\section{EC-3~~Capacitors~and~RC-decay}\label{ec03_cap}
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\latex{Note:  RC I and RC II are future applications. RC I adds the sonic distance ranging probe so that you may rapidly and continuously
measure the distance while moving the plate capacitor and monitoring the voltage.  It is difficult to point the distance probe at the plate and so
the actual results may not be as good as the method outlined below. }

\noindent
OBJECTIVES:  
\begin{enumerate} 
\item \latex{\vspace*{-0.07in}}Through use of a parallel-plate capacitor understand the behavior of capacitors.
\item \latex{\vspace*{-0.07in}}Examine how well a parallel-plate model describes real capacitors. 
\item \latex{\vspace*{-0.07in}}Experiment with the charge and voltage on parallel and series capacitors.
\end{enumerate}

\noindent{\em Preliminary Questions:}
\begin{enumerate} 
\item \latex{\vspace*{-0.07in}}A capacitor plate holds a given charge $Q$. Why is the magnitude of the plate voltage small when a grounded plate is near, but large (for the same $Q$) when it is alone? 
% Answer: Potential represents the ability to do work by or, equivalently, the work done on a charge with respect to a reference point. The work to move this charge from the grouding plate is, relatively speaking, small.  
\item \latex{\vspace*{-0.07in}}If the charge $Q$ on a capacitor is doubled, what is the change in the voltage one measures on the capacitor? (Remember that $Q$ is the magnitude of the charge on each plate, both positive and negative.)
% Answer:  Pose another question first: Imagine two plates, side by side, with $Q$ on each.  Each plate will have the same potential and connecting them with a wire will make no difference.  In this case the potential would remain the same.  However all of the charge here must reside on the same plate and so one must overcome the like-charge repulsion by performing additional work.  Thus the potential must increase. In this case it will double as the capacitance remains constant and Q=CV.  
\item \latex{\vspace*{-0.07in}}How could you double the positive charge on one capacitor plate without changing the negative charge on the other plate?
% Answer:  Bring in a second plate from the other side.  (Seems like a cheat to me.)
\end{enumerate}

\noindent APPARATUS:
\begin{enumerate} 
\item[] \latex{\vspace*{-0.07in}} {\it Conventional equipment:} Parallel plate capacitor; Pasco electrometer \&  power supply; commercial capacitors \& resistors on circuit board; aluminum paddle; low capacitance lead, insulated cup and shield; coaxial lead \& test probe; BNC to bannana plug adaptor;
heat gun.
\item[] \latex{\vspace*{-0.05in}} {\it Computer equipment:}  
Computer, monitor, keyboard, mouse; PASCO interface module; PASCO Voltage Sensor (a pair of leads that plug into PASCO input A).
\end{enumerate}

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\noindent PRECAUTIONS: These are needed for reliable measurements:  \\
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\begin{enumerate}
\item \latex{\vspace*{-.07in}}The output BNC from the electrometer is calibrated to give 5~V when the meter reads full scale.  This means you have to convert the voltage you read from the output in order to get the
correct numerical value for the voltage at the input.
\item \latex{\vspace*{-.07in}}The input BNC from the electrometer has the outer conductor (the shield) connected to ground (this is true for all BNC connectors unless indicated otherwise on the connector.)  This means that you run the risk of discharging your capacitors when you make measurements.  Be careful with the polarity when you make measurements.
\item \latex{\vspace*{-.07in}}Large static charges (common in dry weather) if applied to  electrometer input may damage the sensitive input {\bf f}ield {\bf e}ffect {\bf t}ransistor  (``FET'').   Minimize this possibility by {\em keeping the electrometer input grounded via the SHORT switch during the initial hook up and when you are done with the experiment}.  (The SHORT switch connects the FET input direct to ground.)In otherwords: {\bf Keep the Input switch in the ``Short before making connections'' position whenever there is nothing connected to the input or while you are making a connection.}  (In older models, this switch may be labled ``lock.''  It is the left position in either case.)  After the connection is made, you put the switch in the Input position.  If you need to get rid of charge that may have been collected, you can move the switch to either of the short positions, the ``momentary'' position usually being more convenient.
\item \latex{\vspace*{-.07in}}If your clothing or hair has a net charge, the electrometer reading may change if you move around.  Hence, during a given measurement, change position as little as possible and ground yourself (e.g.~touch a convenient ground) during a measurement.  Making sure you torso is as far as conveniently possible from the measurement point may help as well.  Note that grounding yourself is very often a {\em bad} idea when working with electronic devices that are energized because of the danger of electrocution.  Be careful not to touch any voltage sources while you are grounded.  The exception to this is when low-voltage electronic parts, i.e. many PC components, are removed from their socket because static discharge may damage the sensitive electronics.  Hi-voltage capacitors (i.e., well above 5 V) that are not adequately discharged remain extremely dangerous even when removed from contact with power sources.
\item \latex{\vspace*{-.07in}}To remove all {\em net} charge from the cup, switch the electrometer momentarily to the SHORT position. (This connects the electrometer terminals to each other so that any charge flows from/to ground). If the meter does not read zero, notify your instructor. 
%For the models without outputs: The ZERO knob sets the meter to zero; {\bf adjust to zero 
%only when the input is grounded: }  i.e.~while the switch is in the ``SHORT'' 
%or ``LOCK'' position.
\item \latex{\vspace*{-.07in}}Always discharge paddles and cup before starting an experiment.  To test if an object is charged, put it into the cup and see whether the  electrometer deflects.  Conductors discharge easily by touching them to a grounded conductor.  To discharge an insulator, you must create sufficient ions in the surrounding air. The insulator will then attract ions of the opposite charge until all virtually all charge is neutralized.  An open flame is a simple source  of ionized air; the ions in the flame convect upward with the hot gas. {\bf To avoid damage to the insulator, keep it at least 10 cm above the flame}! 
\end{enumerate}
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\noindent  INTRODUCTION:
\begin{enumerate} 
\item[] \latex{\vspace*{-0.05in}}A capacitor consists of two electrodes separated by an insulator. An electrode is just a piece of metal that can be connected to a voltage or current source.  The capacitance $C$ is a number
that quantifies how much charge $Q$ is required to hold an electrode at a potential difference $\Delta V$ from a second electrode, $Q = C\Delta V$.  The second electrode is usually either an electrode with opposite charge ($-Q$) located some distance $d$ away, or an imaginary surface at potential $V = 0$ located infinitely far away. In Part I you will study a parallel plate capacitor consisting of two circular metal plates separated by air. In Part II you will study simple circuits and the charging behavior of capacitors using conventional commercial capacitors. These are made from two long strips of aluminum foil separated from each other by a thin sheet of plastic, all rolled into a cylinder.  

Modern supercapacitor far exceed the performance of ordinary thin film capacitors.  New designs make use of novel materials such as carbon aerogel.  These devices can exhibit well over 1000 times the capacitance of standard capacitors of equivalent volume.  The capacitance of a supercapacitor the size of your finger can range upward of 50 F and be charged up to 5V (and thus hold 250 Coulombs).   For comparison purposes the net energy density is 10\% that of a nickel-metal hydrid battery.  
\end{enumerate}

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{\bf Part I: THE PARALLEL PLATE CAPACITOR} 
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\noindent{\it EXPERIMENT A:} Potential Difference {\it vs} Separation (for fixed charge)

{\smallskip}
\noindent INTRODUCTION:
\begin{enumerate} 
\item[] \latex{\vspace*{-0.07in}}
For a fixed charge the voltage of a conductor, i.e., the potential difference
between that conductor and ground, depends on what bodies are nearby. If you charge a parallel plate capacitor and then increase the plate spacing--leaving $Q$ unchanged--you will find that the potential difference increases.
\item[Q1.1:] \latex{\vspace*{-0.07in}}How do you reconcile this with the fact
that $Q = CV$ remains constant?  (Two good approaches are either using what you have learned about $C$, or using what you have learned about $\vec{E}$ for capacitors and the integral $\int\vec{E}\cdot d\vec{l}$).
\item[Q1.2:] \latex{\vspace*{-0.07in}}{\bf Preliminary calculation:} Assuming air has a dielectric constant $\kappa=1$ estimate how many excess electrons exist on one plate of the  capacitor in front of you when V is set to 15 V and the plates are separated by 0.5 cm.
\end{enumerate}

\noindent{\it EXPERIMENT A:} Potential Difference {\it vs} Separation for
Fixed Charge on a Capacitor
{\smallskip}

\begin{enumerate}
\item Referring to Figs.~1a and b, connect the electrometer across the capacitor, but use the special low capacitance lead and a separate ground instead of the shielded coaxial cable. 
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Use the movable plate as the grounded one, and turn the apparatus so the
fixed plate faces away from you. The ground removes any new excess charge
that might accidentally come in contact with the plate, in order to
keep the voltage the same. As a result, it provides some shielding of
the system from charges on your hand or clothing. Also, with one plate
at ground, you will only have to touch the other plate with the supply
in order to charge the capacitor in step 2 below.
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\\ Figure 1a: Setting the supply and charging the plate \vspace*{-.15in}}
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\\ Figure 1b: Reading the voltage}
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%\begin{enumerate}
\item Start with an initial plate separation of $d = 0.5\,$cm.  This is large enough to keep charge from leaking across the spacers
on dry days. Use the 30~V scale and output of the PASCO DC power supply.  Connect the negative terminal to the ground terminal of the electrometer.  ({\em Standby switch must be in proper position or no voltage results even though meter reads)}. Set the output of the DC power supply to 15 Volts using the Electrometer (not the meter on the supply; it is not as accurate.  Watch out, the Electrometer can acquire and keep a voltage bias because of its very high input impedance...how do you avoid this before setting the supply output?)  Charge the capacitor to 15~V by touching the appropriate plate with the positive voltage supply lead.

Now change the plate spacing and observe the change of the voltage across
the capacitor.  Record your qualitative answer in your lab book, and then
record readings from the electrometer meter of the voltage for different plate 
spacings.  At large spacings, the capacitance is very sensitive to external effects,  so take more data at small distances.  Span 10\,cm in your measurements.  (Of course, {\em zero} on the cm scale will {\em not} be zero separation).   Devise and report a way to make sure no charge has leaked off or been acquired  by the plates during your measurements.

NOTE: In dry weather stray static charge on your body can adversely affect the charge on the parallel plate capacitor.  Keep body movement to a {\em minimum}. There is an optional shielding screen which you may place in front of your body to minimize this effect.  In addition there is an optional extension handle that attaches to the moving plate which will increase your arm to plate distance.

In humid weather the charge may leak too rapidly off the plate to get reasonable results.  Use the heat gun to {\em gently} warm up the parallel plate capacitor and eliminate some moisture.

\item ANALYSIS:\\
Plot the voltage on the capacitor $V$ vs. the distance $d$ between the plates using the graphical analysis software.  There are a variety of applications.  Currently you may use Microsoft Excel, Vernier and/or DataStudio.  Plot the data and then fit to a model function $V=c/(d-d_0)$ where
$d$ is the actual distance measurement, $d_0$ is an offset and $c$ is a constant (reflecting the materials and geometry of your set-up).
\item[Q 1.3] Which parts of the plot are most consistent with the model of a parallel plate capacitor, and which are not?  
\item[Q 1.4] Explain the deviation from ideal behavior.  
 
Now, create two new columns from your existing data by making a column of 1/V and a column of $1/(d-d_0)$.  Plot $1/V$ vs. $1/(d-d_0)$.  Do you
see a straight line?

%\item[Q1.5] If your capacitor were behaving as a single ideal capacitor, your data would follow a curve $1/V = (\epsilon A/Q)\times 1/d$.  Derive this formula.

\item[Q1.5] This formula will pass through the origin when $d$ gets large [$1/(d-d_0)$ goes to zero].  What do you think is causing your data {\em not} to go through the origin?  (Hint: if the capacitance of your cables is important, then it adds in parallel to the capacitance of the plates: C(plates) + C(cables) $= Q/V$.)

\item[Q1.6]Why do we want the lead to have low capactiance?
\end{enumerate}


\hrule
\latex{\vspace*{.05in}}
\noindent {\em EXPERIMENT B:} (OPTIONAL after completing PART II)
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Surface Charge Distribution on a Parallel Plate Capacitor \\
(at fixed Potential Difference)  \end{center}

\noindent SUGGESTIONS:
\begin{enumerate} 
\item \latex{\vspace*{-.07in}}Ground yourself, the electrometer and the movable plate of the  capacitor. Turn the parallel plate capacitor so that you are behind the movable plate.  Set up electrometer and cup as in {\bf E1} but not close to the capacitor. (Why?)
\item {\em Connect 500 volts to the fixed plate}.  Do NOT apply this voltage to electrometer directly. 
\item Use the aluminum paddle (as in experiment {\bf E1}) to probe the charge density on the capacitor's  surfaces, and then use the electrometer and cup to measure the charge on the  paddle.  To avoid spurious effects from charges on the paddle's  handle, touch the paddle to the bottom inside of the cup and remove the paddle before taking the reading.
\item Record the relative charge density (sign and magnitude) on both  the inner and outer surfaces of the two plates for three radial positions:
center, halfway out, near edge of plate.  Use plate separations of 2.5, 5, and 10~cm. 
\end{enumerate}

\vspace*{.1in}
\noindent QUESTIONS for Experiment B: 
\begin{enumerate}
\item \latex{\vspace*{-.07in}}Why are measurements for separations $<$ 2.5 cm not very  meaningful? 
\item How does relative charge density, $\sigma $, inside and outside the capacitor depend on plate spacing?  On distance from center of the plate?  Explain.
\end{enumerate}
\noindent {\em EXPERIMENT C:} (OPTIONAL)
\begin{enumerate}
\item[] For a fixed spot inside the capacitor, find how $\sigma $ varies 
with voltage.
\end{enumerate}
\latex{\vspace*{.1in}}
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%\newpage

\begin{center} {\bf
Part II: \\
CAPACITORS IN PARALLEL, IN SERIES, AND CONNECTED TO RESISTORS}
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\noindent SUGGESTIONS:
\begin{enumerate}
\item \latex{\vspace*{-.07in}}{\em Use the electrometer to test voltages in Part~II: A and B experiments--{\bf review the Electrometer Precautions under the apparatus section at the beginning of the lab!}}.  Use the lucite circuit board containing different capacitors  and resistors.  Although a push-button switch (plus connectors) permits applying 30~V momentarily to any capacitor, you may prefer just to touch the voltage supply leads directly to the capacitor being charged. 
\end{enumerate}

\noindent {\em EXPERIMENT~A:} CAPACITORS IN SERIES

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{\begin{enumerate}  
\item \latex{\vspace*{-.03in}}Discharge all capacitors first by 
momentarily shorting leads with a banana plug connector.
\item \latex{\vspace*{-.03in}}Connect 20~V across two capacitors in SERIES as the circuit and schematic at shows.  (The schematic is only a {\em suggested} configuration.)
\item \latex{\vspace*{-.03in}}Calculate the potential difference expected across each capacitor.
\item \latex{\vspace*{-.03in}}Now measure the voltage across the individual capacitors.
\item[] {\em OPTIONAL:}  If that's too easy, try this: charge C$_1 = 0.5\,\mu$F to 30~V; discharge C$_2 = 1.0\,\mu$F and {\em then} connect 10~V across C$_1$~+~C$_2$. Now measure the voltage across C$_1$ and across C$_2$.  Is this consistent with the idea of charge conservation in the region between the two capacitors?
\end{enumerate} 
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\noindent {\em EXPERIMENT~B:} CAPACITORS IN PARALLEL  %\medskip\\

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\item \latex{\vspace*{-.07in}}Discharge C$_2 = 1.0\,\mu$F and remove from
circuit. 
\item \latex{\vspace*{-.07in}}Charge C$_1 = 0.5\,$F to 20~V and
then disconnect the D.C.~supply. 
\item \latex{\vspace*{-.07in}}Measure voltage on C$_1$.
\item \latex{\vspace*{-.07in}} Connect C$_2$ in PARALLEL to C$_1$ and measure the final voltage across the capacitors (with the supply still disconnected.)
\item \latex{\vspace*{-.07in}}Calculate the voltage you would expect. 
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\noindent {\em EXPERIMENT~C:}  DISCHARGING  OF CAPACITOR THROUGH A RESISTOR
\begin{enumerate}
\item[] \latex{\vspace*{-0.08in}}In this experiment you will use the electroscope and computer interface to observe the discharging of a capacitor through a resistor.  So far you have only observed the ``steady state'' or DC behavior of capacitors.  Now you will charge the capacitor to a given voltage and then remove the voltage supply.  Because the two plates of the capacitor are actually connected through the resistor, the charge on the capacitor will ``drain,'' or move to the opposite plate through the resistor.  You will  measure the voltage drop across the capacitor, which tells you the amount of charge $Q$ remaining on the capacitor.  This is related to the flow of charge, or current $I$ by the following relations as a function of time $t$ during the discharge:

\[ \frac{V(t)}{R} = I =-\frac{dQ(t)}{dt} \mbox{~~~and~~~} Q(t)=C V(t).  \] 

Solving the above equation gives the relation: $V=V_0 e^{-t/\tau}$ where $\tau \equiv RC $ is the {\em time constant} for a particular circuit. $R$ is the value of the resistor, which is measured in Ohms, or $\Omega$.  Note that the input resistance of the Electrometer is much larger than the values of $R$ you will be using, so the charge flow into the electrometer is negligable.  \latex{\vspace*{-0.05in}}
\end{enumerate} 

\noindent SUGGESTED PROCEDURE:
\begin{enumerate}
\item \latex{\vspace*{-0.1in}}Connect the $10^{7}\,\Omega$ resistor (marked as $10\ Mohm$ or 10~M$\Omega $) across the large  capacitor ($1.0\ \mu F$) and set up the electrometer to measure the voltage on the  capacitor.  Fig.~4 shows the nominal circuit configuration and a possible wiring diagram for the circuit.  Depressing the switch connects the power supply to the circuit, which will rapidly charge the capacitor, and releasing the switch will initiate the discharge.
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\\ Figure 4:  The resistor and capacitor are connected in parallel.}
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\item \latex{\vspace*{-0.08in}}Connect 30~V across the capacitor and observe the voltage V$_{C}$ on the capacitor as a function of time after you disconnect the 30 volts and switch in the resistance $R$.  Qualitatively describe the discharge behavior observed by watching electrometer display.
\item \latex{\vspace*{-0.08in}}After making sure the PASCO interface is connected to the electrometer output (through Channel A), CLICK on the ``Launch RC III'' icon below to initiate the  DataStudio interface software.  There should be a panel, a table and a graphing display for V {\it vs} t.
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\item \latex{\vspace*{-0.08in}} CLICK the REC button, charge the capacitor, and record data while the capacitor is discharging.  CLICK on the STOP button. Record six (or so) evenly spaced data points into your lab book.
\item \latex{\vspace*{-0.08in}} Move the leads to the $1~M\Omega$ resistor and repeat the experiment.  (Configure the interface to show multiple data sets.)
\item  \latex{\vspace*{-0.08in}}Move the leads to the $100~k\Omega$ resistor and repeat the experiment. 
\end{enumerate}

\noindent {\em QUESTIONS:}  
\begin{enumerate}
\item \latex{\vspace*{-0.1in}}Do the curves have the expected functional behavior?
\item \latex{\vspace*{-0.1in}}By moving the curser over the inital voltage and time, and then over the voltage at $1/e$ of the inital value, compare the nominal product $R \times C$ to the time required for the voltage to drop to $1/e$ of the initial value ($e = 2.72 \ldots$).  You may need to adjust the sampling rate to check this for the 100$\,$k$\Omega$ resistor.
\end{enumerate} 
