From the post "So, So What?",
\(\cfrac { \partial ^{ 2 }\, U_{ loss } }{ \partial \, \theta ^{ 2 } } =cos(2\theta )\left[ \mu _{ o }B^{ 2 }\left\{ 1-(\cfrac { \mu _{ o } }{ \mu _{ a } } )^{ 2 } \right\} -\varepsilon _{ o }E^{ 2 }\left\{ 1-(\cfrac { \varepsilon _{ a } }{ \varepsilon _{ o } } )^{ 2 } \right\} \right] \)
then we come to the term,
\(\left[ \mu _{ o }B^{ 2 }\left\{ 1-(\cfrac { \mu _{ o } }{ \mu _{ a } } )^{ 2 } \right\} -\varepsilon _{ o }E^{ 2 }\left\{ 1-(\cfrac { \varepsilon _{ a } }{ \varepsilon _{ o } } )^{ 2 } \right\} \right] \)
which we observe, flips the sign of the expression with increasing \(E\). An increase in \(E\) can switch the system from minimum loss to maximum loss.
So, a pointy antenna has low loss only if \(E\) is keep within a range,
\(E^{ 2 }<B^{ 2 }\cfrac { \mu _{ o } }{ \varepsilon _{ o } } \frac { \left\{ 1-(\cfrac { \mu _{ o } }{ \mu _{ a } } )^{ 2 } \right\} }{ \left\{ 1-(\cfrac { \varepsilon _{ a } }{ \varepsilon _{ o } } )^{ 2 } \right\} } \)
And as long as \(E\) is kept high, a flat antenna has minimum loss,
\(B^{ 2 }<E^{ 2 }\cfrac { \varepsilon _{ o } }{ \mu _{ o } } \frac { \left\{ 1-(\cfrac { \varepsilon _{ a } }{ \varepsilon _{ o } } )^{ 2 } \right\} }{ \left\{ 1-(\cfrac { \mu _{ o } }{ \mu _{ a } } )^{ 2 } \right\} } \)
So, So, So, What?
