PSI Vol.34,No.1 December 2012 Thesis 3. pp.17-26.
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零場をもつ静電界のカタストロフィー現象とフラクタル
小林正典* 青木宏之 祖田直也
Catastrophic Phenomena and Fractal in Electrostatic Fields with Zero Fields
Masanori Kobayashi * Hiroyuki Aoki Naoya Soda
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Abstract
The points of equilibrium on which the magnitudes of magnetic (or electric) field vectors are zero are called the singular points and give the points of the zero magnetic (or electric) fields in the Phi science. The magnetic field produced by the point charges has the unique relation with the electric field produced by the point charges. Therefore, the present paper considers the electric fields produced by EMBED Equation.DSMT4 point charges of same magnitude of EMBED Equation.DSMT4 located at regular intervals on the circle and the point charge EMBED Equation.DSMT4 located at the center. These electric fields have various singular points in depending on the magnitude of EMBED Equation.DSMT4 . The equipotential lines passing through the saddle points among the singular points are called the singular equipotential lines in the present paper. The singular equipotential lines for the case of EMBED Equation.DSMT4 are divided into two figures for EMBED Equation.DSMT4 , are focused to the one figure for EMBED Equation.DSMT4 , are divided into two figures for EMBED Equation.DSMT4 . The outer figure has the cusp for EMBED Equation.DSMT4 and disappears for EMBED Equation.DSMT4 . The catastrophic phenomena appear for the cases in the neighborhoods of the cases of EMBED Equation.DSMT4 and EMBED Equation.DSMT4 . According to the theory of the Poincare-Hopf, the number of Euler and the sum of the indexes of singular points should agree. This is used to derive in the present paper that the cusps in the singular equipotential lines are the singular points of the index of 0. Similarly, the cases of EMBED Equation.DSMT4 and EMBED Equation.DSMT4 are considered and those results are shown. These singular equipotential lines are of equilibrium and beautiful. The figures of the singular equipotential lines are related to the projective figures of the knots and the links of the knot theory in mathematics. Devising the locations of the point charges to let the singular equipotential lines be produced in the nest stage, the three point charges-fractal model is proposed in the present paper. For its model, the figures of singular equipotential lines are illustrated in the nest state. The fractal dimension of its fractal model has been numerically calculated by the box-counting method and given with EMBED Equation.DSMT4 .
Keywords : zero magnetic fields, singular points, singular equipotential lines, catastrophic phenomena, fractal
Received : Feb. 27, 2012
Accepted : Apr. 4, 2012
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