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\begin{document}

\Prova{3}{21 de junho de 2011}

\hrulefill

\vspace{2mm}

\begin{QstsFig}\ {\bf Parte A} -(3.0) Um sistema de duas fendas \'{e} iluminado uniformemente com luz de comprimento de $\,\bf 600\,nm$. Num anteparo, de 
comprimento total $\,\bf  2\,m$ situado a $\,\bf  1\,m\,$ das fendas, observa-se que o m\'{a}nimo de quinta ordem est\'{a} a $\,\bf  5\,cm\,$  do m\'{a}ximo 
central. Determine:

\begin{itemize}
\item{(1.0)}  a) a dist\^{a}ncia entre as fendas;
\item{(1.0)}  b) os m\'{a}ximos e  m\'{\i}nimos observados no anteparo;
\item{(1.0)}  c) a posi\c{c}\~{a}o angular ($sen\theta $) em que a intensidade cai para um quarto.
\end{itemize}
\end{QstsFig}
%\end{QstsFig}
%\vspace{20cm}
\newpage

{\bf Parte B\,}-(3.0) Considere agora que cada fenda tenha largura de  $\,\bf  20\,\mu m$.

\begin{itemize}
\item{(1.0)}  a) Determine quais m\'{a}ximos n\~{a}o ser\~{a}o observados no anteparo.
\item{(1.0)}  b) Calcule para uma \'{u}nica fenda as intensidades dos m\'{a}ximos de primeira e segunda ordem. 
\item{(1.0)}  c) Fa\c{c}a o esbo\c{c}o do gr\'{a}fico final da intensidade em fun\c{c}\~{a}o de $sen\theta $, para o sistema de duas fendas, para $sen\theta \geq 0$.
\end{itemize}
%\end{QstsFig}

\newpage


\begin{QstsFig}\ (2.0) Uma pel\'{\i}cula de vidro ($\,\bf n\,=\,1.6\,$)  \'{e} iluminada com luz branca. Quando a pel\'{\i}cula \'{e} 
observada de cima ( na perpendicular ) sua cor \'{e} laranja avemerlhada,$\,\bf  \lambda \,=\,640\,nm$. Determine a espessura da pel\'{\i}cula.
\end{QstsFig}

\vspace{12cm}

\begin{QstsFig}\ (2.0) Qual deve ser o di\^{a}metro do espelho de um telesc\'{o}pio para que se possa distinguir dois corpos, na superf\'{\i}cie da Lua, distantes 
entre si de $\,\bf  228\,m\,$? A dist\^{a}ncia Terra-Lua \'{e} aproximadamente $\,\bf  3.8\times 10^{8}\,m$.
\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}

$I\,=\,4\,I_{o}\, cos^{2}\,(\frac  {\pi d\,sen\,\theta }{\lambda })$

\vspace{4mm}

Pel\'{\i}cula

$2nt\,=\,(m\,+\,\frac {1}{2})\,\lambda  $
%\ \ \ \ \ m\,=\,0, \pm 1, \pm 2, \pm 3 .....$

$2nt\,=\,m\,\lambda $
%\ \ \ \ \ m\,=\,0, \pm 1, \pm 2, \pm 3 .....$

$d\,cos\theta _{t}\,=\,(2m\,+\,1)\frac {\lambda _{t}}{4} $
%\ \ \ \ \ m\,=\,0, \pm 1, \pm 2, \pm 3 .....$

$d\,cos\theta _{t}\,=\,m\,\frac {\lambda _{t}}{2} $
%\ \ \ \ \ m\,=\,0, \pm 1, \pm 2, \pm 3 .....$


\vspace{4mm}

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

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

\vspace{4mm}

$m\lambda \,=\,a\,sen\,\theta $
% \ \ \ \ m\,=\,1, \pm 2, \pm  3 ..... $

$(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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