So, So What?

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?