
\section{EC-2:  Electric~Fields}\label{ec02_Efields}

\noindent  OBJECTIVES:
\begin{enumerate} \item[] \latex{\vspace*{-.06in}}
To develop an intuitive understanding of relationship between electric fields and equipotential contours and physically map electric fields in two dimensions.
\end{enumerate}

\noindent  APPARATUS:
\begin{enumerate} \item[] \latex{\vspace*{-.06in}}
Pasco power supply, 
(use only 0-30 V DC range); PASCO electrometer 
plus coaxial cables, leads \& two test probes; graphitized paper on 
which are drawn conducting electrode configurations; field plotting 
board; carbon paper; pen with silver 
conductive ink ({\em use cap when not in use!}) shared between 4 experiments;
white paper.
\end{enumerate}
\begin{figure}[h] \centering 
\latex{\includegraphics[height=3.00in]{figs/l104/e02-5.eps}}
\html{\includegraphics[height=4.20in]{figs/l104/e02-5.eps}}
\latex{\vspace*{-.05in}}
\caption{The apparatus.}
\latex{\vspace*{-.05in}}
\end{figure}
{\bf{\em{\noindent Preliminary Questions:}} } \\
\indent  1. Do electric fields extend through a vacuum? \\
\indent  2. Do electric fields extend through the interior of an insulator? \\
\indent  3. Do electric fields extend through the interior of a conductor?
\smallskip

\noindent  INTRODUCTION:
\begin{enumerate} \item[] \latex{\vspace*{-.06in}}
To map electric fields  in a plane near charged conductors we use a 
potential difference to produce a small current in a uniform material 
of high resistivity which surrounds the conductors.  One locates 
equipotential lines between the conductors by means of probes connected 
to an electrometer.  The lines of the electric field are perpendicular to the 
equipotential lines.
The planar resistance material is a graphite treated piece of paper.  
The Pasco 0-30 V supply furnishes the current.  

For the conductors 
we need a paint with resistance negligible compared to that of 
the graphitized paper.  A suspension of silver flakes in a 
carrier which evaporates after painting, is excellent but expensive.
Next best is a suspension of very fine copper or 
nickel flakes; we have mainly used nickel.  The suspending fluid,  an 
insulator, evaporates so slowly that recently painted electrodes are not 
good conductors unless special precautions and procedures are followed:
\end{enumerate}

\begin{rawhtml}  
<IMG SRC="../gif/caution.gif" ALIGN=center>  <br> 
\end{rawhtml}
\noindent PRECAUTIONS:
\begin{enumerate} 
\item[1)] \latex{\vspace*{-.05in}}{\em Use adequate ventilation.  Avoid inhaling the vapor and avoid contact 
of the ink with your skin}.
\item[2)] \latex{\vspace*{-.05in}}Before using, shake the pen vigorously to disperse the particle matter 
suspended in the ink. 
Avoid thick layers.  {\em USE CAP THE PEN WHEN NOT IN USE}.
\item[3)] \latex{\vspace*{-.05in}}Allow 3-5 minutes for the ink to dry (at room temperature:  a 
heat gun will speed the drying, but can easily 
overheat the surface causing blistering if the layer is thick).
\latex{\vspace*{-.03in}}
\end{enumerate}
\begin{rawhtml}  
<IMG SRC="../gif/caution.gif" ALIGN=center>  <br> 
\end{rawhtml}

\noindent GENERAL METHODOLOGY: 
\begin{enumerate}
\item[1)] \latex{\vspace*{-.05in}}Place white paper on the field plotting board, then a piece of carbon 
paper, and finally the graphitized paper with the conducting electrodes on 
the top.  You may use samples already prepared or draw your own.
\item[2)] \latex{\vspace*{-.05in}}Record the shape of the electrodes on the white paper by tracing with a 
hard pencil or a ball point pen.
\item[3)] \latex{\vspace*{-.05in}}With 24 volts across the terminals of the plotting board (as in Fig.~1)
check the quality of your painted electrodes by using the electrometer probe 
to find whether there is an appreciable potential drop {\bf along} the painted 
electrode when you have 24 volts {\bf between} the electrodes.
\begin{figure}[h] \centering 
\latex{\vspace*{-.05in}}
\latex{\includegraphics[width=6.2in]{figs/e2-01.eps}}
\html{\includegraphics[width=7.0in]{figs/e2-01.eps}}
\latex{\vspace*{-.1in}}
\caption {Electrode placement that qualitatively approximates a charge dipole configuration.}
\latex{\vspace*{-.1in}}
\end{figure}
\item[4)] With 24 volts connected as in Figure 1, 
ground the electrometer to the power supply ground. For the 
other electrometer lead use the sharp steel probe.  Check the electrometer 
zero, and then move the sharp steel probe along the graphitized paper from 
the ground terminal A toward B until the electrometer reads +3 volts.  (Hold 
the probe only by the insulated base, and avoid grounding the graphitized 
paper with your hand or you may get nonsensical results.) 
Press down on the sharp probe to record this location as a dot 
on the underlying white paper.

Map the equipotential line which 
corresponds to +3 volts by exploring (with the movable probe) the 
neighborhood of the first recorded point.  Make your recorded points close 
enough together that you can later draw a smooth line through the points:  
the points need to be close together only when the direction of the 
equipotential changes rapidly. 

NOTE: Your results, in terms of the
actual voltages may vary somewhat from the idealized sketch of Fig.~1 because
of contact resistance, leakage currents and other losses.  To compensate you
can measure the voltage at points very close to the A and B electrodes to find values
$V_A$ and $V_B$ respectively and then choosing voltage steps of
$(V_B-V_A)/8$.\latex{\vspace*{-.06in}}

\item[5)] \latex{\vspace*{-.05in}}Map other equipotentials: 6 volts, 9 volts,...~21 volts.
\item[6)] \latex{\vspace*{-.07in}}After you have drawn the equipotentials on the white sheet, dot in a few
electric field lines (seven or so) as demonstrated in the right panel of Figure~1 
(remembering that electric field lines are always perpendicular to equipotential contours.) latex{\vspace*{-.09in}}
\end{enumerate}

\noindent ALTERNATIVE~METHOD: (OPTIONAL) 
\begin{enumerate} \item[] \latex{\vspace*{-.09in}}
For one of your electrode configurations, replace the electrometer ground lead by another coaxial cable connected to a second sharp probe.  Place one exploring probe  about midway between electrodes A and B.  Use the other probe to find positions on the paper which give zero reading on the electrometer.  Since these are electric field lines, as for example Fig.~1, obviously points of the same potential, they can be connected to map an equipotential line.  Map three other equipotential lines  approximately equally spaced on each side of the first.  This method permits use of the electrometer's most sensitive voltage range and hence allows very precise location of the equipotential points.  However, it does not give the potential of the equipotential line. 

(NOTE: Since most existing lab electrometers are referenced to ground this only works if your DC power supply 0 V output terminal itself is not grounded.)   \latex{\vspace*{-.06in}}
\end{enumerate}
\noindent SUGGESTIONS:
\begin{enumerate} \item[] \latex{\vspace*{-.09in}}
Map equipotential lines and electric field lines for several 
of the configurations shown in Figs.~2 or 3 (or create your own test design). 
The distances are merely suggestions.  Return to your instructor the 
graphitized paper with painted electrodes. 
\begin{figure}[h] \centering 
\latex{\vspace*{-.15in}}
\latex{\includegraphics[width=5.4in]{figs/e2-02.eps}}
\html{\includegraphics[width=7.0in]{figs/e2-02.eps}}
\latex{\vspace*{-.15in}}
\caption {Electrode configurations for (a) parallel plates (left) and (b) ``half'' dipole (right).}
\latex{\vspace*{-.15in}}
\end{figure}
%\noindent Fig.~4:  Examples of configurations giving electrostatic focusing\\ \\
%\noindent a) in an X-ray tube
%\noindent b) in a TV or cathode ray tube
%\noindent c) in the ``dee'' structure of a cyclotron
%\noindent Note that the fields for Fig.~4a, 4b, and 4c  differ 
%quantitatively from the actual fields in the three-dimensional cases.  They 
%do, however, give a good idea of how the various fields look.
\begin{figure}[!h] \centering 
\latex{\vspace*{-.15in}}
\latex{\includegraphics[width=5.4in]{figs/e2-03.eps}}
\html{\includegraphics[width=7.0in]{figs/e2-03.eps}}
\latex{\vspace*{-.15in}}
\caption {Two other electrode configurations, (a) ligthning rod, (b) Faraday Cage.}
%\latex{\vspace*{-.1in}}
\end{figure}

In each case map about seven equipotential lines as continuous lines 
and dot in seven to nine electric field lines as dotted lines.  
Make clear which is which.

Turn~off~the~electrometer~when~you~finish~mapping and  
 {\em place} the three position switch in the grounded LOCK  position.
\end{enumerate}
\noindent QUESTIONS:
\begin{enumerate}
\item Explain in a sentence or two how you mapped the equipotential lines.
\item Pick two adjacent equipotential contours in two different electrode configurations and calculate the mean magnitude of the electric field strength where it is largest.
\item Where in Fig.~4(a) do you expect the electric field strength to be strongest?  Weakest?
Why do experts recommend that if you are caught outside during a thunder shower and cannot obtain shelter that you find a low spot and  curl up your body? Note: there are two effects here, one from shape and the other from size, arising from the fact that lightning is caused by excessive electric fields ($> 5000\,$V/cm).

\item For Fig.~4(b) do you expect there to be appreciable electric field strength outside the confines of the box?  If using the web-version of the lab manual click below on the virtual Faraday Cage demo to study 
this configuration in moredetail. 
\input{toolbar/e02_j1}

\item Why must electric field lines be perpendicular to equipotential lines?
Hint:  Consider the relationship between electric potential and electric field: \[V_{b}~-~V_{a}~=~-\int_{a}^{b}  \vec{E} \cdot d \vec{l} \] and note that V$_{b}$~=~V$_{a}$ for any points  $b$ and $a$  on the equipotential.) 

\item For the sketch below see if you can sketch out the electric field 
lines.  Afterwards check your answer (web-version only)  
by clicking below on the Like Charge Pair demo.
\input{toolbar/e02_j2}
\begin{figure}[h] \centering 
\latex{\includegraphics[width=3.2in]{figs/e2-04.eps}}
\html{\includegraphics[width=4.0in]{figs/e2-04.eps}}
\end{figure}

\item If you are you using the web-version of the lab manual, click on the
button below to randomly place five unknown charges in a small region of
space.  Use any information available to determine the relative strength
and sign of these five charges.
\input{toolbar/e02_j3}


%\item Explain how the field you obtained for Fig.~4a or 4b will focus 
%(i.e. concentrate or converge) electrons 
%from the filament toward a small spot on the anode.
%\item In contrast to the fields of Fig.~4a and 4b, the field of Fig.~4c is 
%symmetric about the middle equipotential.  Thus there will be equal and 
%opposite electrical forces pushing 
%a proton to and away from the median plane.  Is there a net focusing force 
%on the proton? Explain.  (Hint:  Are the times during which the opposite
%forces act the same?  If not, why not?)  }
\end{enumerate}
