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

TL;DR

This paper proposes a unified framework for Coulomb's law and Newton's gravity based on membrane oscillations, revealing a quantized parameter that transitions between electrostatic and gravitational forces.

Contribution

It introduces a membrane-based approach that unifies Coulomb's law and Newton's gravity through a common oscillation model with quantized parameters.

Findings

The unified formula retains the inverse square law form.

Different force laws are obtained at specific parameter values.

The parameter $\xi$ quantizes the force formulas.

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 ($F_C$) and the Newton's law of universal gravitation $(F_G)$. 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 the same spatial period $\xi\pi/k$ where $\xi$ is a dimensionless parameter and $k$ 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 $F_C$ when the dimensionless parameter $\xi =274$ and like $F_G$ when $\xi = 1.14198$x$10^{45}$. It was found that the values of the parameter $\xi$ quantize the formula from which $F_C$ and $F_G$ are obtained as particular cases.