The significance of \(p\) being inversely proportional to \(T\) is, in the expression for
\(PE_{ e }=\cfrac { 1 }{ 2 } \cfrac { m_{ e }T^{ 2 }c^{ 2 } }{{ r }^{ 2 }_{ e }} .\cfrac { 1 }{ (2\omega p)^{ 2 }+(\omega ^{ 2 }_{ o }-\omega ^{ 2 })^{ 2 } } \)
If we let \(p=\cfrac{A}{T}\), where \(A\) is any constant.
\(PE_{ e }=\cfrac { 1 }{ 2 } \cfrac { m_{ e }T^{ 2 }c^{ 2 } }{{ r }^{ 2 }_{ e }} .\cfrac { 1 }{ (2\omega \cfrac{A}{T})^{ 2 }+(\omega ^{ 2 }_{ o }-\omega ^{ 2 })^{ 2 } } \)
\(PE_{ e }=\cfrac { 1 }{ 2 } \cfrac { m_{ e }T^{ 4 }c^{ 2 } }{{ r }^{ 2 }_{ e }} .\cfrac { 1 }{ (2\omega A)^{ 2 }+T^2(\omega ^{ 2 }_{ o }-\omega ^{ 2 })^{ 2 } } \)
At near resonance \(\omega_o\simeq \omega\), the term \(T^2(\omega ^{ 2 }_{ o }-\omega ^{ 2 })^{ 2 }\) is small, and
\(PE_{ e }\simeq \cfrac { 1 }{ 2 } \cfrac { m_{ e }T^{ 4 }c^{ 2 } }{{ r }^{ 2 }_{ e }} .\cfrac { 1 }{ (2\omega A)^{ 2 }} \)
\(PE_{ e }\propto T^4\)
And at resonance,
\(\omega=\sqrt{\omega^2_o-2p^2}\), \(\omega^2_o-\omega^2=2p^2\)
\((2wA)^2=4A^2(\omega_o^2-2p^2)\)
\((2wA)^{ 2 }+T^{ 2 }(\omega ^{ 2 }_{ o }-\omega ^{ 2 })^{ 2 }=4A^{ 2 }(\omega ^{ 2 }_{ o }-2\cfrac { A^{ 2 } }{ T^{ 2 } } )+4\cfrac { A^{ 4 } }{ T^{ 2 } } =4A^{ 2 }(\omega ^{ 2 }_{ o }-\cfrac { A^{ 2 } }{ T^{ 2 } } )\)
\(PE_{ e }= \cfrac { 1 }{ 8 }\cfrac{1}{A^{ 2 }} \cfrac { m_{ e }c^{ 2 } }{{ r }^{ 2 }_{ e }}\cfrac{1}{(\omega ^{ 2 }_{ o }-\cfrac { A^{ 2 } }{ T^{ 2 } } ) }.T^4 \)
When \(\omega_o=\omega\),
\(PE_{ e }= \cfrac { 1 }{ 8 }\cfrac{1}{A^{ 2 }} \cfrac { m_{ e }c^{ 2 } }{{ r }^{ 2 }_{ e }}\cfrac{1}{\omega ^{ 2 }_{ o }}.T^4 \)
This relation is at near resonance and at resonance, but not for all frequencies of the driving force. And definitely not for all radiated "frequencies", the two frequency ranges are not the same thing.
