A simple way of unifying the formulas for the Coulomb’s law and Newton’s law of the universal gravitation: An approach based on membranes

Abstract

In this work, a new approach is presented with the aim of showing a simple way of unifying the classical formulas for the forces of the Coulomb’s law of electrostatic interaction <mml:math display="inline"> <mml:mrow> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>C</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> and the Newton’s law of universal gravitation <mml:math display="inline"> <mml:mrow> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>G</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . In this approach, these two forces are of the same nature and are ascribed to the interaction between two membranes that oscillate according to different curvature functions with spatial period <mml:math display="inline"> <mml:mrow> <mml:mfrac> <mml:mrow> <mml:mi>ξ</mml:mi> <mml:mi>π</mml:mi> </mml:mrow> <mml:mi>k</mml:mi> </mml:mfrac> </mml:mrow> </mml:math> , where <mml:math display="inline"> <mml:mi>ξ</mml:mi> </mml:math> is a dimensionless parameter and <mml:math display="inline"> <mml:mi>k</mml:mi> </mml:math> is a wave number. Both curvature functions are solutions of the classical wave equation with wavelength given by the de Broglie relation. This new formula still keeps itself as the inverse square law, and it is like <mml:math display="inline"> <mml:mrow> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>C</mml:mi> </mml:msub> </mml:mrow> </mml:math> when the dimensionless parameter <mml:math display="inline"> <mml:mrow> <mml:mi>ξ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>274</mml:mn> </mml:mrow> </mml:math> and like <mml:math display="inline"> <mml:mrow> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>G</mml:mi> </mml:msub> </mml:mrow> </mml:math> when <mml:math display="inline"> <mml:mrow> <mml:mi>ξ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1.14198</mml:mn> <mml:mo>×</mml:mo> <mml:msup> <mml:mrow> <mml:mn>10</mml:mn> </mml:mrow> <mml:mrow> <mml:mn>45</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> . It was found that the values of the parameter <mml:math display="inline"> <mml:mi>ξ</mml:mi> </mml:math> quantize the formula from which <mml:math display="inline"> <mml:mrow> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>C</mml:mi> </mml:msub> </mml:mrow> </mml:math> and <mml:math display="inline"> <mml:mrow> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>G</mml:mi> </mml:msub> </mml:mrow> </mml:math> are obtained as particular cases.