\input{labpreamble}
\begin{document}
\parskip=.18in

\subsubsection*{PHSX 222L\hfill Refraction and Snell's Law\hfill Name:\underline{\hspace*{2in}}}

\noindent
\underline{Object:} To measure the index of refraction for (a) water, 
(b) a liquid organic compound, (c) a plexiglass or glass block, and (d) a glass prism.

\noindent 
\underline{Theory:} When a light ray crosses the interface 
between two transparent media it bends, or changes direction, 
according to a relationship known as Snell's Law (Decartes' Law in 
France).
\begin{equation}
n_{1}\sin\theta_{1}=n_{2}\sin\theta_{2}
\end{equation}
where $n$ is the index of refraction and $\theta$ is the angle from 
the normal.  Snell's Law can be derived from Fermat's principle of 
least time.  (The index of refraction is actually a function of the 
frequency of the light, which gives rise to the phenomenon of 
dispersion.)

\noindent
\underline{Procedure:} 

For the liquids:
\vspace{-.20in}
\begin{enumerate}
\item The index of refraction for the two liquids will be measured 
by placing them in a tank, attached to which is a scale for measuring 
the incident angle and the refracted angle.  This is done by filling 
the tank half-way and lining up the sights through the liquid and 
then again above the liquid in the air.

\item Draw a diagram of the apparatus and illustrate the angles 
measured and used in Snell's Law.  It may seem that $\theta_{1}$ and 
$\theta_{2}$ are reversed until you think this through very carefully.
\end{enumerate}

For the plexiglass or glass block:
\vspace{-.20in}
\begin{enumerate}
\item The index of refraction for a slab of plexiglass will be
measured by placing it on a sheet of lined paper or a flat ruler and
observing the images of the lines produced by refraction and measuring
the apparent displacement of this image.  First nondestructively (with
a non-permanent marker or pen) make a mark on the top of the block at
point $P_{1}$ and line it up vertically with a line on the paper at
$P_{2}$.  Then move your eye to the position labeled eye and look
simultaneously through the block at another line on the paper and
through the air right next to the block.  $d_{2}$ is the actual
distance from the second line to $P_{2}$, and $d_{1}$ is where the line
\emph{appeared} to be.

\begin{center}
\begin{picture}(350,190)(50,0)
\thicklines
\put(50,50){\framebox(300,100)}  % block
\put (0,49.5){\line(1,0){400}}  % table top
\put(200,150){\line(1,-1){100}}  % undeviated line
\put(200,150){\line(-1,1){50}}  % undeviated line
\put(200,150){\line(1,-2){50}}  % deviated line
\put(180,170){\vector(-1,1){20}}   % arrow for undeviated line
\put(270,80){\vector(-1,1){20}}   % arrow for undeviated line
\put(235,80){\vector(-1,2){10}}   % arrow for deviated line
\thinlines
\put(200,0){\line(0,1){200}}  % normal line
\put(250,30){\line(0,1){16}}  % dimension line
\put(300,10){\line(0,1){36}}  % dimension line
\put(192,170){\makebox(0,0){$\theta_{1}$}}
\put(209,115){\makebox(0,0){$\theta_{2}$}}
\put(210,158){\makebox(0,0){$P_{1}$}}
\put(190,40){\makebox(0,0){$P_{2}$}}
\put(225,38){\makebox(0,0){$d_{2}$}}
\put(250,20){\makebox(0,0){$d_{1}$}}
\put(155,100){\makebox(0,0){$h$}}
\put(70,160){\makebox(0,0){$n_{1}$}}
\put(70,140){\makebox(0,0){$n_{2}$}}
\put(140,192){\makebox(0,0){eye}}
\put(235,88){\shortstack{ray \ \ \ \ \ \ \ \ \ \ \ \ \\ through\\ air}}
\put(25,42){\makebox(0,0){lined\ paper}}
\put(218,38){\vector(-1,0){18}}   % dimension line for d2
\put(232,38){\vector(1,0){18}}   % dimension line for d2
\put(243,20){\vector(-1,0){43}}   % dimension line for d1
\put(257,20){\vector(1,0){43}}   % dimension line for d1
\put(155,110){\vector(0,1){40}}   % dimension line for h
\put(155,90){\vector(0,-1){40}}   % dimension line for h
\end{picture}
\end{center}


\item Verify that the mathematical derivation of $n_{2}$ is given by
\begin{equation}
n_{2}=n_{1}{\sin(\arctan{d_{1}\over h})\over\sin(\arctan{d_{2}\over h})}.
\end{equation}
\end{enumerate}

\goodbreak
For the glass prism:
\vspace{-.20in}
\begin{enumerate}

\item Use the method of minimum deviation on an equilateral prism. 
Adjust both the rotating platform and the telescope of the
spectrometer until a minimum deviation is achieved.  Minimum deviation
occurs when $\theta_{1}$ on the left face is equal to $\theta_{1}$ on
the right face (the ray inside the prism will then be parallel to the
base).  Equation \ref{eqn:min} for $n$ is only valid at minimum
deviation.

\item Measure $n$ for a couple of different wavelengths of light.  
As shown in Example 35.7 in Serway, 
\begin{equation}\label{eqn:min}
n={\sin({\Phi +\delta_{\rm min}\over 2})\over \sin(\Phi/2)}.
\end{equation}

\end{enumerate}

\begin{center}
\begin{picture}(300,220)
\thicklines
\put(150,200){\line(2,-3){100}}  % prism face
\put(150,200){\line(-2,-3){100}}  % prism face
\put(50,50){\line(1,0){200}}  % prism face
\put(100,125){\line(1,0){100}}  % ray
\put(100,125){\line(-3,-2){60}}  % ray
\put(200,125){\line(3,-2){60}}  % ray
\thinlines
\multiput(150,91.667)(18,12){6}{\line(3,2){12}}
\multiput(150,91.667)(-18,12){6}{\line(-3,2){12}}
% \put(150,91.667){\line(-3,2){120}}  % normal
% \put(150,91.667){\line(3,2){120}}  % normal
\multiput(40,85)(18,12){10}{\line(3,2){12}}  % extended ray
% \put(40,85){\line(3,2){200}}  % extended ray
\put(40,85){\vector(3,2){40}}  % arrow
\multiput(260,85)(-18,12){7}{\line(-3,2){12}} % extended ray
% \put(260,85){\line(-3,2){110}}  % extended ray
\put(200,125){\vector(3,-2){30}}  % arrow
\put(100,125){\vector(1,0){50}}  % arrow
\put(150,158.33){\line(0,1){60}}  % normal line
\put(80,125){\makebox(0,0){$\theta_{1}$}}
\put(220,125){\makebox(0,0){$\theta_{1}$}}
\put(124,118){\makebox(0,0){$\theta_{2}$}}
\put(176,118){\makebox(0,0){$\theta_{2}$}}
\put(168,158){\makebox(0,0){$\delta_{\rm min}$}}
\put(145,180){\makebox(0,0){$\Phi\over 2$}}
\put(150,65){\makebox(0,0){$n$}}
\end{picture}
\end{center}


\noindent
\underline{Conclusions:} 
\vspace{-.25in}
\begin{enumerate}
\item Compare the indices of refration as measured above with the 
standard handbook values.

\item Discuss sources of error in each case.  What clever things did 
you do to reduce your error and to determine its size?
\end{enumerate}

\end{document}
