From the previous post "Loss, So What",
\(U_{loss}=\cfrac{1}{2}\left[\varepsilon_oE^{ 2 }cos^{ 2 }(\theta )\left\{ 1-(\cfrac { \varepsilon _{ a } }{ \varepsilon _{ o } } )^{ 2 } \right\}+\mu_oB^{ 2 }sin^{ 2 }(\theta )\left\{ 1-(\cfrac { \mu _{ o } }{ \mu _{ a } } )^{ 2 } \right\} \right]\)
When the point is very sharp, \(\theta\rightarrow0\), \(sin(\theta)=0\) and all the loss is wholly due to \(E\). When the point flatten, \(\theta\rightarrow90^o\), \(cos(\theta)=0\) and all the loss is due to \(B\).
Consider,
\(\cfrac{\partial\, U_{loss}}{\partial\,\theta}=\cfrac{1}{2}\left[-\varepsilon_oE^{ 2 }2cos(\theta )sin(\theta)\left\{ 1-(\cfrac { \varepsilon _{ a } }{ \varepsilon _{ o } } )^{ 2 } \right\}+\mu_oB^{ 2 }2cos(\theta)sin(\theta )\left\{ 1-(\cfrac { \mu _{ o } }{ \mu _{ a } } )^{ 2 } \right\} \right]\)
\(\cfrac { \partial \, U_{ loss } }{ \partial \, \theta } =cos(\theta )sin(\theta )\left[ \mu _{ o }B^{ 2 }\left\{ 1-(\cfrac { \mu _{ o } }{ \mu _{ a } } )^{ 2 } \right\} -\varepsilon _{ o }E^{ 2 }\left\{ 1-(\cfrac { \varepsilon _{ a } }{ \varepsilon _{ o } } )^{ 2 } \right\} \right] \)
\(=\cfrac{1}{2}sin(2\theta )\left[ \mu _{ o }B^{ 2 }\left\{ 1-(\cfrac { \mu _{ o } }{ \mu _{ a } } )^{ 2 } \right\} -\varepsilon _{ o }E^{ 2 }\left\{ 1-(\cfrac { \varepsilon _{ a } }{ \varepsilon _{ o } } )^{ 2 } \right\} \right] \)
\(\cfrac { \partial ^{ 2 }\, U_{ loss } }{ \partial \, \theta ^{ 2 } } =cos(2\theta )\left[ \mu _{ o }B^{ 2 }\left\{ 1-(\cfrac { \mu _{ o } }{ \mu _{ a } } )^{ 2 } \right\} -\varepsilon _{ o }E^{ 2 }\left\{ 1-(\cfrac { \varepsilon _{ a } }{ \varepsilon _{ o } } )^{ 2 } \right\} \right] \)
\(\cfrac { \partial ^{ 2 }\, U_{ loss } }{ \partial \, \theta ^{ 2 } }\gt0\) when \(\theta=0^o\), this implies a sharp point has minimum loss and is mainly due to \(E\).
\(\cfrac { \partial ^{ 2 }\, U_{ loss } }{ \partial \, \theta ^{ 2 } }\lt0\) when \(\theta=90^o\), this implies a flat surface has maximum loss and is mainly due to \(B\).
Note: For material \(\cfrac{\mu_a}{\mu_o}\gt\gt1\) and \(\cfrac{\varepsilon_a}{\varepsilon_o}\gt\gt1\).
So, So What?
