6 電場の導出

$\displaystyle \vB\xt$	$\displaystyle = \Nabla \times \vA \xt = \frac{\mu_0 q}{4\pi} \int_{V'} d^3\vx' ... ...left( \dfrac{ \delta\left( t-t' -\frac{R(\vx,t')}{c}\right)}{R(\vx,t')} \right)$
	$\displaystyle = \frac{ q}{4\pi\vepsilon_0}\frac{1}{c^2} \Int dt' \, \left(\vn(t... ...t')} \left(\frac{\delta\left(t-t'-\frac{R(\vx,t')}{c}\right)}{R(\vx,t')}\right)$
	$\displaystyle =\frac{ q}{4\pi\vepsilon_0} \frac{1}{c} \Int dt'\, \left(\vn(t') ... ...{R(\vx,t')} \deL{R(\vx,t')} \delta \left(t-t'-\frac{R(\vx,t')}{c}\right) \Bigg]$

$\displaystyle \vB\xt = \frac{q}{4\pi\vepsilon_0} \frac{1}{c} \left[ \frac{\bm{\... ...t')}\deL{t'} \frac{\bm{\beta}(t') \times \vn(t')}{R(\vx,t')\kappa(t')} \right]$

$\displaystyle \vE\xt$

$\displaystyle = \frac{q}{4\pi \vepsilon_0} \Bigg[ \frac{\vn(t')}{R^2(\vx,t')\ka... ...a(t')} \deL{t'} \left(\frac{\bm{\beta}(t')}{\kappa(t') R(\vx,t')}\right) \Bigg]$

$\displaystyle \frac{\vn(t')}{R^2(\vx,t')\kappa(t')} + \frac{\vn(t')\times \vn(t')\times \bm{\beta}(t')}{R^2(\vx,t')\kappa^2(t')}$	$\displaystyle =\frac{\vn(t') \kappa(t')}{R^2(\vx,t')\kappa^2(t')} + \frac{\vn(t')\times \vn(t')\times \bm{\beta}(t')}{R^2(\vx,t')\kappa^2(t')}$
	$\displaystyle =\frac{\vn(t')}{\kappa^2(t')R^2(\vx,t')} -\frac{\vn(t') \left(\vn... ...) -\vn(t') \left(\vn(t')\times \bm{\beta}(t')\right)}{\kappa^2(t') R^2(\vx,t')}$
	$\displaystyle =\frac{\vn(t')}{\kappa^2(t')R^2(\vx,t')} -\frac{\bm{\beta}(t')}{\kappa^2(t') R^2(\vx,t')}$

$\displaystyle \vE\xt = \frac{q}{4\pi \vepsilon_0} \Bigg[ \frac{\vn(t')}{R^2(\vx... ...a(t')} \deL{t'} \left(\frac{\bm{\beta}(t')}{\kappa(t') R(\vx,t')}\right) \Bigg]$

(36)

$\displaystyle \dI{t'} \bm{\beta}(t')$	$\displaystyle = \dot{\bm{\beta}}(t')$
$\displaystyle \dI{t'}\left(\kappa(t') R(\vx,t')\right)$	$\displaystyle = \di{\kappa(t')}{t'} R(\vx,t') + \kappa(t') \di{R(\vx,t')}{t'} =... ...}(t')}{c}\right)\right\}R(\vx,t') -\kappa(t') \left(\vn(t')\cdot \vu(t')\right)$
	$\displaystyle = -\frac{1}{c}\left(\frac{\vn(t') \times \left(\vn(t')\times \vu(... ...eft(1-\frac{1}{c} \vn(t')\cdot \vu(t')\right)\left(\vn(t') \cdot \vu(t')\right)$
	$\displaystyle =-\frac{1}{c}\left(\frac{\vn(t')\left(\vn(t')\cdot \vu(t') -\vu(t... ...eft(1-\frac{1}{c} \vn(t')\cdot \vu(t')\right)\left(\vn(t') \cdot \vu(t')\right)$
	$\displaystyle =c \bm{\beta}^2(t') -c \left(\vn(t')\cdot \bm{\beta}(t')\right) -R(\vx,t')\left(\vn(t')\cdot \dot{\bm{\beta}}(t')\right)$

$\displaystyle \vE\xt$	$\displaystyle = \frac{q}{4\pi \vepsilon_0} \Bigg[ \frac{\vn}{R^2 \kappa^2} + \f... ...frac{1}{c \kappa} \deL{t'} \left(\frac{\bm{\beta}}{\kappa R}\right) \Bigg]_{t'}$
	$\displaystyle =\frac{q}{4\pi \vepsilon_0} \Bigg[ \frac{\vn}{R^2 \kappa^2} + \fr... ...eft(\vn\cdot \bm{\beta}\right) -R\left(\vn\cdot \dot{\bm{\beta}}\right) \right)$
	$\displaystyle \hspace{30mm} -\frac{\bm{\beta}}{R^2 \kappa^2} -\frac{1}{c \kappa... ...,t')\left(\vn(t')\cdot \dot{\bm{\beta}}(t')\right) \right) \right\} \Bigg]_{t'}$
	$\displaystyle =\frac{q}{4\pi\vepsilon_0} \Bigg[ \frac{ \left(1-\vn\cdot \bm{\be... ...bm{\beta}\bm{\beta}^2-\left(\vn\cdot \bm{\beta}\right)\bm{\beta}}{\kappa^3 R^2}$
	$\displaystyle \hspace{80mm} + \frac{ \vn \left(\vn \cdot \dot{\bm{\beta}}\right... ...)-\bm{\beta}\left(\vn \cdot \dot{\bm{\beta}}\right) }{c \kappa^3 R} \Bigg]_{t'}$
	$\displaystyle =\frac{q}{4\pi \vepsilon_0} \left[ \frac{1}{\kappa^3} \frac{ \lef... ...bm{\beta}} \left(\vn -\bm{\beta}\right) -\kappa\dot{\bm{\beta}}}{R}\right]_{t'}$	(37)
	$\displaystyle =\frac{q}{4\pi \vepsilon_0} \left[ \frac{\left(\vn -\bm{\beta}\ri... ...\left(\vn \cdot \left(\vn -\bm{\beta}\right)\right) }{c\kappa^3 R} \right]_{t'}$
	$\displaystyle = \frac{q}{4\pi \vepsilon_0} \left[ \frac{\left(1-\bm{\beta}^2\ri... ...vn -\bm{\beta}\right) \times \dot{\bm{\beta}}\right\}}{\kappa^3 R} \right]_{t'}$	(38)