Consider an EMW propagating down a sharp point of half angle \(\theta\),
Electric displacement, \(\varepsilon E\) perpendicular to the slop boundary is the same. The tangential component of \(E\) is also the same. So, the \(E\) field after the boundary is,
\(E^2_f=\left\{ Esin(\theta ) \right\} ^{ 2 }+\left\{ \cfrac { \varepsilon_a }{ \varepsilon _{ o } } Ecos(\theta ) \right\} ^{ 2 }\)
Loss in \(E\) is,
\(E^{ 2 }_{ loss }=E^{ 2 }-E^2_f\)
\(E^{ 2 }_{ loss }=E^{ 2 }\left\{ 1-sin^{ 2 }(\theta )-(\cfrac { \varepsilon_a }{ \varepsilon _{o } } )^{ 2 }cos^{ 2 }(\theta ) \right\} \)
\(E^{ 2 }_{ loss }=E^{ 2 }cos^{ 2 }(\theta )\left\{ 1-(\cfrac { \varepsilon _{ a } }{ \varepsilon _{ o } } )^{ 2 } \right\} \)
The perpendicular component of the \(B\) field is the same, and the tangential \(H\) field is the same
\(\cfrac{B_c}{\mu_o}=\cfrac{Bsin(\theta)}{\mu_a}\)
The \(B\) field after the boundary is,
\(B^2_f=\left\{Bcos(\theta)\right\}^2+\left\{\cfrac{\mu_o}{\mu_a}Bsin(\theta)\right\}^2\)
Loss in \(B\) is,
\(B^{ 2 }_{ loss }=B^{ 2 }-B^2_f\)
\(B^{ 2 }_{ loss }=B^{ 2 }-(\left\{Bcos(\theta)\right\}^2+\left\{\cfrac{\mu_o}{\mu_a}Bsin(\theta)\right\}^2)\)
\(B^{ 2 }_{ loss }=B^{ 2 }\left\{ 1-cos^{ 2 }(\theta )-(\cfrac { \mu _{ o } }{ \mu _{ a } } )^{ 2 }sin^{ 2 }(\theta ) \right\} \)
\(B^{ 2 }_{ loss }=B^{ 2 }sin^{ 2 }(\theta )\left\{ 1-(\cfrac { \mu _{ o } }{ \mu _{ a } } )^{ 2 } \right\} \)
Total loss in energy crossing the boundary,
\(U_{loss}=\cfrac{1}{2}\varepsilon_oE^2_{loss}+\cfrac{1}{2}\mu_oB^2_{loss}\)
\(U_{loss}=\cfrac{1}{2}\left[\varepsilon_oE^{ 2 }cos^{ 2 }(\theta )\left\{ 1-(\cfrac { \varepsilon _{ a } }{ \varepsilon _{ o } } )^{ 2 } \right\}+\mu_oB^{ 2 }sin^{ 2 }(\theta )\left\{ 1-(\cfrac { \mu _{ o } }{ \mu _{ a } } )^{ 2 } \right\} \right]\)
So what?
