\section{LC-2: Mirrors and Lenses}\label{lc02_geomop}

\noindent {OBJECTIVES}: 
\begin{enumerate}
\item[] \latex{\vspace*{-.07in}}   To study image formation and focal lengths of mirrors and lenses.
\end{enumerate}

\noindent {APPARATUS}:   
\begin{enumerate}
\item[] \latex{\vspace*{-.07in}}   Optical bench; optical components (10) kit: lenses \& mirrors (5), telescope, illuminated arrow light source \& 12V power supply, sharply pointed rod, desk lamp, white card or screen, plane mirror. \\
NOTE: {\em the objects on the optical bench slides are not necessarily centered with respect to the index on the slide}.
\end{enumerate}

\noindent {INTRODUCTION}: 
\begin{enumerate}
\item[] \latex{\vspace*{-.07in}}  We will design, test and then measure mirror and lens assemblies by several techniques.  In all instances it will be possible to use a simple interactive Java applet to perform a virtual pre-lab exercise.  For the physical set-ups it will require testing location of image by absence of {\em parallax}, and others require {\em focusing a 
telescope for parallel rays}.  For both techniques see \hyperref{parallax}{Appendix~}{)}{parallax} on page~\pageref{parallax}.  Please master this material: your instructor is  available for help.
\end{enumerate}

\noindent {\bf Experiment I. Radius~of~curvature~and~focal~length~of~a~concave~mirror:}

\htmlrule
\noindent VIRTUAL PRE-LAB:
\begin{enumerate}
%\latex{\begin{enumerate}}
%\html{\begin{htmllist}\htmlitemmark{BlueBall} }
\item \latex{\vspace*{-.1in}}If using the web-lab manual launch 
the virtual application by clicking on the Concave Mirror Application button below.
\input{toolbar/l02_j1}
\item \latex{\vspace*{-.1in}}After clicking on ``Start Me'' you should observe a concave mirror, an object and its image.  On the lower left corner is a cursor position readout. 
\item \latex{\vspace*{-.1in}}With respect to the figure below and the mirror equation.  Find the focal length.
\item \latex{\vspace*{-.1in}}Adjust the object to get $p=q$, and obtain the focal length again. 
\item \latex{\vspace*{-.1in}}Move the object so that $p=f$. Do you see an image?
\item \latex{\vspace*{-.1in}} Move the object so that $p<f$. Is there an image? Is it real or virtual? Is the image height smaller or larger than the object?
%\html{\end{htmllist}} 
%\latex{\end{enumerate}}
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\html{ \begin{tabular}{p{3.5in}l} }
\latex{\parbox{0.4\linewidth}}
{The Mirror Equation is given by:  \[\frac{1}{p}~+~\frac{1}{q}=\frac{2}{R}  = \frac{1}{f} \] where $f$ is the focal length and $R$ is the mirror radius curvature. Note that when the object and image are equally distant from the mirror, $p~=~q~=~R$.  You can use this condition to get an approximate value for $R$ and $f$.}
\html{ & }
\latex{\parbox{0.50\linewidth}}
{\hspace*{.05in}
\latex{\includegraphics[height=1.4in]{figs/l2-01.eps}}
\html{\includegraphics[height=1.5in]{figs/l2-01.eps}}
\begin{center} {\bf Figure~1:} Layout of concave lens experiment.
\end{center}}
\html{ \cr \end{tabular}  }
\end{enumerate}


\noindent SUGGESTED PROCEDURE (Note: The mirror holder contains a concave and a convex mirror on opposite sides. 
Be sure you use the {\em concave} mirror!)
\begin{enumerate}
\item[] \latex{\vspace*{-.07in}}I. By using object and image distances:
\begin{enumerate}
\item[1.] \latex{\vspace*{-.04in}}Resolve the image of the illuminated arrow formed by the concave mirror on the white card.  Experiment with varying object distances, $p$, until you able to follow how the image distance, $q$, varies with the object distance.
\item[2.] \latex{\vspace*{-.04in}}Measure the image distance at several different positions of the illuminated arrow object. From each pair of conjugate object and image positions calculate $f$ for the mirror.  Make a table of the results in your lab book, but leave space to compare with the results of II.~and III.~below.
\end{enumerate}
\end{enumerate}


\html{\begin{tabular}{p{4.5in}l} }
\latex{\parbox{0.45\linewidth}}
{\noindent II. Obtaining $f = R/2 $ by imaging at center of curvature ($q=p=R$):
Replace the illuminated arrow and screen with the rod and place the rod at $R$.  If the tip of the rod is at the center of curvature, a real inverted image will appear just above it with the tip and its image coinciding.  An absence of parallax (see \hyperref{parallax}{Appendix~}{)}{parallax}) between the tip and image is the most sensitive test for coincidence. When the {\em parallax vanishes}, the distance between the tip and the mirror is the radius of curvature $R$.  From several determinations of $R$, (switching roles with your lab partners estimate the reliability of your results.)}
\html{ & }
\latex{\parbox{0.55\linewidth}}{
\latex{\includegraphics[height=1.6in]{figs/l2-02.eps}}
\html{\includegraphics[height=2.1in]{figs/l2-02.eps}}
\begin{center} {\bf Figure~2:} Object and image at center of curvature, $R$.
\end{center}  }
\html{ \cr \end{tabular}  }
\medskip

\html{\begin{tabular}{p{4.5in}l} }
\latex{\parbox{0.45\linewidth}}
{\noindent III. By placing object at the principal focus ($p=f$): If the tip of the rod is at $f$, rays from the tip will reflect from the mirror as a {\em parallel} bundle and give a sharp image of the tip in a telescope {\em focused for parallel rays} (see appendix D). Keep the distance between tip and telescope small or the reflected light may {\em miss} the telescope.  The best test for locating the focal point is {\em absence of parallax between the tip's image and the cross hairs} in the telescope, but the telescope {\em must} already be focused for parallel rays (e.g., focus on something out the window).  From several settings estimate the reliability of your $f$ measurement. }
\html{ & }
\latex{\parbox{0.55\linewidth}}
{\latex{\includegraphics[height=1.6in]{figs/l2-03.eps}}
 \html{\includegraphics[height=2.1in]{figs/l2-03.eps}}
\begin{center} {\bf Figure~3:} Object at $f$, image at infinity.\end{center}}
\html{ \cr \end{tabular}  }

\noindent {\bf Experiment II: LENSES} \latex{\hrule \smallskip} 

\htmlrule
\noindent VIRTUAL PRE-LAB:
\begin{enumerate}
%\latex{\begin{enumerate}}
%html{\begin{htmllist} \htmlitemmark{BlueBall}}
\item \latex{\vspace*{-.07in}}If using the web-lab manual launch the virtual application by clicking on the Converging Lens Application button below.
\input{toolbar/l02_j2}
\item \latex{\vspace*{-.1in}}After clicking on ``Start Me'' you should observe a convex lens, an object and its
image.  On the lower left corner is a cursor position readout. 
\item \latex{\vspace*{-.1in}}With respect to the figure below and the lens equation.  Find the focal length.
\item \latex{\vspace*{-.1in}}Move the object so that its inverted image has the same height.  Find the focal length at this point.
\item \latex{\vspace*{-.1in}}Move the object so that $p=f$. Do you see an image?
\item \latex{\vspace*{-.1in}}Move the object so that $p<f$. Is there an image? Is it real or virtual? Is the image height smaller or larger than the object? What is the sign of $q$?
%\html{\end{htmllist}} \latex{\end{enumerate}}
\end{enumerate}
\latex{\hrule \medskip}

\htmlrule

\noindent 1. {\em Converging lens with short~focal~length}: 

\begin{enumerate}
\item[] \latex{\vspace*{-.09in}}
From the thin lens formula $$ \frac{1}{\mbox{object distance}} + \frac{1}{\mbox{image distance}} =
\frac{1}{p} + \frac{1}{q} = \frac{1}{f} $$
Measure $f$ using the three methods diagrammed below (by adjustments in the object distance). 
Compare the results from the various methods.\\
I. \vspace*{-.25in}\begin{center} 
\includegraphics[height=1.6in]{figs/l2-04.eps} \end{center} 
II. \vspace*{-.25in}\begin{center} 
\includegraphics[height=1.9in]{figs/l2-05.eps} \end{center} 
III. OPTIONAL \vspace*{-.25in}\begin{center} 
\includegraphics[height=1.6in]{figs/l2-06.eps} \end{center} 
\end{enumerate}

\latex{\vspace*{-.1in}}\noindent 2. {\em Focal~length~of~a~diverging~lens}:  

\begin{enumerate}
\item[]\latex{\vspace*{-.15in}}Since the image is always virtual it is necessary to combine it with a converging lens and
configured so that the final image is real.  
\end{enumerate}

\htmlrule

\noindent VIRTUAL PRE-LAB:

\begin{enumerate}
%\latex{\begin{enumerate}} %{}{\parsep 0ex \labelsep -5.0ex \itemsep -5.0ex}}
%\html{\begin{htmllist} \htmlitemmark{BlueBall}}
\item \latex{\vspace*{-.1in}}If using the web-lab manual launch the virtual application by clicking on the Diverging Lens Application button below.
\input{toolbar/l02_j3}
\item \latex{\vspace*{-.1in}}After clicking on ``Start Me'' you should observe a concave lens, an object and its virtual image.  On the lower left corner is a cursor position readout. 
\item \latex{\vspace*{-.1in}}With respect to lens equation and starting positions,
find the focal length. Can you find an object distant in which the image height and object height are the same?
\item \latex{\vspace*{-.1in}}Now restart the simulation with a combination of a convex and concave lens by clicking on the ``Add 2nd lens'' link.
\item \latex{\vspace*{-.1in}}This 2nd lens will render a final image which is real.  The image with a ``1'' by it is that of just the convex lens while the image with a ``2'' next to it is that of the pair.  Move the object and alter its height, if necessary, so that the first image has the relationship $q_1 \approx 2 p_2$.  (In this simulation $|f_2| = 3 |f_1|$ and qualitatively resembles the figure shown below.)
\item \latex{\vspace*{-.1in}}Now repeat the last step with $|f_2| = |f_1|$ by clicking on the ``Another 2nd lens'' link.  Finally reset to the ``Another 2nd lens'' and then  click on the {\em convex} lens and drag it past the diverging lens.
%\html{\end{htmllist}} \latex{\end{enumerate}}
\end{enumerate}

\htmlrule
\latex{\hrule}

\begin{enumerate}
\item[I.] \latex{\vspace*{-.03in}}Use the set-up sketched below. First adjust 
Lens$_1$ so that with Lens$_2$ removed, $q_1 \sim 2p_1$.  Measure $q_1$, insert the diverging Lens$_2$ reasonably close to Lens$_1$ and then locate the new image 
distance $q_2$.  From $p_2 = d- q_1$ (a virtual object) and $q_2$ calculate 
$f_2$.
\begin{center} 
\includegraphics[height=2.0in]{figs/l2-07.eps} \end{center} 

\item[II.] OPTIONAL: Find $f_2$ by this more sensitive method:  With the pointed rod as object now detect the image by a telescope focused for parallel rays. (As shown in the following figure.)

\indent If using the web-lab manual, launch the virtual application by clicking on the Dual Lens Test Application button below to obverse a point source in action.  
\input{toolbar/l02_j4}

\noindent To align all the objects:  tape a piece of paper with a lot of markings on it to the pointed rod.  Look through the telescope and move the diverging lens forward and backward until you can see some markings clearly in the telescope.  Then move the rod up or down, as necessary, and rotate the telescope from side to side, as necessary, until you can see the tip of the pointed rod.  Then move the diverging lens a small amount until the tip is in focus.

\noindent As a last step you will move the pointed rod (i.e.~vary $p_1$ and hence $q_1$) until a sharp image appears in the telescope with {\em no parallax} relative to the cross hairs.  (This occurs only if parallel rays leave the diverging lens.)  The image from the converging lens at $q_1$ serves as a virtual object for the concave lens, Lens$_2$.   After properly adjusting $p_1$ this virtual object will be at the focal point of the diverging lens, Lens$_2$.  Obtain q$_1$ by setting  p$_1$ and using the known value of $f_1$. (Make sure that $q_1$ is large enough to accommodate $d+|f_2|$).  Now look for the image in the telescope and adjust $p_1$.  Note that since $q_2= \infty $, $f_2 = p_2$ and $p_2 = d-q_1$.
\end{enumerate}
\begin{center} 
\includegraphics[height=2.0in]{figs/l2-08.eps} \end{center} 

\noindent OPTIONAL \\ {\bf Experiment III: Focal~Length~of~a~Convex~Mirror}
\begin{enumerate}
\item[] \latex{\vspace*{-.07in}}
Once again, since convex mirrors give virtually images it is necessary to study the mirror in combination with a converging lens.

%\noindent VIRTUAL PRE-LAB: 
%\begin{enumerate}
%\item \vspace*{-.1in}If using the web-lab manual launch 
%the virtual application by clicking on the Convex Mirror Application button below.
%\input{toolbar/l02_j2}
%\item \vspace*{-.1in} After clicking on ``Start Me'' you should observe a convex lens, 
%a convex mirror and a point source. On the lower left corner is a cursor position readout. 
%\item \vspace*{-.1in} Move the source so that its ``image'' coincide with the source.
%\item \vspace*{-.1in} With respect to the lens equation  
%find the focal length of the convex mirror.  Note that $f_1$ (for the lens) is 1.75 distance
%units.
%
%\end{enumerate}

Measure the focal length of the {\em convex} mirror by combining it with a converging lens. Set the pointed rod at twice the focal length of the 
lens ($p_1=2 f_1$). Next adjust the mirror position until the inverted image position shows no parallax with the object.  Then $R = d-2f_1$ and 

\[f_{\mbox{\small mirror}}~=~f_2~=~R/2.\] 

\noindent Of course the lens must have $f_1 > f_2$, so use a long focal length lens. 

\begin{center} 
\includegraphics[height=2.2in]{figs/l2-09.eps} 
\end{center} 
\end{enumerate}
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