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\begin{document}

\Prova{2}{10 de maio de 2011}

\hrulefill

\vspace{2mm}

\begin{QstsFig}\ (3.5) O campo el\'{e}trico de uma onda  eletromagn\'{e}tica \'{e} se propagando num meio de 
 permitividade  $\,\bf  \epsilon \,$ e  permeabilidade $\,\bf  \mu \,$ \'{e} dado por

\begin{center}
%$\,\vec {\bf  E}\,=\,2E_{o}\,\cos (kz\,-\,\omega t)\vec {\bf  a}_{x}\,+\,3E_{o}\,\sen (kz\,-\,\omega t)\vec {\bf  a}_{y}$

$\,\vec {\bf  E}\,=\,2\,cos [2\times 10^{4}(z\,-\,t)]\vec {\bf  a}_{x}\,+\,3\,sen [2\times 10^{4}(z\,-\,t)]\vec {\bf  a}_{y}$
\end{center}

\noindent Determine:

\begin{itemize}
\item{(1.0)}  a) o vetor de onda, o comprimento de onda, a frequ\^{e}ncia e a velocidade da onda;   
\item{(1.0)}  b) o vetor indu\c{c}\~{a}o magn\'{e}tica da onda;
\item{(1.0)}  c) o vetor de Poynting e a intensidade da onda;
\item{(0.5)}  d) o estado de polariza\c{c}\~{a}o da onda. Justifique sua resposta;
\end{itemize}
\end{QstsFig}
\newpage

\begin{QstsFig}{q1.eps}\ (4.0) Uma fonte esf\'{e}rica de $\bf 4\pi \,\,kw\,$ de pot\^{e}ncia emite radia\c{c}\~{a}o, n\~{a}o polarizada, que 
atinge uma plaquinha de $\bf 10\,\,cm^{2}\,$ de \'{a}rea, cujo coeficiente de reflex\~{a}o \'{e} $\bf 0.8$ e que est\'{a} a $\bf 10\,\,m\,$ da 
fonte. Sendo $\bf c\,=\,3\times 10^{8}\,\,m/s$, determine: 

\begin{itemize}
\item{(1.0)}  a) a intensidade da radia\c{c}\~{a}o ao atingir a plaquinha;
\item{(1.0)}  b) a quantidade de movimento transferido \`{a} plaquinha;
\item{(1.0)}  c) a press\~{a}o da radia\c{c}\~{a}o sobre a plaquinha.
\end{itemize}

Suponha, agora, que radia\c{c}\~{a}o da fonte do enunciado incida sobre dois polar\'{o}ides, cujas eixos de polariza\c{c}\~{a}o 
formem entre si um \^{a}gulo de  $\bf 45^{o}$ e determine:

\begin{itemize}
\item{(1.0)}  d) a intensidade da radia\c{c}\~{a}o ap\'{o}s passar no segundo polar\'{o}ide.
\end{itemize}
\end{QstsFig}

\newpage

\begin{QstsFig}\ (2.5) O campo el\'{e}trico e o campo magn\'{e}tico, em coordenadas cil\'{\i}ndricas, em uma certa regi\~{a}o de 
permitividade  $\bf \epsilon $ e permeabilidade $\bf \mu $ s\~{a}o dados por:

\begin{center}     


$\,\vec {\bf  E}\,=\,\frac {4\rho _{o}}{\epsilon R^{2}}\,r^{2}\,\vec {\bf  a}_{r}$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
$\,\vec {\bf  B}\,=\,\frac {4\mu J_{o}}{R^{2}}\,r^{2}\,\vec {\bf  a}_{\phi }$   \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \                                                                                                                                

                                                                           
\end{center}

\noindent Determine:

\begin{itemize}
\item{(1.25)}  a) a densidade da carga nas regi\~{o}es $\bf r\,<\,R$ e $\bf r\,>\,R$;
\item{(1.25)}  b) a densidade de corrente na regi\~{a}o $\bf r\,<\,R$.

\end{itemize}
\end{QstsFig}

%\vspace{1cm}

%\begin{QstsFig}\ (1.0) 

%\begin{itemize}
%\item{(0.5)}  a) 
%\item{(1.5)}  b) 
%\item{(1.0)}  c) 
%\item{(1.0)}  d) 
%\end{itemize}

%\end{QstsFig}



%\end{document}

\newpage

{\Large  Formul\'{a}rio

\vspace{10mm}

%\noindent  $\vec  {D}\,=\,\epsilon _{o}\,\vec  {E}\,+\,\vec  {P} $ \ \ \ \ \ \ \ \
%$\vec  {B}\,=\,\mu _{o}\,{\,\vec  {H}\,+\,\vec  {M}}$ \ \ \ \ \ \ \ \
%$\vec  {D}\,=\,\epsilon \,\vec {E} $ \ \ \ \ \ \ \ \
%$\vec  {B}\,=\,\mu \,\vec {H} $

%\vspace{3mm}

\noindent  $\nabla \vec {D}\,=\,\rho $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
$\nabla \times \,\vec  {H}\,=\,\vec {J}\,+\,\frac {\partial \vec {D}}{\partial  t}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
%\sigma _{p}\,=\,\vec {P}.\vec {n} $

\vspace{3mm}

\noindent  $\nabla {\vec {D}}\,=\,\frac {\partial D_{x}}{\partial  x}\,+\,\frac {\partial D_{y}}{\partial y}\,
+\,\frac  {\partial D_{z}}{\partial z}$

\vspace{3mm}

\noindent  $\nabla {\vec {D}}\,=\,\frac {1}{r}\frac {\partial }{\partial  r}(rD_{r})\,+\,
\frac {1}{r}\frac {\partial D_{\phi}}{\partial \phi}\,
+\,\frac {\partial D_{z}}{\partial z}$ (coord. cil\'{\i}ndricas)

%\noindent  $\nabla {\vec {D}}\,=\,\frac {1}{r^2}\frac {\partial }{\partial  r}(r^2D_{r})\,+\,
%\frac {1}{rsen\theta }\frac {\partial}{\partial \theta }(sen\theta D_{\theta })\,
%+\,\frac {1}{rsen\theta }\frac {\partial D_{\phi }}{\partial \phi }$


\vspace{3mm}

\noindent  $\nabla  \times \vec {H}\,=\,(\frac {\partial  H_{z}}{\partial  y}\, - \,\frac {\partial H_{y}}{\partial  z})\vec {i}\,+ \,(
\frac {\partial H_{x}}{\partial  z}\, -\,\frac {\partial H_{z}}{\partial  x})\,\vec {j}\,+
 \,(\frac {\partial H_{y}}{\partial  x}\,- \,\frac {\partial H_{x}}{\partial  y})\,\vec {k} $

\vspace{3mm}

\noindent  $\nabla  \times \vec {H}\,=\,(\frac {1}{r}\frac {\partial  H_{z}}{\partial  \phi }\, - \,\frac {\partial H_{\phi }}{\partial  z})\vec {a}_{r}\,+ 
\,(\frac {\partial H_{r}}{\partial  z}\, -\,\frac {\partial H_{z}}{\partial  r})\,\vec {a}_{\phi}\,+
\frac {1}{r} \,(\frac {\partial {(rH_{\phi }})}{\partial  r}\,-\,\frac {\partial H_{r}}{\partial  \phi })\,\vec {a}_{z} $ (coord. cil\'{\i}ndricas)


\vspace{3mm}


$\vec {k}\,\times \,\vec {E}\,=\,\omega \vec {B}
\ \ \ \ \ \ \ \ \ \ \ \ \  \vec {S}\,=\,\frac {1}{\mu _{o}}\,\vec {E}\,\times \,\vec {B}
\ \ \ \ \ \ \ \ \ \ \ \ \  P\,=\,f_{a}\,\frac {<S>}{c}\,+\,2f_{r}\,\frac {<S>}{c}\ \ \ \ \ \ \ \ \ \ \ \ \ E_{o}\,=\,cB_{o} $

\vspace{3mm}

$ <p>\,=\,f_{a}\,\frac  {<U>}{c} \,+\,2\,f_{r}\,\frac  {<U>}{c} \ \ \ \ \ \ \ \ \ \ \ \ \  P_{ot}\,=\, \int \vec {S}.\vec {dA}
\ \ \ \ \ \ \ \ \ \ \ \ \  I\,=\,I_{o}\,cos^{2}\,\theta $

\vspace{3mm}
$ D_{1n}\,=\,D_{2n} \ \ \ \ \ \ \ \ \ \ \ \ \  E_{1t}\,=\,E_{2t} \ \ \ \ \ \ \ \ \ \ \ \ \ B_{1n }\,=\,B_{2n}  \ \ \ \ \ \ \ \ \ \ \ \ \
H_{1t}\,=\,H_{2t}$

\vspace{4mm}

Interfer\^{e}ncia

$m\lambda \,=\,d\,sen\,\theta \ \ \ \ \ \ \ \ \ \ \ \ (m\,+\,\frac{1}{2})\lambda \,=\,d\,sen\,\theta \ \ \ \ \ m\,=\,0, \pm 1, \pm 2, \pm 3 .....$

\vspace{4mm}

$I\,=\,I_{o}\, cos^{2}\,(\frac  {\pi d\,sen\,\theta }{\lambda })$

\vspace{4mm}

Difra\c{c}\~{a}o

$sen\,\theta\,=\,1.22\,\frac {\lambda }{a}$

\vspace{4mm}

$m\lambda \,=\,a\,sen\,\theta \ \ \ \ \ \ \ \ \ \ \ \ (m\,+\,\frac{1}{2})\lambda \,=\,a\,sen\,\theta \ \ \ \ \
m\,=\,0, \pm 1, \pm 2, \pm  3 ..... $

\vspace{4mm}

$I\,=\,I_{o}\, (\frac {sen\,\frac {\pi a\,sen\,\theta }{\lambda }}{ \frac {\pi a\,sen\,\theta }{\lambda }})^{2}        $

\vspace{4mm}

R\^{e}de de difra\c{c}\~{a}o

\vspace{4mm}

$ m\lambda \,=\,d\,sen\,\theta \ \ \ \ \ \ \ \ \ \ \ \ (m\,+\,\frac{1}{2})\lambda \,=\,d\,sen\,\theta \ \ \ \ \ m\,=\,0, \pm 1, \pm 2, \pm 3 ..... $

\vspace{4mm}

$I\,=\,I_{o}\, cos^{2}\,(\frac  {\pi d\,sen\,\theta }{\lambda })(\frac {sen\,\frac {\pi a\,sen\,\theta }{\lambda }}{ \frac {\pi a\,sen\,\theta }{\lambda }})^{2} $

\vspace{4mm}

$R\,=\,\frac {\lambda }{\lambda _{2}\,-\,\lambda _{1}}\,=\,\frac {\Delta \lambda }{\lambda } \ \ \ \ \ \ \ \ \ \ \ \  R\,=\,m\,N   $

}


\vspace{3mm}

\noindent $\, $
\noindent $\, $
\noindent $\, $




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