\(\cfrac { m_{ e }v^{ 2 } }{ r_{ e } } =\cfrac { q^{ 2 } }{ 4\pi \varepsilon _{ o }r^{ 2 }_{ e } } -qE^{'}_{or}\)
\(E_{or}=\cfrac{\tau_o}{2}\cfrac{Tv_{\small{T}}}{\{r^2_{\small{T}}+(r_{or}-r_{e})^2\}^{3/2}}.({r_{or}-r_{e}})\)
If we again assume that \(r_{e}\lt\lt r_{or}\),
\(qE_{or}=A.T\) and,
\(A=\cfrac{\tau_o}{2}\cfrac{qv_{\small{T}}}{\{r^2_{\small{T}}+r_{or}^2\}^{3/2}}.{r_{or}}\)
then
\( m_{ e }v^{ 2 }.r_{ e }=\cfrac { q^{ 2 } }{ 4\pi \varepsilon _{ o } } -AT.r^{ 2 }_{ e }\)
\( AT.r^{ 2 }_{ e }+m_{ e }v^{ 2 }.r_{ e }-\cfrac { q^{ 2 } }{ 4\pi \varepsilon _{ o } } =0\)
\( r_{ e }=\cfrac { -m_{ e }v^{ 2 }\pm \sqrt { (m_{ e }v^{ 2 })^{ 2 }+4AT\cfrac { q^{ 2 } }{ 4\pi \varepsilon _{ o } } } }{ 2AT } \)
Consider only \( r_{ e }\gt0\),
\( r_{ e }=\cfrac { m_{ e }v^{ 2 } }{ 2AT } \left( \sqrt { 1+AT\cfrac { q^{ 2 } }{ \pi \varepsilon _{ o }(m_{ e }v^{ 2 })^{ 2 } } } -1 \right) \)
A illustrative plot shows that \(r_e\) decreases monotonously with \(T\)
In this case the pull of the weak field due to a spinning positive temperature particle is stronger. The electron rolls inside the proton orbit.
Does this actually happen?
