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\Prova{1}{27 de mar\c{c}o de 2012}

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

\begin{itemize}
\item{(1.0)}  a) os vetores $\,   \vec {\bf  D}\,$, $\,\vec {\bf  E}\,$  e $\,\vec {\bf  P}\,$  regi\~{a}o $\,\bf  2R>r>R$;
\item{(1.0)}  b) os vetores $\,\vec {\bf  D}\,$, $\,\vec {\bf  E}\,$  e $\,\vec {\bf  P}\,$  na regi\~{a}o $\,\bf  r>2R$;
\item{(1.0)}  c) as densidades de carga de polariza\c{c}\~{a}o na super\'{\i}cie entre os dois diel\'{e}tricos;
\item{(1.0)}  d) a carga total.
\end{itemize}
\end{QstsFig}

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\begin{QstcFig}{q1.eps}\ (2.0) Duas cargas  $\,\bf +\,q\,$ e $\,\bf  -\,2q$  est\~{a}o situadas no plano
$\,\bf  XY\,$ nos pontos  $\,\bf  (2a,2\sqrt {3}\,a)\,$ e $\,\bf  (2a,-2\sqrt {3}\,a)\,$ respectivamente. 
A regi\~{a}o $\,\bf   x\,< \,0\,$ \'{e} v\'{a}cuo e a regi\~{a}o $\,\bf   x\,\geq\,0\,$ permitividade $\,\bf  \epsilon $.

\begin{itemize}

\item{(1.0)}  a) Determine os vetores $\,\vec {\bf  D}\,$, $\,\vec {\bf  E}\,$ e $\,\vec {\bf  P}\,$ na regi\~{a}o
$\,\bf  x\,\geq  \,0\,$ para cada carga.
\item{(1.0)}  b) Determine no ponto $\,\bf  P(0,0)\,$ a densidade de carga de polariza\c{c}\~{a}o resultante  na
superf\'{\i}cie $\,\bf x=\,0$.

\end{itemize}
\end{QstcFig}

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\begin{QstsFig}\ (4.0) Um tor\'{o}ide de sec\c{c}\~{a}o retangular de lados $\,\bf a\,$ e $\,\bf b$, com $\,\bf N\,$ espiras,
tem raio interno $\,\bf R\,$ e raio externo $\,\bf 3R\,$. A regi\~{a}o  $\,\bf R<r< 2R\,$ est\'{a} preenchida por um material
de permeabilidade $\,\bf \mu_{1}\,$ e a regi\~{a}o  $\,\bf 2R<r< 3R\,$ com  um material de
permeabilidade $\,\bf \mu_{2}$. A corrente que circula no tor\'{o}ide \'{e} $\,\bf i_{o}$.

\begin{itemize}
\item{(0.5)}  a) Mostre que o m\'{o}dulo do vetor intensidade magn\'{e}tica de um tor\'{o}ide \'{e}
\large {$\,\bf \frac {N\,i_{o}}{2\pi \,r}$}.
\end{itemize}

\noindent Indique a condi\c{c}\~{a}o de contorno usada e determine:

\begin{itemize}
\item{(1.5)}  b) Determine os vetores $\,\vec {\bf  H}\,$, $\,\vec {\bf  B}\,$ e $\,\vec {\bf  M}\,$ regi\~{a}o $\,\bf  2R>r>R$;
\item{(1.0)}  c) Determine os vetores $\,\vec {\bf  H}\,$, $\,\vec {\bf  B}\,$ e $\,\vec {\bf  M}\,$ na regi\~{a}o $\,\bf  3R>r>2R$;
\item{(1.0)}  d) Determine a auto indut\^{a}ncia do tor\'{o}ide.
\end{itemize}

\end{QstsFig}
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{\Large  Formul\'{a}rio

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\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} $

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

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

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


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\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} $

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$\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} $

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$ <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 $

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

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Interfer\^{e}ncia

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

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

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

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

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

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

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

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

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$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} $

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

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