Given a configuration of charges in two or more groups, MxN the total number of force lines between them are fixed. Let's define a forces line per unit area term, \(\Phi\) over the area of a sphere around one of the group
\({\Phi_1}.{4\pi r^2_1}={\Phi_2}.{4\pi r^2_2}=M.N\)
and we have,
\(\Phi_1=\cfrac{{\Phi_2}.{4\pi r^2_2}}{4\pi r^2_1}=\cfrac{M.N}{4\pi r^2_1}\)
In general,
\(\Phi=\cfrac{M.N}{4\pi r^2}\)
\({\Phi}\) = number of force line per unit area = flux,
What is this force lines per unit area over a sphere? Image a force being divided over small points covering a sphere. Each of these points joins the center of the sphere along a radius. The force at each of these points is a force along the radial line of the sphere.
Compare this with,
\(F_c=\cfrac{q_M.q_N}{4\pi {\varepsilon_o} r^2}=\cfrac{Me.Ne}{{\varepsilon_o}}\cfrac{1}{4\pi r^2}=\cfrac{e^2}{{\varepsilon_o}}\cfrac{M.N}{4\pi r^2}=\cfrac{e^2}{{\varepsilon_o}}\Phi\)
We found that the \(\cfrac{1}{r^2}\) dependence is the result of the total flux being conserved and that the source of this total flux is,
\(F_{co}=\cfrac{e^2}{{\varepsilon_o}}\) and so,
\(F_c=F_{co}\Phi=\cfrac{e^2}{{\varepsilon_o}}\Phi\)
In a similar way,
\(F_m=\cfrac{Gm_Mm_N}{r^2}=4\pi G\cfrac{Mm_s.Nm_s}{4\pi r^2}=4\pi Gm^2_s\cfrac{MN}{4\pi r^2}\)
\(F_m=4\pi Gm^2_s\Phi\)
where \(m_s\) is an elemental mass, \(G\) is the gravitational constant. Let
\(F_{mo}=4\pi Gm^2_s\) and so,
\(F_m=F_{mo}\Phi=4\pi Gm^2_s\Phi\)
The point is the basic number counting in such formula. They assume that charges or mass are made up of countable elemental point quantities and there are distinct countable force lines. In the first place, discrete. And that the formulation can be separated into a source part and a flux part (flux are just lines).
If such formula are experimental and valid, then an elemental mass \(m_s\) exists already, thanks to Newton.
