Fat Electrons and the Learned Inverse Square

Areas of two concentric spheres of radii ${R}_{1}$ and ${R}_{2}$,

 $Area_1 = 4\pi {R}^2_{1}$

$Area_2 = 4 \pi {R}^2_{2}$

If we consider flux emanating from a constant source of $W$, then

$flux_{R1} = \cfrac{W}{4\pi {R}^2_{1}}$

$flux_{R2} = \cfrac{W}{4\pi {R}^2_{2}}$

from which we have the inverse square law.

$flux  \propto  \cfrac{1}{{R}^{2}}$

However, if

$W \propto \cfrac{4}{3}\pi R^3$

where the volume immediately below is of concern, like space density, then

$flux \propto \cfrac{ R}{3}$ as in the case of dielectric in electrostatic.

This is the case when electrons are trapped in a oil drop, a dielectric.  The factor of $\cfrac{1}{3}$ is serious.  The term $R$ may cancel in the final formulation, especially if the same droplet is swinging about, but the factor $\cfrac{1}{3}$ does not go away no matter how you orientate the E flux.

This is how the fat electrons slimmed down.