点電荷

$\phi=(r^2+z^2)^{-1/2}$、 $\phi_z=-z(r^2+z^2)^{-3/2}$、

$$\Phi=\int_0^r \rho z(\rho^2+z^2)^{-3/2} d\rho =\left.-\fra... ...t _{\rho=0}^r =-\frac z{\sqrt{r^2+z^2}}+\frac z{\vert z\vert}$$ (1.11)

「定数項」である第2項を省けば $\Phi=-\frac z{\sqrt{r^2+z^2}}$。

図 1.8: $z=\pm 3$、$r=0$に符号が反対の点電荷が1つずつ。

gnuplot> set contour base
gnuplot> unset surface
gnuplot> set view 0,0
gnuplot> set size square
gnuplot> set isosample 100
gnuplot> unset key
gnuplot> mf(r,z)=z/sqrt(r**2+z**2)
gnuplot> set cntrparam levels incremental -1.9,0.2,0
gnuplot> splot mf(x,y-3)-mf(x,y+3)
gnuplot> set term postscript enhanced color eps
Terminal type set to 'postscript'
Options are 'eps enhanced color colortext \
   dashed dashlength 1.0 linewidth 1.0 defaultplex \
   palfuncparam 2000,0.003 \
   butt "Helvetica-Ryumin" 14'
gnuplot> set output "2pointcharges3d.eps"
gnuplot> replot