\(F_a=-\cfrac { n_{ e }q^{ 2 } }{ 4\varepsilon _{ o } } \cfrac { r-r_e }{ \left\{ (2a_{ e })^{ 2 }+(r-r_e)^{ 2 } \right\} ^{ 3/2 } } d\,r\)
is
\(\cfrac{d\,F_a}{d\,r}=-\cfrac { n_{ e }q^{ 2 } }{ 4\varepsilon _{ o } }\cfrac{(2a_e)^2-2(r-r_e)^2}{\left[(2a_e)^2+(r-r_e)^2\right]^{5/2}}\)
An illustrative plot is given below,
The first derivative peaks at \(r=r_e\) with a value of
\(F^{'}_{a\, max}=-\cfrac { n_{ e }q^{ 2 } }{ 4\varepsilon _{ o } }\cfrac{1}{(2a_e)^3}\)
If we approximate this with a simple oscillation system,
\(F=ma=-kx\)
we have resonance at,
\(\omega=\sqrt{\cfrac { n_{ e }q^{ 2 } }{ 4\varepsilon _{ o } m_e}\cfrac{1}{(2a_e)^3}}\)
where \(n_e\) is the number of electrons in the stack orbits and \(a_e\) the electron's radius.
This is the frequency at which the electron in its \(B\) orbit will oscillate naturally. A driving force due to heat or magnetic field varying at this frequency will drive the system into resonance. A splendid display of light and magic!
