\(U_{loss}=\cfrac{1}{2}\left[\varepsilon_oE^{ 2 }cos^{ 2 }(\theta )\left\{ 1-(\cfrac { \varepsilon _{ a } }{ \varepsilon _{ o } } )^{ 2 } \right\}+\mu_oB^{ 2 }sin^{ 2 }(\theta )\left\{ 1-(\cfrac { \mu _{ o } }{ \mu _{ a } } )^{ 2 } \right\} \right]\)
\(\cfrac { \partial \, U_{ loss } }{ \partial \, \theta } =\cfrac{1}{2}sin(2\theta )\left[ \mu _{ o }B^{ 2 }\left\{ 1-(\cfrac { \mu _{ o } }{ \mu _{ a } } )^{ 2 } \right\} -\varepsilon _{ o }E^{ 2 }\left\{ 1-(\cfrac { \varepsilon _{ a } }{ \varepsilon _{ o } } )^{ 2 } \right\} \right] \)
\(\cfrac { \partial ^{ 2 }\, U_{ loss } }{ \partial \, \theta ^{ 2 } } =cos(2\theta )\left[ \mu _{ o }B^{ 2 }\left\{ 1-(\cfrac { \mu _{ o } }{ \mu _{ a } } )^{ 2 } \right\} -\varepsilon _{ o }E^{ 2 }\left\{ 1-(\cfrac { \varepsilon _{ a } }{ \varepsilon _{ o } } )^{ 2 } \right\} \right] \)
when \(\theta=90^o\), \(\cfrac { \partial \, U_{ loss } }{ \partial \, \theta } =0\) and \(cos(2\theta)=-1\). The term from the second derivative,
\(\mu _{ o }B^{ 2 }\left\{ 1-(\cfrac { \mu _{ o } }{ \mu _{ a } } )^{ 2 } \right\} -\varepsilon _{ o }E^{ 2 }\left\{ 1-(\cfrac { \varepsilon _{ a } }{ \varepsilon _{ o } } )^{ 2 } \right\} \)
can determines whether it is maximum or minimum loss.
For copper,
\(\cfrac { \varepsilon _{ a } }{ \varepsilon _{ o } } \rightarrow\infty\)
\(\cfrac { \mu _{ o } }{ \mu _{ a } }\)=1/0.999994= 1.000006
This term simplifies to
\(\varepsilon _{ o }E^{ 2 }\left\{ (\cfrac { \varepsilon _{ a } }{ \varepsilon _{ o } } )^{ 2 }-1 \right\} -\mu _{ o }B^{ 2 }\left\{ (\cfrac { \mu _{ o } }{ \mu _{ a } } )^{ 2 }-1 \right\} \)
for minimum loss,
\(\mu _{ o }B^{ 2 }\left\{ (\cfrac { \mu _{ o } }{ \mu _{ a } } )^{ 2 }-1 \right\} \gt\varepsilon _{ o }E^{ 2 }\left\{ (\cfrac { \varepsilon _{ a } }{ \varepsilon _{ o } } )^{ 2 }-1 \right\} \)
\(B^{ 2 }\gt\cfrac { \varepsilon _{ o } }{ \mu _{ o } } E^{ 2 }\frac { \left\{ (\cfrac { \varepsilon _{ a } }{ \varepsilon _{ o } } )^{ 2 }-1 \right\} }{ \left\{ (\cfrac { \mu _{ o } }{ \mu _{ a } } )^{ 2 }-1 \right\} } \)
such that over all,
\(\cfrac { \partial ^{ 2 }\, U_{ loss } }{ \partial \, \theta ^{ 2 } }\gt0 \)
In general,
\(\cfrac { \varepsilon _{ o } }{ \mu _{ o } } = (c\varepsilon _{ o } )^2\)= (299792458*8.8541878176e-12)^2 = 7.046e-6
which makes the inequality possible even without \(B\) very large. Any material with a relative permittivity greater than one, but a relative permeability less than one (diamagnetic), graphite and sealed mercury, are suitable as flat antenna material. (Sliver is not suitable because of its high permittivity which would require a high \(B\).)
A coil is a flat antenna. And from the post "The Third Wave",
\(\nabla \times Y=-\cfrac { 4\pi }{ c^{ 2 } } \cfrac { \partial \, j_{ e } }{ \partial t } \)
\(j_e\) set off a wave perpendicular to the direction of \(Y\).
And the loss of this type of antenna,
\(U_{loss}=\mu_oB^{ 2 }sin^{ 2 }(\theta )\left\{ 1-(\cfrac { \mu _{ o } }{ \mu _{ a } } )^{ 2 } \right\} \)
since,
\(1-(\cfrac { \mu _{ o } }{ \mu _{ a } } )^{ 2 } \lt0\)
we have a small gain.
Why forgone the gain from the term
\( 1-(\cfrac { \varepsilon _{ a } }{ \varepsilon _{ o } } )^{ 2 } \)?
Because a high \(E\) will give you cancer. In practice, whether such an antenna will be effective is speculative. Try else never know.
