Moving Trough

The electron orbital radius,  \(r_e\)  was obtained from equating forces, together with the drag of free space at near terminal velocity.   \(r_e\)  was then made to vary with temperature,  \(T\) as the drag factor,  \(A\) varies with  temperature.  For all values of  \(r_e\),  the resultant force on the electron is zero.

\(\because \sum F=0\)

Force per unit inertia,  \(F=-\cfrac{\partial\,PE_{re}}{\partial\,r_e}\)

as in

\(E=-\cfrac{d\,V}{d\,x}\)

So,

\(\cfrac{\partial  PE_{re}}{\partial r_e}=0\)

 As such its potential,  \(PE_{re}\)  is at an extrema for all values of  \(r_e\).    And we understand it to be a minimum as the electrons are in stable orbits.  (If it is a maxima, a small perturbation will result in a positive force that is the negative of the gradient, pushing the electron further away from the extreme point.  ie unstable.)  This potential trough moves along the  \(r_e\)  vs  \(T\)  profile as  \(T\)  changes.

In other words,  \(PE_{re}\)  is always a minimum on the  \(r_e\)  vs  \(T\)  profile.

Have a nice day.