The respective posts where this mistake propagates has been changed.
From the post "Opps! Lucky Me", we suggested that,
\(F=-\cfrac{d\,E}{d\,r}=-\psi_A\)
where \(E\) is the energy in a infinitely thin spheric shell passing through a point \(r\), \(r\) distance from a point particle center. And
\(\cfrac{d\,E}{d\,r}\)
is the change in \(E\) along the radial line.
This is again wrong. This force must be divided by \(4\pi r^2\), as this total force is redistributed over the surface area of the sphere, to obtain \(F\), the force along a radial line.
\(F=-\cfrac{1}{4\pi r^2}\cfrac{d\,E}{d\,r}=-\cfrac{\psi_A}{4\pi r^2}\)
and we have,
\(F=-\psi\)
when
\(\psi_A=\psi.4\pi r^2\)
And for \(F\) to have a inverse square law dependence,
\(\psi\propto\cfrac{1}{r^2}\)
This hopefully is the last of the issue.
