HOME ＞ 特殊相対性理論 ＞ ガリレイの相対性原理 ＞マクスウェル方程式のローレンツ変換
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本ページでは…
本ページでは、マクスウェル方程式の形がローレンツ変換の下で不変であることを見る。
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前ページまで⋯
前ページでは、局所時間とローレンツ収縮からローレンツ変換を導き、ローレンツ変換で速度や加速度、さらには微分演算子がどのように変換されるかを確認した。
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内容
マクスウェル方程式のローレンツ変換
本ページでは、マクスウェル方程式の形はローレンツ変換
\begin{align}\frac{\partial}{\partial x}&=\gamma\left(\frac{\partial}{\partial x’}-\frac{v}{c^2}\frac{\partial}{\partial t’}\right)\\\frac{\partial}{\partial y}&=\frac{\partial}{\partial y’}\\\frac{\partial}{\partial z}&=\frac{\partial}{\partial z’}\\\frac{\partial}{\partial t}&=\gamma\left(\frac{\partial}{\partial t’}-v\frac{\partial}{\partial x’}\right)
\end{align}
\end{align}
のみを行なっても不変とならないが、電場·磁場を次のように変換
\begin{align*}E_x&=E’_x\\E_y&=\gamma(E’_y+v\mu_0H’_z)\\E_z&=\gamma(E’_z-v\mu_0H’_y)\\H_x&=H’_x\\H_y&=\gamma(H’_y-v\varepsilon_0E’_z)\\H_z&=\gamma(H’_z+v\varepsilon_0 E’_y)\end{align*}
すれば不変となることを見る。
成分ごとに表したマクスウェル方程式(以前のページを参照)
\begin{align}
\frac{\partial H_x}{\partial x}
+\frac{\partial H_y}{\partial y}
+\frac{\partial H_z}{\partial z}
&=0\tag{A}\\
\frac{\partial E_x}{\partial x}
+\frac{\partial E_y}{\partial y}
+\frac{\partial E_z}{\partial z}
&=0\tag{B}\\
\frac{\partial E_z}{\partial y}
-\frac{\partial E_y}{\partial z}
&=-\mu_0\frac{\partial H_x}{\partial t}\tag{C}\\
\frac{\partial E_x}{\partial z}
-\frac{\partial E_z}{\partial x}
&=-\mu_0\frac{\partial H_y}{\partial t}\tag{D}\\
\frac{\partial E_y}{\partial x}
-\frac{\partial E_x}{\partial y}
&=-\mu_0\frac{\partial H_z}{\partial t}\tag{E}\\
\frac{\partial H_z}{\partial y}
-\frac{\partial H_y}{\partial z}
&=\varepsilon_0\frac{\partial E_x}{\partial t}\tag{F}\\
\frac{\partial H_x}{\partial z}
-\frac{\partial H_z}{\partial x}
&=\varepsilon_0\frac{\partial E_y}{\partial t}\tag{G}\\
\frac{\partial H_y}{\partial x}
-\frac{\partial H_x}{\partial y}
&=\varepsilon_0\frac{\partial E_z}{\partial t}\tag{H}
\end{align}
\frac{\partial H_x}{\partial x}
+\frac{\partial H_y}{\partial y}
+\frac{\partial H_z}{\partial z}
&=0\tag{A}\\
\frac{\partial E_x}{\partial x}
+\frac{\partial E_y}{\partial y}
+\frac{\partial E_z}{\partial z}
&=0\tag{B}\\
\frac{\partial E_z}{\partial y}
-\frac{\partial E_y}{\partial z}
&=-\mu_0\frac{\partial H_x}{\partial t}\tag{C}\\
\frac{\partial E_x}{\partial z}
-\frac{\partial E_z}{\partial x}
&=-\mu_0\frac{\partial H_y}{\partial t}\tag{D}\\
\frac{\partial E_y}{\partial x}
-\frac{\partial E_x}{\partial y}
&=-\mu_0\frac{\partial H_z}{\partial t}\tag{E}\\
\frac{\partial H_z}{\partial y}
-\frac{\partial H_y}{\partial z}
&=\varepsilon_0\frac{\partial E_x}{\partial t}\tag{F}\\
\frac{\partial H_x}{\partial z}
-\frac{\partial H_z}{\partial x}
&=\varepsilon_0\frac{\partial E_y}{\partial t}\tag{G}\\
\frac{\partial H_y}{\partial x}
-\frac{\partial H_x}{\partial y}
&=\varepsilon_0\frac{\partial E_z}{\partial t}\tag{H}
\end{align}
に、ローレンツ変換
\begin{align}\frac{\partial}{\partial x}&=\gamma\left(\frac{\partial}{\partial x’}-\frac{v}{c^2}\frac{\partial}{\partial t’}\right)\\\frac{\partial}{\partial y}&=\frac{\partial}{\partial y’}\\\frac{\partial}{\partial z}&=\frac{\partial}{\partial z’}\\\frac{\partial}{\partial t}&=\gamma\left(\frac{\partial}{\partial t’}-v\frac{\partial}{\partial x’}\right)
\end{align}
\end{align}
を行なうと
\begin{align}
\gamma\frac{\partial H_x}{\partial x’}
+\frac{\partial H_y}{\partial y’}
+\frac{\partial H_z}{\partial z’}
&=\frac{v}{c^2}\gamma\frac{\partial H_x}{\partial t’}\tag{$\bar {\text A}$}\\
\gamma\frac{\partial E_x}{\partial x’}
+\frac{\partial E_y}{\partial y}
+\frac{\partial E_z}{\partial z}
&=\frac{v}{c^2}\gamma\frac{\partial E_x}{\partial t’}\tag{$\bar {\text B}$}\\
-v\mu_0\gamma\frac{\partial H_x}{\partial x’}+\frac{\partial E_z}{\partial y’}
-\frac{\partial E_y}{\partial z’}
&=-\mu_0\gamma\frac{\partial H_x}{\partial t’}\tag{$\bar {\text C}$}\\
\frac{\partial E_x}{\partial z’}
-\frac{\partial }{\partial x’}(\gamma E_z+v\mu_0\gamma H_y)
&=-\frac{\partial }{\partial t’}\left(\mu_0\gamma H_y+\frac{v}{c^2}\gamma E_z\right)\tag{$\bar {\text D}$}\\
\frac{\partial }{\partial x’}(\gamma E_y-v\mu_0\gamma H_z)-\frac{\partial E_x}{\partial y’}&=-\frac{\partial }{\partial t’}\left(\mu_0\gamma H_z-\frac{v}{c^2}\gamma E_y\right)\tag{$\bar {\text E}$}\\
v\varepsilon_0\gamma\frac{\partial E_x}{\partial x’}+\frac{\partial H_z}{\partial y’}
-\frac{\partial H_y}{\partial z’}
&=\varepsilon_0\gamma\frac{\partial E_x}{\partial t’}\tag{$\bar {\text F}$}\\
\frac{\partial H_x}{\partial z’}
-\frac{\partial}{\partial x’}(\gamma H_z-v\varepsilon_0\gamma E_y)
&=\frac{\partial }{\partial t’}\left(\varepsilon_0\gamma E_y-\frac{v}{c^2}\gamma H_y\right)\tag{$\bar {\text G}$}\\
\frac{\partial }{\partial x’}(\gamma H_y+v\varepsilon_0\gamma E_z)-\frac{\partial H_x}{\partial y’}&=\frac{\partial E_z}{\partial t’}\left(\varepsilon_0\gamma E_z+\frac{v}{c^2}\gamma H_z\right)\tag{$\bar {\text H}$}
\end{align}
\gamma\frac{\partial H_x}{\partial x’}
+\frac{\partial H_y}{\partial y’}
+\frac{\partial H_z}{\partial z’}
&=\frac{v}{c^2}\gamma\frac{\partial H_x}{\partial t’}\tag{$\bar {\text A}$}\\
\gamma\frac{\partial E_x}{\partial x’}
+\frac{\partial E_y}{\partial y}
+\frac{\partial E_z}{\partial z}
&=\frac{v}{c^2}\gamma\frac{\partial E_x}{\partial t’}\tag{$\bar {\text B}$}\\
-v\mu_0\gamma\frac{\partial H_x}{\partial x’}+\frac{\partial E_z}{\partial y’}
-\frac{\partial E_y}{\partial z’}
&=-\mu_0\gamma\frac{\partial H_x}{\partial t’}\tag{$\bar {\text C}$}\\
\frac{\partial E_x}{\partial z’}
-\frac{\partial }{\partial x’}(\gamma E_z+v\mu_0\gamma H_y)
&=-\frac{\partial }{\partial t’}\left(\mu_0\gamma H_y+\frac{v}{c^2}\gamma E_z\right)\tag{$\bar {\text D}$}\\
\frac{\partial }{\partial x’}(\gamma E_y-v\mu_0\gamma H_z)-\frac{\partial E_x}{\partial y’}&=-\frac{\partial }{\partial t’}\left(\mu_0\gamma H_z-\frac{v}{c^2}\gamma E_y\right)\tag{$\bar {\text E}$}\\
v\varepsilon_0\gamma\frac{\partial E_x}{\partial x’}+\frac{\partial H_z}{\partial y’}
-\frac{\partial H_y}{\partial z’}
&=\varepsilon_0\gamma\frac{\partial E_x}{\partial t’}\tag{$\bar {\text F}$}\\
\frac{\partial H_x}{\partial z’}
-\frac{\partial}{\partial x’}(\gamma H_z-v\varepsilon_0\gamma E_y)
&=\frac{\partial }{\partial t’}\left(\varepsilon_0\gamma E_y-\frac{v}{c^2}\gamma H_y\right)\tag{$\bar {\text G}$}\\
\frac{\partial }{\partial x’}(\gamma H_y+v\varepsilon_0\gamma E_z)-\frac{\partial H_x}{\partial y’}&=\frac{\partial E_z}{\partial t’}\left(\varepsilon_0\gamma E_z+\frac{v}{c^2}\gamma H_z\right)\tag{$\bar {\text H}$}
\end{align}
\begin{align*}
\frac{\partial H_x}{\partial x}
+\frac{\partial H_y}{\partial y}
+\frac{\partial H_z}{\partial z}
&=0\tag{A}\\\rightarrow\gamma\left(\frac{\partial}{\partial x’}-\frac{v}{c^2}\frac{\partial}{\partial t’}\right)H_x
+\frac{\partial H_y}{\partial y’}
+\frac{\partial H_z}{\partial z’}
&=0\\\rightarrow\gamma\frac{\partial H_x}{\partial x’}
+\frac{\partial H_y}{\partial y’}
+\frac{\partial H_z}{\partial z’}
&=\frac{v}{c^2}\gamma\frac{\partial H_x}{\partial t’}\tag{$\bar {\text A}$}\end{align*}
\frac{\partial H_x}{\partial x}
+\frac{\partial H_y}{\partial y}
+\frac{\partial H_z}{\partial z}
&=0\tag{A}\\\rightarrow\gamma\left(\frac{\partial}{\partial x’}-\frac{v}{c^2}\frac{\partial}{\partial t’}\right)H_x
+\frac{\partial H_y}{\partial y’}
+\frac{\partial H_z}{\partial z’}
&=0\\\rightarrow\gamma\frac{\partial H_x}{\partial x’}
+\frac{\partial H_y}{\partial y’}
+\frac{\partial H_z}{\partial z’}
&=\frac{v}{c^2}\gamma\frac{\partial H_x}{\partial t’}\tag{$\bar {\text A}$}\end{align*}
\begin{align*}
\frac{\partial E_x}{\partial x}
+\frac{\partial E_y}{\partial y}
+\frac{\partial E_z}{\partial z}
&=0\tag{B}\\\rightarrow\gamma\left(\frac{\partial}{\partial x’}-\frac{v}{c^2}\frac{\partial}{\partial t’}\right)E_x
+\frac{\partial E_y}{\partial y}
+\frac{\partial E_z}{\partial z}
&=0\\\rightarrow\gamma\frac{\partial E_x}{\partial x’}
+\frac{\partial E_y}{\partial y}
+\frac{\partial E_z}{\partial z}
&=\frac{v}{c^2}\gamma\frac{\partial E_x}{\partial t’}\tag{$\bar {\text B}$}\end{align*}
\frac{\partial E_x}{\partial x}
+\frac{\partial E_y}{\partial y}
+\frac{\partial E_z}{\partial z}
&=0\tag{B}\\\rightarrow\gamma\left(\frac{\partial}{\partial x’}-\frac{v}{c^2}\frac{\partial}{\partial t’}\right)E_x
+\frac{\partial E_y}{\partial y}
+\frac{\partial E_z}{\partial z}
&=0\\\rightarrow\gamma\frac{\partial E_x}{\partial x’}
+\frac{\partial E_y}{\partial y}
+\frac{\partial E_z}{\partial z}
&=\frac{v}{c^2}\gamma\frac{\partial E_x}{\partial t’}\tag{$\bar {\text B}$}\end{align*}
\begin{align*}
\frac{\partial E_z}{\partial y}
-\frac{\partial E_y}{\partial z}
&=-\mu_0\frac{\partial H_x}{\partial t}\tag{C}\\\rightarrow\frac{\partial E_z}{\partial y’}
-\frac{\partial E_y}{\partial z’}
&=-\mu_0\gamma\left(\frac{\partial}{\partial t’}-v\frac{\partial}{\partial x’}\right)H_x\\\rightarrow-v\mu_0\gamma\frac{\partial H_x}{\partial x’}+\frac{\partial E_z}{\partial y’}
-\frac{\partial E_y}{\partial z’}
&=-\mu_0\gamma\frac{\partial H_x}{\partial t’}\tag{$\bar {\text C}$}\end{align*}
\frac{\partial E_z}{\partial y}
-\frac{\partial E_y}{\partial z}
&=-\mu_0\frac{\partial H_x}{\partial t}\tag{C}\\\rightarrow\frac{\partial E_z}{\partial y’}
-\frac{\partial E_y}{\partial z’}
&=-\mu_0\gamma\left(\frac{\partial}{\partial t’}-v\frac{\partial}{\partial x’}\right)H_x\\\rightarrow-v\mu_0\gamma\frac{\partial H_x}{\partial x’}+\frac{\partial E_z}{\partial y’}
-\frac{\partial E_y}{\partial z’}
&=-\mu_0\gamma\frac{\partial H_x}{\partial t’}\tag{$\bar {\text C}$}\end{align*}
\begin{align*}
\frac{\partial E_x}{\partial z}
-\frac{\partial E_z}{\partial x}
&=-\mu_0\frac{\partial H_y}{\partial t}\tag{D}\\\rightarrow\frac{\partial E_x}{\partial z’}
-\gamma\left(\frac{\partial}{\partial x’}-\frac{v}{c^2}\frac{\partial}{\partial t’}\right)E_z
&=-\mu_0\gamma\left(\frac{\partial}{\partial t’}-v\frac{\partial}{\partial x’}\right)H_y\\\rightarrow\frac{\partial E_x}{\partial z’}
-\frac{\partial }{\partial x’}(\gamma E_z+v\mu_0\gamma H_y)
&=-\frac{\partial }{\partial t’}\left(\mu_0\gamma H_y+\frac{v}{c^2}\gamma E_z\right)\tag{$\bar {\text D}$}\end{align*}
\frac{\partial E_x}{\partial z}
-\frac{\partial E_z}{\partial x}
&=-\mu_0\frac{\partial H_y}{\partial t}\tag{D}\\\rightarrow\frac{\partial E_x}{\partial z’}
-\gamma\left(\frac{\partial}{\partial x’}-\frac{v}{c^2}\frac{\partial}{\partial t’}\right)E_z
&=-\mu_0\gamma\left(\frac{\partial}{\partial t’}-v\frac{\partial}{\partial x’}\right)H_y\\\rightarrow\frac{\partial E_x}{\partial z’}
-\frac{\partial }{\partial x’}(\gamma E_z+v\mu_0\gamma H_y)
&=-\frac{\partial }{\partial t’}\left(\mu_0\gamma H_y+\frac{v}{c^2}\gamma E_z\right)\tag{$\bar {\text D}$}\end{align*}
\begin{align*}\frac{\partial E_y}{\partial x}
-\frac{\partial E_x}{\partial y}
&=-\mu_0\frac{\partial H_z}{\partial t}\tag{E}\\\rightarrow\gamma\left(\frac{\partial}{\partial x’}-\frac{v}{c^2}\frac{\partial}{\partial t’}\right)E_y-\frac{\partial E_x}{\partial y’}&=-\mu_0\gamma\left(\frac{\partial}{\partial t’}-v\frac{\partial}{\partial x’}\right)H_z
\\\rightarrow\frac{\partial }{\partial x’}(\gamma E_y-v\mu_0\gamma H_z)-\frac{\partial E_x}{\partial y’}&=-\frac{\partial }{\partial t’}\left(\mu_0\gamma H_z-\frac{v}{c^2}\gamma E_y\right)\tag{$\bar {\text E}$}\end{align*}
-\frac{\partial E_x}{\partial y}
&=-\mu_0\frac{\partial H_z}{\partial t}\tag{E}\\\rightarrow\gamma\left(\frac{\partial}{\partial x’}-\frac{v}{c^2}\frac{\partial}{\partial t’}\right)E_y-\frac{\partial E_x}{\partial y’}&=-\mu_0\gamma\left(\frac{\partial}{\partial t’}-v\frac{\partial}{\partial x’}\right)H_z
\\\rightarrow\frac{\partial }{\partial x’}(\gamma E_y-v\mu_0\gamma H_z)-\frac{\partial E_x}{\partial y’}&=-\frac{\partial }{\partial t’}\left(\mu_0\gamma H_z-\frac{v}{c^2}\gamma E_y\right)\tag{$\bar {\text E}$}\end{align*}
\begin{align*}
\frac{\partial H_z}{\partial y}
-\frac{\partial H_y}{\partial z}
&=\varepsilon_0\frac{\partial E_x}{\partial t}\tag{F}\\\rightarrow\frac{\partial H_z}{\partial y’}
-\frac{\partial H_y}{\partial z’}
&=\varepsilon_0\gamma\left(\frac{\partial}{\partial t’}-v\frac{\partial}{\partial x’}\right)E_x\\\rightarrow v\varepsilon_0\gamma\frac{\partial E_x}{\partial x’}+\frac{\partial H_z}{\partial y’}
-\frac{\partial H_y}{\partial z’}
&=\varepsilon_0\gamma\frac{\partial E_x}{\partial t’}\tag{$\bar {\text F}$}\end{align*}
\frac{\partial H_z}{\partial y}
-\frac{\partial H_y}{\partial z}
&=\varepsilon_0\frac{\partial E_x}{\partial t}\tag{F}\\\rightarrow\frac{\partial H_z}{\partial y’}
-\frac{\partial H_y}{\partial z’}
&=\varepsilon_0\gamma\left(\frac{\partial}{\partial t’}-v\frac{\partial}{\partial x’}\right)E_x\\\rightarrow v\varepsilon_0\gamma\frac{\partial E_x}{\partial x’}+\frac{\partial H_z}{\partial y’}
-\frac{\partial H_y}{\partial z’}
&=\varepsilon_0\gamma\frac{\partial E_x}{\partial t’}\tag{$\bar {\text F}$}\end{align*}
\begin{align*}
\frac{\partial H_x}{\partial z}
-\frac{\partial H_z}{\partial x}
&=\varepsilon_0\frac{\partial E_y}{\partial t}\tag{G}\\\rightarrow\frac{\partial H_x}{\partial z’}
-\gamma\left(\frac{\partial}{\partial x’}-\frac{v}{c^2}\frac{\partial}{\partial t’}\right)H_z
&=\varepsilon_0\gamma\left(\frac{\partial}{\partial t’}-v\frac{\partial}{\partial x’}\right)E_y\\\rightarrow\frac{\partial H_x}{\partial z’}
-\frac{\partial}{\partial x’}(\gamma H_z-v\varepsilon_0\gamma E_y)
&=\frac{\partial }{\partial t’}\left(\varepsilon_0\gamma E_y-\frac{v}{c^2}\gamma H_y\right)\tag{$\bar {\text G}$}\end{align*}
\frac{\partial H_x}{\partial z}
-\frac{\partial H_z}{\partial x}
&=\varepsilon_0\frac{\partial E_y}{\partial t}\tag{G}\\\rightarrow\frac{\partial H_x}{\partial z’}
-\gamma\left(\frac{\partial}{\partial x’}-\frac{v}{c^2}\frac{\partial}{\partial t’}\right)H_z
&=\varepsilon_0\gamma\left(\frac{\partial}{\partial t’}-v\frac{\partial}{\partial x’}\right)E_y\\\rightarrow\frac{\partial H_x}{\partial z’}
-\frac{\partial}{\partial x’}(\gamma H_z-v\varepsilon_0\gamma E_y)
&=\frac{\partial }{\partial t’}\left(\varepsilon_0\gamma E_y-\frac{v}{c^2}\gamma H_y\right)\tag{$\bar {\text G}$}\end{align*}
\begin{align*}
\frac{\partial H_y}{\partial x}
-\frac{\partial H_x}{\partial y}
&=\varepsilon_0\frac{\partial E_z}{\partial t}\tag{H}\\\rightarrow\gamma\left(\frac{\partial}{\partial x’}-\frac{v}{c^2}\frac{\partial}{\partial t’}\right)H_y
-\frac{\partial H_x}{\partial y’}
&=\varepsilon_0\gamma\left(\frac{\partial}{\partial t’}-v\frac{\partial}{\partial x’}\right)E_z\\\rightarrow\frac{\partial }{\partial x’}(\gamma H_y+v\varepsilon_0\gamma E_z)-\frac{\partial H_x}{\partial y’}&=\frac{\partial E_z}{\partial t’}\left(\varepsilon_0\gamma E_z+\frac{v}{c^2}\gamma H_z\right)\tag{$\bar {\text H}$}\end{align*}
\frac{\partial H_y}{\partial x}
-\frac{\partial H_x}{\partial y}
&=\varepsilon_0\frac{\partial E_z}{\partial t}\tag{H}\\\rightarrow\gamma\left(\frac{\partial}{\partial x’}-\frac{v}{c^2}\frac{\partial}{\partial t’}\right)H_y
-\frac{\partial H_x}{\partial y’}
&=\varepsilon_0\gamma\left(\frac{\partial}{\partial t’}-v\frac{\partial}{\partial x’}\right)E_z\\\rightarrow\frac{\partial }{\partial x’}(\gamma H_y+v\varepsilon_0\gamma E_z)-\frac{\partial H_x}{\partial y’}&=\frac{\partial E_z}{\partial t’}\left(\varepsilon_0\gamma E_z+\frac{v}{c^2}\gamma H_z\right)\tag{$\bar {\text H}$}\end{align*}
となる。次に、\(v\varepsilon_0\)を掛けた式(\(\bar {\text C})\)を式(\(\bar {\text A})\)に足して、\(-v\mu_0\)を掛けた式(\(\bar {\text F})\)を式(\(\bar {\text B})\)に足して、\(v\mu_0\)を掛けた式(\(\bar {\text A})\)を式(\(\bar {\text C})\)に足して、\(-v\varepsilon_0\)を掛けた式(\(\bar {\text B})\)を式(\(\bar {\text F})\)に足すと
\begin{align}
\frac{\partial H_x}{\partial x’}+\frac{\partial }{\partial y’}(\gamma H_y+v\varepsilon_0\gamma E_z)+\frac{\partial }{\partial z’}(\gamma H_z-v\varepsilon_0 \gamma E_x)&=0\tag{$\bar {\text A}’$}\\\frac{\partial E_x}{\partial x’}
+\frac{\partial}{\partial y}(\gamma E_y-v\mu_0 \gamma H_z)
+\frac{\partial}{\partial z}(\gamma E_z-v\mu_0 \gamma H_y)
&=0\tag{$\bar {\text B}’$}\\
\frac{\partial}{\partial y’}(\gamma E_z+v\mu_0\gamma H_y)
-\frac{\partial }{\partial z’}(\gamma E_y-v\mu_0\gamma H_z)
&=-\mu_0\frac{\partial H_x}{\partial t’}\tag{$\bar {\text C}’$}\\
\frac{\partial E_x}{\partial z’}
-\frac{\partial }{\partial x’}(\gamma E_z+v\mu_0\gamma H_y)
&=-\frac{\partial }{\partial t’}\left(\mu_0\gamma H_y+\frac{v}{c^2}\gamma E_z\right)\tag{$\bar {\text D}$}\\
\frac{\partial }{\partial x’}(\gamma E_y-v\mu_0\gamma H_z)-\frac{\partial E_x}{\partial y’}&=-\frac{\partial }{\partial t’}\left(\mu_0\gamma H_z-\frac{v}{c^2}\gamma E_y\right)\tag{$\bar {\text E}$}\\
\frac{\partial}{\partial y’}(\gamma H_z-v\varepsilon_0\gamma E_y)
-\frac{\partial}{\partial z’}(\gamma H_y+v\varepsilon_0\gamma E_z)
&=\varepsilon_0\frac{\partial E_x}{\partial t’}\tag{$\bar {\text F}’$}\\
\frac{\partial H_x}{\partial z’}
-\frac{\partial}{\partial x’}(\gamma H_z-v\varepsilon_0\gamma E_y)
&=\frac{\partial }{\partial t’}\left(\varepsilon_0\gamma E_y-\frac{v}{c^2}\gamma H_y\right)\tag{$\bar {\text G}$}\\
\frac{\partial }{\partial x’}(\gamma H_y+v\varepsilon_0\gamma E_z)-\frac{\partial H_x}{\partial y’}&=\frac{\partial E_z}{\partial t’}\left(\varepsilon_0\gamma E_z+\frac{v}{c^2}\gamma H_z\right)\tag{$\bar {\text H}$}
\end{align}
\frac{\partial H_x}{\partial x’}+\frac{\partial }{\partial y’}(\gamma H_y+v\varepsilon_0\gamma E_z)+\frac{\partial }{\partial z’}(\gamma H_z-v\varepsilon_0 \gamma E_x)&=0\tag{$\bar {\text A}’$}\\\frac{\partial E_x}{\partial x’}
+\frac{\partial}{\partial y}(\gamma E_y-v\mu_0 \gamma H_z)
+\frac{\partial}{\partial z}(\gamma E_z-v\mu_0 \gamma H_y)
&=0\tag{$\bar {\text B}’$}\\
\frac{\partial}{\partial y’}(\gamma E_z+v\mu_0\gamma H_y)
-\frac{\partial }{\partial z’}(\gamma E_y-v\mu_0\gamma H_z)
&=-\mu_0\frac{\partial H_x}{\partial t’}\tag{$\bar {\text C}’$}\\
\frac{\partial E_x}{\partial z’}
-\frac{\partial }{\partial x’}(\gamma E_z+v\mu_0\gamma H_y)
&=-\frac{\partial }{\partial t’}\left(\mu_0\gamma H_y+\frac{v}{c^2}\gamma E_z\right)\tag{$\bar {\text D}$}\\
\frac{\partial }{\partial x’}(\gamma E_y-v\mu_0\gamma H_z)-\frac{\partial E_x}{\partial y’}&=-\frac{\partial }{\partial t’}\left(\mu_0\gamma H_z-\frac{v}{c^2}\gamma E_y\right)\tag{$\bar {\text E}$}\\
\frac{\partial}{\partial y’}(\gamma H_z-v\varepsilon_0\gamma E_y)
-\frac{\partial}{\partial z’}(\gamma H_y+v\varepsilon_0\gamma E_z)
&=\varepsilon_0\frac{\partial E_x}{\partial t’}\tag{$\bar {\text F}’$}\\
\frac{\partial H_x}{\partial z’}
-\frac{\partial}{\partial x’}(\gamma H_z-v\varepsilon_0\gamma E_y)
&=\frac{\partial }{\partial t’}\left(\varepsilon_0\gamma E_y-\frac{v}{c^2}\gamma H_y\right)\tag{$\bar {\text G}$}\\
\frac{\partial }{\partial x’}(\gamma H_y+v\varepsilon_0\gamma E_z)-\frac{\partial H_x}{\partial y’}&=\frac{\partial E_z}{\partial t’}\left(\varepsilon_0\gamma E_z+\frac{v}{c^2}\gamma H_z\right)\tag{$\bar {\text H}$}
\end{align}
\begin{align*}\left(1-\frac{v^2}{c^2}\right)\gamma\frac{\partial H_x}{\partial x’}
+\frac{\partial }{\partial y’}(H_y+v\varepsilon_0 E_z)
+\frac{\partial }{\partial z’}(H_z-v\varepsilon_0 E_x)
&=0\\\rightarrow\frac{1}{\gamma}\frac{\partial H_x}{\partial x’}
+\frac{\partial }{\partial y’}(H_y+v\varepsilon_0 E_z)
+\frac{\partial }{\partial z’}(H_z-v\varepsilon_0 E_x)
&=0\\\rightarrow\frac{\partial H_x}{\partial x’}+\frac{\partial }{\partial y’}(\gamma H_y+v\varepsilon_0\gamma E_z)+\frac{\partial }{\partial z’}(\gamma H_z-v\varepsilon_0 \gamma E_x)&=0\tag{$\bar {\text A}’$}\end{align*}
+\frac{\partial }{\partial y’}(H_y+v\varepsilon_0 E_z)
+\frac{\partial }{\partial z’}(H_z-v\varepsilon_0 E_x)
&=0\\\rightarrow\frac{1}{\gamma}\frac{\partial H_x}{\partial x’}
+\frac{\partial }{\partial y’}(H_y+v\varepsilon_0 E_z)
+\frac{\partial }{\partial z’}(H_z-v\varepsilon_0 E_x)
&=0\\\rightarrow\frac{\partial H_x}{\partial x’}+\frac{\partial }{\partial y’}(\gamma H_y+v\varepsilon_0\gamma E_z)+\frac{\partial }{\partial z’}(\gamma H_z-v\varepsilon_0 \gamma E_x)&=0\tag{$\bar {\text A}’$}\end{align*}
\begin{align*}\left(1-\frac{v^2}{c^2}\right)\gamma\frac{\partial E_x}{\partial x’}
+\frac{\partial}{\partial y}(E_y-v\mu_0 H_z)
+\frac{\partial}{\partial z}(E_z-v\mu_0 H_y)
&=0\\\rightarrow\frac{1}{\gamma}\frac{\partial E_x}{\partial x’}
+\frac{\partial}{\partial y}(E_y-v\mu_0 H_z)
+\frac{\partial}{\partial z}(E_z-v\mu_0 H_y)
&=0\\\frac{\partial E_x}{\partial x’}
+\frac{\partial}{\partial y}(\gamma E_y-v\mu_0 \gamma H_z)
+\frac{\partial}{\partial z}(\gamma E_z-v\mu_0 \gamma H_y)
&=0\tag{$\bar {\text B}’$}\end{align*}
+\frac{\partial}{\partial y}(E_y-v\mu_0 H_z)
+\frac{\partial}{\partial z}(E_z-v\mu_0 H_y)
&=0\\\rightarrow\frac{1}{\gamma}\frac{\partial E_x}{\partial x’}
+\frac{\partial}{\partial y}(E_y-v\mu_0 H_z)
+\frac{\partial}{\partial z}(E_z-v\mu_0 H_y)
&=0\\\frac{\partial E_x}{\partial x’}
+\frac{\partial}{\partial y}(\gamma E_y-v\mu_0 \gamma H_z)
+\frac{\partial}{\partial z}(\gamma E_z-v\mu_0 \gamma H_y)
&=0\tag{$\bar {\text B}’$}\end{align*}
\begin{align*}\frac{\partial}{\partial y’}(E_z+v\mu_0H_y)
-\frac{\partial }{\partial z’}(E_y-v\mu_0H_z)
&=-\mu_0\left(1-\frac{v^2}{c^2}\right)\gamma\frac{\partial H_x}{\partial t’}\\\rightarrow\frac{\partial}{\partial y’}( E_z+v\mu_0 H_y)
-\frac{\partial }{\partial z’}( E_y-v\mu_0 H_z)
&=-\mu_0\frac{1}{\gamma}\frac{\partial H_x}{\partial t’}\\\rightarrow\frac{\partial}{\partial y’}(\gamma E_z+v\mu_0\gamma H_y)
-\frac{\partial }{\partial z’}(\gamma E_y-v\mu_0\gamma H_z)
&=-\mu_0\frac{\partial H_x}{\partial t’}\tag{$\bar {\text C}’$}\end{align*}
-\frac{\partial }{\partial z’}(E_y-v\mu_0H_z)
&=-\mu_0\left(1-\frac{v^2}{c^2}\right)\gamma\frac{\partial H_x}{\partial t’}\\\rightarrow\frac{\partial}{\partial y’}( E_z+v\mu_0 H_y)
-\frac{\partial }{\partial z’}( E_y-v\mu_0 H_z)
&=-\mu_0\frac{1}{\gamma}\frac{\partial H_x}{\partial t’}\\\rightarrow\frac{\partial}{\partial y’}(\gamma E_z+v\mu_0\gamma H_y)
-\frac{\partial }{\partial z’}(\gamma E_y-v\mu_0\gamma H_z)
&=-\mu_0\frac{\partial H_x}{\partial t’}\tag{$\bar {\text C}’$}\end{align*}
\begin{align*}\frac{\partial}{\partial y’}( H_z-v\varepsilon_0 E_y)
-\frac{\partial}{\partial z’}(H_y+v\varepsilon_0 E_z)
&=\varepsilon_0\gamma\left(1-\frac{v^2}{c^2}\right)\frac{\partial E_x}{\partial t’}\\\rightarrow\frac{\partial}{\partial y’}( H_z-v\varepsilon_0 E_y)
-\frac{\partial}{\partial z’}(H_y+v\varepsilon_0 E_z)
&=\varepsilon_0\frac{1}{\gamma}\frac{\partial E_x}{\partial t’}\\\rightarrow\frac{\partial}{\partial y’}(\gamma H_z-v\varepsilon_0\gamma E_y)
-\frac{\partial}{\partial z’}(\gamma H_y+v\varepsilon_0\gamma E_z)
&=\varepsilon_0\frac{\partial E_x}{\partial t’}\tag{$\bar {\text F}’$}\end{align*}
-\frac{\partial}{\partial z’}(H_y+v\varepsilon_0 E_z)
&=\varepsilon_0\gamma\left(1-\frac{v^2}{c^2}\right)\frac{\partial E_x}{\partial t’}\\\rightarrow\frac{\partial}{\partial y’}( H_z-v\varepsilon_0 E_y)
-\frac{\partial}{\partial z’}(H_y+v\varepsilon_0 E_z)
&=\varepsilon_0\frac{1}{\gamma}\frac{\partial E_x}{\partial t’}\\\rightarrow\frac{\partial}{\partial y’}(\gamma H_z-v\varepsilon_0\gamma E_y)
-\frac{\partial}{\partial z’}(\gamma H_y+v\varepsilon_0\gamma E_z)
&=\varepsilon_0\frac{\partial E_x}{\partial t’}\tag{$\bar {\text F}’$}\end{align*}
となり、もし電場·磁場が変化しないとするとマクスウェル方程式の形は不変とならない。しかし、電場·磁場を次のように変換
\begin{align*}E’_x&=E_x\\E’_y&=\gamma(E_y-v\mu_0H_z)\\E’_z&=\gamma(E_z+v\mu_0H_y)\\H’_x&=H_x\\H’_y&=\gamma(H_y+v\varepsilon_0E_z)\\H’_z&=\gamma(H_z-v\varepsilon_0 E_y)\end{align*}
すると、
\begin{align}
\frac{\partial H’_x}{\partial x’}
+\frac{\partial H’_y}{\partial y’}
+\frac{\partial H’_z}{\partial z’}
&=0\tag{A$’$}\\
\frac{\partial E’_x}{\partial x’}
+\frac{\partial E’_y}{\partial y’}
+\frac{\partial E’_z}{\partial z’}
&=0\tag{B$’$}\\
\frac{\partial E’_z}{\partial y’}
-\frac{\partial E’_y}{\partial z’}
&=-\mu_0\frac{\partial H’_x}{\partial t’}\tag{C$’$}\\
\frac{\partial E’_x}{\partial z’}
-\frac{\partial E’_z}{\partial x’}
&=-\mu_0\frac{\partial H’_y}{\partial t’}\tag{D$’$}\\
\frac{\partial E’_y}{\partial x’}
-\frac{\partial E’_x}{\partial y’}
&=-\mu_0\frac{\partial H’_z}{\partial t’}\tag{E$’$}\\
\frac{\partial H’_z}{\partial y’}
-\frac{\partial H’_y}{\partial z’}
&=\varepsilon_0\frac{\partial E’_x}{\partial t’}\tag{F$’$}\\
\frac{\partial H’_x}{\partial z’}
-\frac{\partial H’_z}{\partial x’}
&=\varepsilon_0\frac{\partial E’_y}{\partial t’}\tag{G$’$}\\
\frac{\partial H’_y}{\partial x’}
-\frac{\partial H’_x}{\partial y’}
&=\varepsilon_0\frac{\partial E’_z}{\partial t’}\tag{H$’$}
\end{align}
\frac{\partial H’_x}{\partial x’}
+\frac{\partial H’_y}{\partial y’}
+\frac{\partial H’_z}{\partial z’}
&=0\tag{A$’$}\\
\frac{\partial E’_x}{\partial x’}
+\frac{\partial E’_y}{\partial y’}
+\frac{\partial E’_z}{\partial z’}
&=0\tag{B$’$}\\
\frac{\partial E’_z}{\partial y’}
-\frac{\partial E’_y}{\partial z’}
&=-\mu_0\frac{\partial H’_x}{\partial t’}\tag{C$’$}\\
\frac{\partial E’_x}{\partial z’}
-\frac{\partial E’_z}{\partial x’}
&=-\mu_0\frac{\partial H’_y}{\partial t’}\tag{D$’$}\\
\frac{\partial E’_y}{\partial x’}
-\frac{\partial E’_x}{\partial y’}
&=-\mu_0\frac{\partial H’_z}{\partial t’}\tag{E$’$}\\
\frac{\partial H’_z}{\partial y’}
-\frac{\partial H’_y}{\partial z’}
&=\varepsilon_0\frac{\partial E’_x}{\partial t’}\tag{F$’$}\\
\frac{\partial H’_x}{\partial z’}
-\frac{\partial H’_z}{\partial x’}
&=\varepsilon_0\frac{\partial E’_y}{\partial t’}\tag{G$’$}\\
\frac{\partial H’_y}{\partial x’}
-\frac{\partial H’_x}{\partial y’}
&=\varepsilon_0\frac{\partial E’_z}{\partial t’}\tag{H$’$}
\end{align}
となって、マクスウェル方程式の形が不変となることがわかる。ここで、先程の電場·磁場の変換は次のような変換
\begin{align*}E_x&=E’_x\\E_y&=\gamma(E’_y+v\mu_0H’_z)\\E_z&=\gamma(E’_z-v\mu_0H’_y)\\H_x&=H’_x\\H_y&=\gamma(H’_y-v\varepsilon_0E’_z)\\H_z&=\gamma(H’_z+v\varepsilon_0 E’_y)\end{align*}
\begin{align*}E’_y+v\mu_0 H’_z&=\gamma\left(1-\frac{v^2}{c^2}\right)E_y\\\rightarrow E_y&=\gamma(E’_y+v\mu_0 H’_z)\end{align*}
\begin{align*}E’_z-v\mu_0 H’_y&=\gamma\left(1-\frac{v^2}{c^2}\right)E_z\\\rightarrow E_z&=\gamma(E’_z-v\mu_0 H’_y)\end{align*}
\begin{align*}H’_y-v\varepsilon_0 E’_z&=\gamma\left(1-\frac{v^2}{c^2}\right)H_y\\\rightarrow H_y&=\gamma(H’_y-v\varepsilon_0 E’_z)\end{align*}
\begin{align*}H’_z+v\varepsilon_0 E’_y&=\gamma\left(1-\frac{v^2}{c^2}\right)H_z\\\rightarrow H_z&=\gamma(H’_z-v\varepsilon_0 E’_y)\end{align*}
\begin{align*}\mu_0\varepsilon_0&=\frac{1}{c^2}\\\gamma&=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\end{align*}
としても書くことができる。
電場と磁場
ここで、なぜ慣性系を変えると電場と磁場が互いに混ざるのだろうかという疑問が生じる。
これは、電場と磁場が本質的には独立した別々の存在ではなく、電磁場という1つの現象を異なる慣性系から見た結果だからである。
たとえば、静止している電荷のまわりでは電場のみが存在し、磁場は現れない。しかし、その電荷を運動している慣性系から見ると、電荷は動いているように見える。すると、その運動する電荷は電流として振る舞い、磁場が現れる。つまり、ある慣性系では「電場だけ」が存在していたとしても、別の慣性系では「電場と磁場の両方」が現れるのである。このことは、電場と磁場が完全に独立した存在ではなく、観測者の運動状態によって互いに変換されうる量であることを示している。
実際、マクスウェル方程式を異なる慣性系で比較すると、慣性系を移る際に電場と磁場を混ぜるような変換を行わなければ、方程式の形を保つことができなかった。つまり、電場と磁場が混ざるという性質は、マクスウェル方程式そのものに含まれているのである。
電場·磁場の低速極限
マクスウェル方程式は低速極限のとき、ガリレイ変換の下で次のような電場·磁場の変換
\begin{align}E_x &= E’_x\\E_y &= E’_y + v\mu_0 H’_z \\
E_z&= E’_z{}- v\mu_0 H’_y\\H_x &= H’_x\\
H_y &= H’_y -v \varepsilon_0 E’_z \\
H_z &= H’_z + v\varepsilon_0 E’_y
\end{align}
E_z&= E’_z{}- v\mu_0 H’_y\\H_x &= H’_x\\
H_y &= H’_y -v \varepsilon_0 E’_z \\
H_z &= H’_z + v\varepsilon_0 E’_y
\end{align}
が行なわれると形は不変となっていた(以前のページを参照)。実はこの電場·磁場の変換は、先程の電場·磁場の変換
\begin{align*}E_x&=E’_x\\E_y&=\gamma(E’_y+v\mu_0H’_z)\\E_z&=\gamma(E’_z-v\mu_0H’_y)\\H_x&=H’_x\\H_y&=\gamma(H’_y-v\varepsilon_0E’_z)\\H_z&=\gamma(H’_z+v\varepsilon_0 E’_y)\end{align*}
を低速極限にしたものである。
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HOME ＞ 特殊相対性理論 ＞ ガリレイの相対性原理 ＞マクスウェル方程式のローレンツ変換