}_{z}\right)-\frac{\partial {E}_{x}}{\partial {y}^{\prime }}=-\gamma \frac{\partial }{\partial x}\left({{B}^{\prime }}_{z}+\frac{v}{{c}^{2}}{{E}^{\prime }}_{y}\right)$ (34)

${E}_{x}={{E}^{\prime }}_{x}$ ,

${E}_{y}=\gamma \left({{E}^{\prime }}_{y}+v{{B}^{\prime }}_{z}\right)$ (35)

${B}_{z}=\gamma \left({{B}^{\prime }}_{z}+\frac{v}{{c}^{2}}{{E}^{\prime }}_{y}\right)$ (36)

$\frac{\partial {E}_{z}}{\partial {y}^{\prime }}-\frac{\partial {E}_{y}}{\partial {z}^{\prime }}=-\gamma \left(\frac{\partial {B}_{x}}{\partial {t}^{\prime }}-\frac{\partial {B}_{x}}{\partial {x}^{\prime }}\right)$ (37)

$\gamma \left(\frac{\partial {B}_{x}}{\partial {x}^{\prime }}-\frac{v}{{c}^{2}}\frac{\partial {B}_{x}}{\partial {t}^{\prime }}\right)+\frac{\partial {B}_{y}}{\partial {y}^{\prime }}+\frac{\partial {B}_{z}}{\partial {z}^{\prime }}=0$ (38)

(39)与(9)之间，(38)与(12)之间可以进行方程对比和场量协变关系设定。根据文献 [1]，把(29)、(35)代入(37)，得到

$\frac{\partial {B}_{x}}{\partial {x}^{\prime }}+\frac{\partial {{B}^{\prime }}_{y}}{\partial {y}^{\prime }}+\frac{\partial {{B}^{\prime }}_{z}}{\partial {z}^{\prime }}=\frac{1}{v}\left(\frac{\partial {{E}^{\prime }}_{z}}{\partial {y}^{\prime }}-\frac{\partial {{E}^{\prime }}_{y}}{\partial {z}^{\prime }}+\frac{\partial {B}_{x}}{\partial {t}^{\prime }}\right)$ (39)

$\frac{\partial {B}_{x}}{\partial {x}^{\prime }}+\frac{\partial {{B}^{\prime }}_{y}}{\partial {y}^{\prime }}+\frac{\partial {{B}^{\prime }}_{z}}{\partial {z}^{\prime }}=\frac{v}{{c}^{2}}\left(\frac{\partial {{E}^{\prime }}_{z}}{\partial {y}^{\prime }}-\frac{\partial {{E}^{\prime }}_{y}}{\partial {z}^{\prime }}+\frac{\partial {B}_{x}}{\partial {t}^{\prime }}\right)$ (40)

$\frac{\partial {B}_{x}}{\partial {x}^{\prime }}+\frac{\partial {{B}^{\prime }}_{y}}{\partial {y}^{\prime }}+\frac{\partial {{B}^{\prime }}_{z}}{\partial {z}^{\prime }}=0$ (41)

$\frac{\partial {{E}^{\prime }}_{z}}{\partial {y}^{\prime }}-\frac{\partial {{E}^{\prime }}_{y}}{\partial {z}^{\prime }}=-\frac{\partial {B}_{x}}{\partial {t}^{\prime }}$ (42)

${B}_{x}={{B}^{\prime }}_{x}$

${{E}^{\prime }}_{y}={E}_{y}-v{B}_{z}$ , ${{B}^{\prime }}_{z}={B}_{z}$ ;

${{B}^{\prime }}_{y}={B}_{y}$ , ${{E}^{\prime }}_{z}={E}_{z}-v{B}_{y}$ ;

3. 洛伦兹变换下的平面波波动方程

$\frac{{\partial }^{2}}{\partial {x}^{2}}=\frac{1}{{c}^{2}}\frac{{\partial }^{2}}{\partial {t}^{2}}$ (43)

$\frac{{\partial }^{2}}{\partial {x}^{2}}={\gamma }^{2}{\left(\frac{\partial }{\partial {x}^{\prime }}-\frac{v}{{c}^{2}}\frac{\partial }{\partial {t}^{\prime }}\right)}^{2}={\gamma }^{2}\frac{{\partial }^{2}}{\partial {{x}^{\prime }}^{2}}-{\gamma }^{2}\frac{2v}{{c}^{2}}\frac{\partial }{\partial {x}^{\prime }}\frac{\partial }{\partial {t}^{\prime }}+{\gamma }^{2}\frac{{v}^{2}}{{c}^{4}}\frac{{\partial }^{2}}{\partial {{t}^{\prime }}^{2}}$

$\frac{1}{{c}^{2}}\frac{{\partial }^{2}}{\partial {t}^{2}}={\gamma }^{2}\frac{1}{{c}^{2}}{\left(\frac{\partial }{\partial {t}^{\prime }}-v\frac{\partial }{\partial {x}^{\prime }}\right)}^{2}={\gamma }^{2}\frac{1}{{c}^{2}}\frac{{\partial }^{2}}{\partial {{t}^{\prime }}^{2}}-{\gamma }^{2}\frac{2v}{{c}^{2}}\frac{\partial }{\partial {x}^{\prime }}\frac{\partial }{\partial {t}^{\prime }}+{\gamma }^{2}\frac{{v}^{2}}{{c}^{2}}\frac{{\partial }^{2}}{\partial {{t}^{\prime }}^{2}}$

${\gamma }^{2}\left(1-\frac{{v}^{2}}{{c}^{2}}\right)\frac{{\partial }^{2}}{\partial {{x}^{\prime }}^{2}}={\gamma }^{2}\left(1-\frac{{v}^{2}}{{c}^{2}}\right)\frac{{\partial }^{2}}{\partial {{t}^{\prime }}^{2}}$

$\frac{{\partial }^{2}}{\partial {{x}^{\prime }}^{2}}=\frac{1}{{c}^{2}}\frac{{\partial }^{2}}{\partial {{t}^{\prime }}^{2}}$ (44)

(44)就是平面波在坐标系K'中的平面波波动方程。

$\frac{{\partial }^{2}{E}_{y\left(z\right)}}{\partial {x}^{2}}=\frac{1}{{c}^{2}}\frac{\partial {E}_{y\left(z\right)}}{\partial {t}^{2}}$ , $\frac{{\partial }^{2}{E}_{z\left(y\right)}}{\partial {x}^{2}}=\frac{1}{{c}^{2}}\frac{\partial {E}_{z\left(y\right)}}{\partial {t}^{2}}$ .

$\frac{{\partial }^{2}{{E}^{\prime }}_{y\left(z\right)}}{\partial {x}^{2}}=\frac{1}{{c}^{2}}\frac{\partial {{E}^{\prime }}_{y\left(z\right)}}{\partial {t}^{2}}$ , $\frac{{\partial }^{2}{{E}^{\prime }}_{z\left(y\right)}}{\partial {x}^{2}}=\frac{1}{{c}^{2}}\frac{\partial {{E}^{\prime }}_{z\left(y\right)}}{\partial {t}^{2}}$ .

${v}^{\prime }=v\sqrt{\frac{c±v}{c\mp v}}$

${x}^{\prime }=\gamma \left(x±vt\right)$ (45)

${t}^{\prime }=\gamma \left(t±\frac{vx}{{c}^{2}}\right)$ (46)

$x=ct$ (47)

${x}^{\prime }=x\sqrt{\frac{c\mp v}{c±v}}$

${t}^{\prime }=t\sqrt{\frac{c\mp v}{c±v}}$

(48)和(49)与洛伦兹变换之间是可相互转换的，在平面波波动方程的坐标系协变中是完全等价的。

4. 结论

[1] (英) W.G.V.罗瑟. 相对论导论[M]. 北京: 科学出版社, 1980.

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