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\begin{document}

\Prova{1}{26 de mar\c{c}o  de 2015}

\hrulefill

\vspace{2mm}




\begin{QstsFig}\ (4.0) Um condutor cil\'{\i}ndrico de raio $\bf \,R\,$, carregado com densidade superficial de carga $\,\bf  -\,\mat{\sigma }$, est\'{a} 
envolto por  uma casca cil\'{\i}ndrica  de raio interno $\,\bf  R\,$ e raio externo $\,\bf  2R\,$ de permissividade 
$\,\bf  \mat{\epsilon }_{1}\,$  e a regi\~{a}o  $\,\bf  r\geq 2R\,$ tem outro material  de permissividade  $\,\bf  \mat{\epsilon }_{2}\,$.

\begin{itemize}
\item{(0.5)}  a) Mostre que o vetor densidade de fluxo el\'{e}trico \'{e} $\vec {\bf D}\,=\,-\,\frac {\bf \mat{\sigma }R}{\bf r}\;\vec {\bf a}_{\bf r} $.
\item{(1.0)}  a) Determine os vetores $\,\vec {\bf  D}\,$, $\,\vec {\bf  E}\,$  e $\,\vec {\bf  P}\,$ na regi\~{a}o $\,\bf  2R>r>R$.
\item{(1.0)}  b) Determine os vetores $\,\vec {\bf  D}\,$, $\,\vec {\bf  E}\,$  e $\,\vec {\bf  P}\,$  na regi\~{a}o $\,\bf  r>2R$.
\item{(1.0)}  c) Determine a densidade de carga de polariza\c{c}\~{a}o na super\'{\i}cie $\,\bf  r\,=\,2R\,$ entre os dois diel\'{e}tricos.
\item{(0.5)}  d) Determine a carga total por unidade de comprimento do condutor.
\end{itemize}
\end{QstsFig}

\newpage

\begin{QstcFig}{15q2p1_ice.eps}\ (3.0) Uma placa diel\'{e}trica de constante  diel\'{e}trica $\,\bf k_{2}\,$  est\'{a} imersa  em um meio cuja 
constante  diel\'{e}trica \'{e} $\,\bf k_{1}$, onde existe um campo el\'{e}trico  $\,\vec {\bf E}_{o}\,$, que forma um \^{a}ngulo 
 $\,\bf \mat{\alpha }\,$ com o eixo  $\,\bf x$, como mostra a figura. Determine:
\begin{itemize}
\item{(1.5)}  a) os vetores $\,\vec {\bf  D}\,$, $\,\vec {\bf  E}\,$ e $\,\vec {\bf  P}\,$ dentro da placa, em fun\c{c}\~{a}o de   $\,\bf k_{1}$, $\,\bf k_{2}\,$, 
$\,\bf \mat{\epsilon }_{o}\,$, $\,\bf E_{o}\,$ e $\,\bf \mat{\alpha }$;
\item{(1.5)}  b) a densidade de carga de polariza\c{c}\~{a}o resultante  na superf\'{\i}cie (1) da placa. 
\end{itemize}
\end{QstcFig}

\newpage

\begin{QstsFig}\ (3.0) Um tor\'{o}ide de sec\c{c}\~{a}o retangular de lados $\,\bf R\,$ e $\,\bf a$, com $\,\bf N\,$ espiras, tem raio interno $\,\bf R\,$ 
e raio externo $\,\bf 2R\,$. O interior do tor\'{o}ide est\'{a} preenchido com um material de permeabilidades  magn\'{e}tica  
$\,\bf \mat{\mu }$. A corrente que circula no tor\'{o}ide \'{e} $\,\bf i_{o}$.

\begin{itemize}
\item{(1.0)}  a) Fa\c{c}a um desenho do tor\'{o}ide e do sistema de coordenadas adotado, indique no seu desenho a corrente  $i_{o}$ e obtenha usando a lei de Amp\`{e}re 
o m\'{o}dulo do vetor intensidade magn\'{e}tica dentro e fora do tor\'{o}ide.
\item{(1.0)}  b) Obtenha os vetores indu\c{c}\~{a}o magn\'{e}tica  e magnetiza\c{c}\~{a}o no interior do tor\'{o}ide. 
\item{(1.0)}  c) Obtenha a corrente de magnetiza\c{c}\~{a}o. 
\end{itemize}
\end{QstsFig}

\newpage



{\bf 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}

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

\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)

%\vspace{3mm}

%\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{5mm}

Ondas eletromagn\'{e}ticas


\vspace{3mm}


$\vec {k}\,\times \,\vec {E}\,=\,\omega \vec {B}
\ \ \ \ \ \ \ \ \ \ \ \ \  \vec {S}\,=\,\frac {1}{\mu }\,\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 \ \ \ \ \ \ \ \ \    c\,=\,3\times 10^{8}\,m/s$


\vspace{3mm}

%$ D_{1n}\,=\,D_{2n} \ \ \ \ \ \ E_{1t}\,=\,E_{2t} \ \ \ \ \ \ B_{1n }\,=\,B_{2n}  \ \ \ \ \ \ H_{1t}\,=\,H_{2t}  \ \ \ \ \ \  


$\tan \theta _{i}_{B}\,=\,\frac {n_{t}}{n_{i}} \ \ \ \ \ \ 
\theta _{i}\,+\,\theta _{t}\,=\,\frac {\pi }{2}$

\vspace{5mm}

Interfer\^{e}ncia

\vspace{3mm}

$m\lambda \,=\,d\,sen\vspace{1mm}\,\theta  \ \ \ \ \ \ \  (m\,+\,\frac{1}{2})\lambda \,=\,d\,sen\,\theta \ \ \ \ \ \ \ \ $m\lambda \,=\,a\,sen\,\theta $ \ \ \ \ \ \ \  
$(m\,+\,\frac{1}{2})\lambda \,=\,a\,sen\,\theta  $ \ \ \ \ \ \ \  $sen\,\theta\,=\,1.22\,\frac {\lambda }{a}$ 

\lambda _{n}\,=\,\frac {\lambda }{n}$ 

\vspace{3mm}

$m\lambda \,=\,a\,sen\,\theta $ \ \ \ \ \ \ \  $(m\,+\,\frac{1}{2})\lambda \,=\,a\,sen\,\theta  $ \ \ \ \ \ \ \  $sen\,\theta\,=\,1.22\,\frac {\lambda }{a}$

\vspace{3mm}

$I\,=\,I_{o}\,(\frac {sen\,\frac {\beta }{2}}{\frac{\beta }{2}})^{2}\,\,\,(\frac {sen \,\frac {N\phi}{2}}{sen \,\frac{\phi }{2}})^{2} \ \ \ \ \ \ \  
 \phi \,=\,{\frac  {2\pi }{\lambda }{\,d\,sen \,\theta } \ \ \ \ \ \ \ \beta \,=\,{\frac  {2\pi }{\lambda }{\,a\,sen \,\theta } $ \ \ \ \ \ \ \ 
$R\,=\,\frac {\lambda _{m} }{\lambda _{2}\,-\,\lambda _{1}}\,=\,\frac {\lambda _{m}}{\Delta \lambda } \ \ \ \ \ \ \   R\,=\,m\,N $

\vspace{5mm}

Pel\'{\i}cula

\vspace{3mm}

$2nt\,=\,(m\,+\,\frac {1}{2})\,\lambda  \ \ \ \ \ \ \ \ \ \ \ m\,=\,0, 1, 2, ...... $

\vspace{3mm}

$2nt\,=\,m\,\lambda \ \ \ \ \ \ \ \ \ \ \ m\,=\,0, 1, 2, ......$


\vspace{3mm}

$t\,cos\theta _{t}\,=\,(2m\,+\,1)\frac {\lambda _{t}}{4} \ \ \ \ \ \ \ \ \ \ \ m\,=\,0, 1, 2, ......$

\vspace{3mm}

$t\,cos\theta _{t}\,=\,m\,\frac {\lambda _{t}}{2} \ \ \ \ \ \ \ \ \ \ \ m\,=\,0, 1, 2, ......$

\vspace{5mm}

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

\vspace{3mm}

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

\vspace{3mm}

$m\lambda \,=\,a\,sen\,\theta $

\vspace{3mm}

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

\vspace{4mm}

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

\vspace{5mm}

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

\vspace{4mm}

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

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







\end{document}
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Formul\'{a}rio

\vspace{1mm}

\noindent $\oint \vec {D}.\vec {dS}\,=\,q$ \ \ \ \ \ \ \ $\oint \vec {P}.\vec {dS}\,=\,-\,q_{pol}$ \ \ \ \ \ \ \ $\oint \vec {H}.\vec {dl}\,=\,i$ 
\ \ \ \ \ \ \ $\oint \vec {M}.\vec {dl}\,=\,i_{m}$ \ \ \ \ \ \ \ $\vec  {D}\,=\,\epsilon _{o}\,\vec  {E}\,+\,\vec  {P}$ \ \ \ \ \ \ \ 
$\vec  {D}\,=\,\epsilon \,\vec {E}$
 
\noindent $\sigma _{p}\,=\,\vec {P}.\vec {n}  \ \ \ \ \ \ \ \vec  {B}\,=\,\mu _{o}\,({\,\vec  {H}\,+\,\vec  {M}})$ \ \ \ \ \ \ \ $\vec  {B}\,=\,\mu \,\vec {H}  
\ \ \ \ \ \ \ D_{1n}\,=\,D_{2n} \ \ \ \ \ \ \   E_{1t}\,=\,E_{2t} \ \ \ \ \ \ \  B_{1n }\,=\,B_{2n}  \ \ \ \ \ \ \ H_{1t}\,=\,H_{2t}$

\end{document}



\vspace{3mm}

\noindent  $\nabla .\vec {D}\,=\,\rho $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
$\nabla \times \,\vec  {H}\,=\,\vec {J}\,+\,\frac {\partial \vec {D}}{\partial  t} $

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


\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}


$\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{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 $\, $

}

\end{document}

