\(F_{ L }sin(\theta )=F_{ a }\)
The attractive force between the electrons is the Lorentz's force. \(F_a\) is the force in the plane of the orbit that keeps the orbital radius constant due to the attraction of the neighboring orbit.
\( tan(\theta )=\cfrac {\Delta r }{ 2a_{ e } } \)
\(F_L=\cfrac { n_{ e }q^{ 2 } }{ 4\varepsilon _{ o }(2a_{ e })^{ 2 } } cos^2(\theta )=\cfrac { n_{ e }q^{ 2 } }{ 4\varepsilon _{ o }(2a_{ e })^{ 2 } } \cfrac { 4a^2_e }{ (2a_{ e })^{ 2 }+(\Delta r)^{ 2 } } =\cfrac { n_{ e }q^{ 2 } }{ 4\varepsilon _{ o } } \cfrac { 1 }{ (2a_{ e })^{ 2 }+(\Delta r)^{ 2 } } \)
Therefore,
\(F_{ a }=\cfrac { n_{ e }q^{ 2 } }{ 4\varepsilon _{ o }(2a_{ e })^{ 2 } } \cfrac { (2a_{ e })^{ 2 } }{ (2a_{ e })^{ 2 }+(\Delta r)^{ 2 } } \cfrac {(\Delta r)}{ \sqrt { (2a_{ e })^{ 2 }+(\Delta r)^{ 2 } } } \)
\(F_{ a }=\cfrac { n_{ e }q^{ 2 } }{ 4\varepsilon _{ o } } \cfrac { (\Delta r)}{ \left\{ (2a_{ e })^{ 2 }+(\Delta r)^{ 2 } \right\} ^{ 3/2 } } \)
\(\Delta r= r-r_e\)
\(F_{ a }=\cfrac { n_{ e }q^{ 2 } }{ 4\varepsilon _{ o } } \cfrac { r-r_e }{ \left\{ (2a_{ e })^{ 2 }+(r-r_e)^{ 2 } \right\} ^{ 3/2 } } \)
A brand new attractive force! And we have,
\(\cfrac { m_{ e }c^{ 2 } }{ r_{ e } } =Ac^{ 2 }+G\cfrac { m_{ a }m_{ e } }{ r^{ 2 }_{ e } }+\cfrac { n_{ e }q^{ 2 } }{ 4\varepsilon _{ o } } \cfrac { r-r_e }{ \left\{ (2a_{ e })^{ 2 }+(r-r_e)^{ 2 } \right\} ^{ 3/2 } } -\cfrac { Zq^{ 2 } }{ 4\varepsilon _{ o }(r_{ e }-r_{ p })^{ 2 } } -\cfrac { T_{ n }T_{ e } }{ 4\pi \tau_{ o }r^{ 2 }_{ e } } \)
a new expression for the forces in action on an orbiting electron. Under normal circumstances,
\(r=r_e\) and \(F_a=0\)
But this force will definitely play a part in ionization.
\(2a_e\) is twice the radius of an electron, and \(n_e\) the number of electrons in orbit minus one. We are assuming that the orbiting electrons are stacked closest possible in parallel orbits.
