Resonance Strikes Again

The first derivative of the force,

$F_a=-\cfrac { n_{ e }q^{ 2 } }{ 4\varepsilon _{ o } } \cfrac { r-r_e }{ \left\{ (2a_{ e })^{ 2 }+(r-r_e)^{ 2 } \right\} ^{ 3/2 } } d\,r$

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!