4 磁場の導出

$\displaystyle \vB\rt$	$\displaystyle =\Nabla \times \vA\rt =\Nabla\times \left(\frac{q\bm{\beta}(t')}{... ...m{\beta}(t')-q\bm{\beta}(t')\times \Nabla\left(\frac{1}{\kappa(t')R(t')}\right)$
	$\displaystyle =\frac{q}{\kappa(t')R(t')} \left( \Nabla_{\vr_0}-\frac{vn(t')}{c\... ...abla\left\{ \kappa(t')R(t')\right\}}{\kappa^2(t')R^2(t')} \times \bm{\beta}(t')$	(28)
	$\displaystyle =\frac{q}{\kappa(t')R(t')} \Nabla_{\vr_0}\times \bm{\beta}(t')-\frac{q}{c\kappa^2(t')R(t')}\vn(t')\times \dot{\bm{\beta}}(t')$
	$\displaystyle \hspace{30mm} - q \frac{\vn(t')-\bm{\beta}(t')+ \dfrac{\left(\vn(... ...a}}(t')\right)}{c\kappa(t')}\vn(t')}{c\kappa^2(t')R^2(t')}\times \bm{\beta}(t')$
	$\displaystyle = 0 +q\vn(t')\times \frac{-\kappa(t')\dot{\bm{\beta}}(t')-\left(\... ...\beta}(t')\right)\bm{\beta}(t')+\beta^2(t')\bm{\beta}(t')}{\kappa^3(t')R^2(t')}$
	$\displaystyle =\vn(t')\times q \frac{\left(\vn(t')\cdot\dot{\bm{\beta}}(t')\rig... ...left(\vn(t')\cdot\dot{\bm{\beta}}(t')\right)\bm{\beta}(t')}{c\kappa^3(t')R(t')}$
	$\displaystyle \hspace{20mm} \vn(t')\times q \frac{\kappa(t')\vn(t')-\kappa(t')\... ...\beta}(t')\right)\bm{\beta}(t')+\beta^2(t')\bm{\beta}(t')}{\kappa^3(t')R^2(t')}$
	$\displaystyle = \vn(t')\times q\frac{\left(1-\beta^2(t')\right)\left(\vn(t')-\b... ...)-\bm{\beta}(t')\right)\times \dot{\bm{\beta}}(t')\right\}}{c\kappa^3(t')R(t')}$
	$\displaystyle = q \vn(t')\times \left[ \frac{\left(1-\beta^2\right)\left(\vn-\b... ...\dot{\bm{\beta}}\right\}}{c\kappa^3 R} \right] =\left[ \vn\times \vE\rt \right]$	(29)
	$\displaystyle \qquad \because\,\, \vn(t')\times \vn(t')=0$

$\displaystyle \vn(t')\times \left(-\Nabla \phi\rt\right)$	$\displaystyle = \vn(t') \times \left\{ -q \Nabla_{\vr_0}\left(\frac{1}{\kappa(t... ...times \frac{ \Nabla_{\vr_0}\left\{\kappa(t')R(t')\right\}}{\kappa^2(t')R^2(t')}$
	$\displaystyle = q \vn(t') \times \frac{\kappa(t')\vn(t')+ \left(\vn(t')\cdot\bm... ...kappa^2(t')R^2(t')} =-q\frac{\vn(t')\times \bm{\beta}(t')}{\kappa^2(t')R^2(t')}$	(30)

$\displaystyle \vB\rt$	$\displaystyle =\Nabla\times \vA\rt = \Nabla_{\vr_0}\times \vA\rt -\frac{\vn(t')... ...bm{\beta}}{\kappa R}\right] -\frac{\vn(t')}{c\kappa(t')}\times \del{\vA\rt}{t'}$
	$\displaystyle =q \frac{\left( \Nabla_{\vr_0}\times \bm{\beta}(t')\right)\kappa(... ...ight)}{\kappa^2(t')R^2(t')} -\frac{\vn(t')}{c\kappa(t')}\times \del{\vA\rt}{t'}$
	$\displaystyle = q \frac{ 0+ \bm{\beta}(t')\times \left\{ \kappa(t')\vn(t')+ \le... ...(t') }{\kappa^2(t')R^2(t')} -\frac{\vn(t')}{c\kappa(t')}\times \del{\vA\rt}{t'}$
	$% latex2html id marker 1468 $\displaystyle = \vn(t')\times \left\{ -\Nabla\phi\rt -\frac{1}{c}\del{\vA\rt}{t} \right\}\, \quad \because \,Eq.(\ref{eq:nabla-phi})$$
	$\displaystyle =\left[ \vn \times \vE\rt \right]$	(31)