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\title{Draft Lab Manual\\
Chapters for Electricity \& Magnetism}

\author{Murray Thompson \& Mike Winokur\\
made with LaTeX 2e/AMS-LaTeX in the Oztex 3.0 Package\\
Most changes from TEDI to LaTeX made by Miriam Schmidt\\
-- this draft omits the designs of some of some experiments --\\
-- this is on ZIP A2 --\\
-- earlier draft on A6 and Arawa's APS4 --\\
file=DraftE\&M.tex}
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\tableofcontents



\chapter{Electricity and Magnetism}
\section{E1.~~Electrostatics}
\subsection{Relevant Physics Theory}

\begin{center}
PART I. THE ELECTROSCOPE
\end{center}

\subsection{OBJECT}  To use an electroscope to study electrostatic phenomena.

\subsection{Preliminary Questions}
\begin{enumerate}
\item
An isolated object A with zero total charge, is brought from a distance to a 
position close to a an isolated but positively charged object.
Is there any electrostatic force upon the object A?
\item
Describe any charges on the object A.
\item
Will the movement of object A, to this new position, increase or lower the 
potential energy of object B?
\item
?
\end{enumerate}
\subsection{APPARATUS}  A wooden box containing: 
aluminum-leaf electroscope, insulated hollow sphere
proofball, hard rubber rod, rabbit fur, lucite rod, silk cloth. 

\subsection{SUGGESTIONS} 
\begin{enumerate}
\item In humid air insulators may adsorb enough moisture 
that charges leak off rapidly.  If so, dry the insulator with a heat gun.
\item In very dry or cold weather the humidity is so low that 
clothing, table tops, etc.~become good insulators.  Their large
surface charges may influence nearby unshielded instruments.
If so, ask your instructor 
for help; e.g.~use grounded foil to shield against them.
Also remove any clinging loose bits of fiber (e.g.~silk or rabbit fur) that may 
disturb results. 
\item The fragile leaves of the electroscope may tear if charged too heavily.
{\bf Do not disassemble electroscope} to attempt repair:  see instructor.
\item To remove charges on the glass windows of the electroscope, 
rub your hands over the windows while grounding your body.
\item {\rightskip=2in Charges on the insulating handle of a proofball 
can cause serious measuring errors. Test the
handle by grasping 
the ball with one hand (while the other hand touches the electroscope
ground), and then bring parts of the
insulating handle close to the electroscope
knob.  If the leaves move, the handle 
is charged.\smallskip\par}

\noindent To discharge it, hold the handle in a source of ionized air.  The charged 
insulator will attract ions of the opposite sign until it is neutral.  An 
open flame is a simple source of ions of both signs.  The heated air 
convects the ions upward so {\bf hold insulator at least 10 cm above open flame 
to avoid heat damage to the insulator}.
Avoid unnecessary handling of insulators because 
handling may impair their insulating capability.
\item Ground the electroscope case to a grounding jack near 
the electrical outlet.  Leave it grounded throughout the experiment.  
\end{enumerate}

\subsection{EXPERIMENTS} 
\begin{enumerate}
\item Charge a rubber rod negatively (-) by rubbing with fur, 
and transfer some of the 
charge to the electroscope leaves by touching the rod to the electroscope 
knob.  See diagram A.  Note what happens when a 
(-) charged rod approaches the knob (diagram B).  Without grounding the 
electroscope, also observe and diagram what happens 
when a lucite rod rubbed with silk approaches the knob.  

\vspace*{2.5in}

\item {\bf Charging~by~induction}   (First discharge the electroscope 
by touching the knob and case 
simultaneously.)  To {\bf charge by induction} make the leaves diverge by 
bringing up a (-) charged rod:  keeping the rod fixed, {\bf ground the 
knob} to the case, {\bf break the ground} and then {\bf remove the rod}.  Explain with 
diagrams each step of the process.  Show by diagrams what happens when a 
(-) charge is brought near the knob; a positive (+) charge.  Explain each 
step briefly.
\item In the following give the proofball a charge of known sign by 
contact with a charged lucite or rubber rod.  {\bf Use diagrams to explain and 
record results}. 
\begin{enumerate}
\item[a.] Connect the hollow conductor to electroscope knob by fine wire.  
Discharge them.  Charge proofball by touching\footnote{Avoid rubbing which
may result in charge separation of opposite sign (depending on the 
proofball metal and type of insulator: ``triboelectric series'', 
see exp \#6 of Part II).}
it to a charged 
rubber or lucite rod; then introduce it into the hollow 
conductor but without contacting the conductor.  Ground the hollow 
conductor by touching it.  Note behavior of electroscope.  Break 
the ground and remove proofball.  Test the sign of charge on the 
electroscope.
 \item[b.] Repeat part a, but now ground the hollow conductor 
by touching it {\bf only on the inside} with your finger or a short conductor.  
Explain.
\end{enumerate}
\end{enumerate}
\newpage

\begin{center}
PART II. Use of ELECTROMETER (a sensitive voltmeter) 
\end{center}
\subsection{OBJECT}   To measure potential differences 
(i.e.~the work/charge to move a 
small test charge between two conductors) and thus
(indirectly) charges.

\subsection{Preliminary Questions}
\begin{enumerate}
\item
Why does the placement of a charge on the electroscope cause the thin hanging foil 
of the electroscope to move away from the rest of the electroscope?
\item
Why does the movement away occur for both positive and
negative charges?
\item
?
\item
?
\end{enumerate}
\subsection{APPARATUS}   Pasco electrometer \& power supply; 
two closed spheres, one open hollow sphere; two charge producing 
paddles (white \& blue), one aluminum paddle; 
proofball; insulated cup and shield:  {\bf do not carry the cup and 
shield by the 
top cover} (which may separate, the cup drop and the insulator break); 
heat gun; alcohol lamp.

\vspace*{2.8in}

\subsection{INTRODUCTION}   The electrometer reads directly the difference of 
potential (in volts) between the ground and input terminal.  The 
electrometer's input resistance ($\sim 10^{14}\ ohm $) 
is so high that negligible current flows 
and hence discharging effects are negligible.  (For circuit, see 
instructor.)  From the several voltage ranges,
choose one that gives a large but less than full scale deflection.   
Accuracy is about $\pm $~3\% of full scale.
Connect the insulated cup and shield to the electrometer as shown in the 
sketch.  The electrometer measures the potential difference between the cup 
and grounded shield.  {\bf This potential is proportional to the 
quantity of charge on the cup if the aluminum can furnishes a perfect 
shield from external charged objects}.

\subsection{PRECAUTIONS} Precautions are needed for reliable measurements:
\label{ElectrometerPrecaution}
\begin{enumerate}
\item Large static charges (common in dry weather) if applied to 
electrometer input may damage the sensitive input field effect transistor (``FET'').  
Minimize this possibility by {\bf keeping the electrometer input grounded via 
the LOCK switch during the initial hook up and when you are done with the 
experiment}.  Also use a banana plug lead to connect 
the GND terminal of the electrometer to the ground jack near an 
electrical outlet.
\item Check batteries by switching to B$_{1}$ and then B$_{2}$. 

*****EXPLAIN B1 and B2.***** 

(If meter readings don't drift, batteries in the REPLACE range may still be 
usable.)
\item If your clothing 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 
the grounded shield all the time during a measurement or connect a bare 
wire from your body to ground).
\item To remove all charge from the cup, switch the electrometer 
momentarily to the CHECK position. (This connects the electrometer terminals 
to each other so that any charge flows to ground).  
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 ``CHECK'' 
or ``LOCK'' position.
\item 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, create lots of 
ions in the surrounding air. The insulator will then attract 
ions of the opposite 
charge until 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}
\subsection{SUGGESTED~EXPERIMENTS} 
\begin{enumerate}
\item Rub the blue and white paddles together and determine which paddle 
has a net positive charge and which has a net negative charge.
Explain the following experiments with diagrams showing charge 
distributions, e.g.~appropriate diagrams for experiment \#2 below are:

\vspace*{2.8in}

\item Discharge both the blue and the white paddles.  Gently rub the white and 
the blue surfaces of the paddles together.  Then: 
\begin{enumerate}
\item[a)]  Hold one of them near 
the bottom of the cup, but don't let it touch. 
\item[b)]  Take the paddle out.  
\item[c)]  Put it back in, touching the cup this time.  
\item[d)]  Remove the paddle.  
\end{enumerate}
Record the electrometer reading after each step.  Explain the results.  Was 
there any charge left on the paddle at the end?
\item {\bf Charging~by~induction}  Momentarily ground the cup.  Charge one of 
the paddles, then: 
\begin{enumerate}
\item[a)]  Place in cup without {\bf touching}.  
\item[b)]  Again 
momentarily ground the cup.  
\item[c)]  Remove the paddle.  
\end{enumerate}
Note the reading after each step.  Explain what happened at each step.
\item Start with both paddles discharged. Then rub them together.  
Measure the charge on each.  Explain.  Compare the amount of charge produced 
by rubbing two paddles of the {\bf same} kind (borrow one from another group), or 
by rubbing a metal paddle on a metal paddle, etc.  (Surface ``dirt'' on one paddle 
may mean you are rubbing dissimilar paddles.  Sometimes cleaning the paddles 
with alcohol makes a big difference.)
\item Discharge the cup and the blue and white paddles.  What happens if you 
place the paddles inside the cup (without touching the cup) and, 
\begin{enumerate}
\item[a)]  you charge them by rubbing while they both are inside the cup?  
\item[b)]  Take out one paddle?  
\item[c)]  Put it back in?  
\item[d)]  Take out the other?  
\item[e)]  Take them both out?  

\noindent Observe electrometer reading after each step.  Explain.
\end{enumerate}
\item Do only insulators acquire charge by rubbing?  Rub the aluminum 
paddle on the white (or blue) paddle.  Determine sign of effect.  Arrange 
the white, blue and aluminum paddles in a series such that the rubbed one with 
a positive charge is always higher (``triboelectric'' series).  
\begin{enumerate}
\item[]{\bf NOTE}  In the remaining parts you may measure charge and still avoid spurious 
effects from charges on the insulating handles if you will touch the 
charged proofball (or paddle) to the bottom of the cup and then remove it 
from the cup before reading the electrometer.  But remember to discharge 
the cup (by momentarily grounding) before taking the next reading!  However, 
if the potential of the insulating handle is too large (e.g.~way off the 
least sensitive scale), one can still get spurious effects from leakages.
\end{enumerate}
\item Instead of frictional charging, use the Pasco power supply.  Ground 
the black binding post and connect the {\bf green} binding post to the hollow 
sphere with a copper wire.  Adjust the voltage to $\cong $~500V.
{\bf The switch must be in the correct position or no voltage results even though 
the meter may
read}!  Discharge the 
proofball and test that it (and handle!) have zero charge; then touch it 
to the outside of the hollow sphere and measure its charge.  Do the same 
experiment but touch inside the sphere.  What do you conclude?
\item Connect the power supply to an isolated solid metal sphere.  Measure 
the relative charge density $\sigma $ at various points of the sphere.  Charge 
density $\sigma $ is $\Delta $Q/$\Delta $A where $\Delta $Q is the charge on a 
small element $\Delta $A of the 
surface.  Since it is not practical to remove a piece of the surface, we 
place the aluminum paddle {\bf flat} against the surface and measure the charge 
on the paddle after it is removed.  This measurement gives a number only 
approximately proportional to the charge density (why?) but does give a 
good idea of the relative charge distribution.  

OPTIONAL:  Repeat 8 but 
with paddle perpendicular to the surface instead of flat.  Can you explain 
the differences?
\item Discharge the second solid sphere when it is far from the charged 
sphere. Then move it  within a few centimeters of the charged sphere.  
Explore the charge distribution on both spheres with the paddle.  Observe 
carefully the polarity of charge on different parts of the second sphere.
\item Does a grounded conductor necessarily have no net charge?  
Momentarily ground the second sphere when it is near the first one, then 
move it away and see whether it is charged.  What is the polarity?  ~Explain.
\item How much deflection does one get if one applies 1000V to the 
aluminum leaf electroscope?  Compare the sensitivity to that of the 
electrometer (but do {\bf not} connect 1000V to the electrometer or to anything 
connected to the electrometer e.g.~the cup.)
\item {\bf Turn~off~electrometer~so~batteries~will~not~run~down} and put switch 
in the LOCK position.
\end{enumerate}
\newpage

\section{E2~Electric~Fields}

\subsection{OBJECT}  To map electric fields in two dimensions.

\subsection{Preliminary Questions}
\begin{enumerate}
\item Why are the electric fields across uniform conductors (as in this experiment)
similar to the electric fields across vacuums, insulating gases and insulators if the 
shapes of the conductors are similar?
\item ?
\item ?
\item ?
\end{enumerate}

\subsection{APPARATUS} 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 ({\bf cap when not in use!}) shared between 4 experiments; white paper.

\subsection{INTRODUCTION}  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:

\noindent {\bf PRECAUTIONS}
\begin{enumerate}
\item[1)] {\bf Use adequate ventilation.  Avoid inhaling the vapor and avoid contact 
of the ink with your skin}.
\item[2)] Before using, shake the pen vigorously to disperse the particle matter 
suspended in the ink. 
Avoid thick layers.  {\bf CAP THE PEN WHEN NOT IN USE}.
\item[3)] 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).
\end{enumerate}

\noindent {\bf GENERAL METHOD} 
\begin{enumerate}
\item[1)] 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)] Record the shape of the electrodes on the white paper by tracing with a 
hard pencil or a ball point pen.
\item[3)] 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.

\vspace*{2in}

\item[4)] With 24 volts connected as in Fig.~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 moveable 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.
\item[5)] Map other equipotentials: 6 volts, 9 volts,...21 volts.
\item[6)] After you have drawn the equipotentials on the white sheet, dot in a few
electric field lines ($\simeq $~7) as for example Fig.~2.
\end{enumerate}

\noindent {\bf ALTERNATIVE~METHOD}  

For one of the electrode configurations, replace
the electrometer ground lead by another 
coaxial cable connected to 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 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.

\noindent {\bf SUGGESTIONS}  

Map equipotential lines and electric field lines for several 
of the configurations shown in Fig.~3 and Fig.~4 (or create your own design). 
The distances are merely suggestions.  Return to your instructor the 
graphitized paper with painted electrodes. 

\vspace*{.5in}

\noindent Fig.~3

\vspace*{1in}

\noindent Fig.~4:  Examples of configurations giving electrostatic focusing\\ \\
\noindent a) in an X-ray tube

\vspace*{1.5in}

\noindent b) in a TV or cathode ray tube

\vspace*{1.5in}

\noindent c) in the ``dee'' structure of a cyclotron

\vspace*{1.5in}

\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.

\noindent 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.

\noindent{\bf Turn~off~the~electrometer~when~you~finish~mapping.}  
Leave the zero check in the locked position.

\noindent {\bf QUESTIONS}
\begin{enumerate}
\item Explain in a sentence or two how you mapped the equipotential lines.
\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 {\bf b} and {\bf a} 
on the equipotential.)
\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}
\newpage

\section{E3~Capacitors~and~RC-decay}
\noindent {\bf OBJECTIVES} I.~For a parallel plate capacitor explore how (for fixed 
charge) the potential difference varies with spacing. 
II. For parallel and series capacitors study the charging; 
also the discharge of C thru a resistor.

\subsection{Preliminary Questions}
\begin{enumerate}
\item
How can one surface of a capacitor hold a large charge with a modest voltage 
while a similar charge upon an isolated object would have a much higher voltage?
\item
Why is the leakage of charge less on a capacitor with a particular type of 
insulation than on an isolated object with same type of insulator?? Revise!
\item
If the positive and negative charges upon a capacitor are doubled, 
is the relative voltage increased or decreased and by what factor?
\item
Is it practical to double the the positive charge on a capacitor without 
changing the negative charge? Give your reasons.
\end{enumerate}

\noindent {\bf APPARATUS} 

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; 
timer.

\noindent {\bf PRECAUTION} 

{\bf Remember electrometer precaution 
\#1, on page \pageref{ElectrometerPrecaution}}.

\noindent {\bf INTRODUCTION}  

A capacitor consists of two conducting electrodes 
separated by an insulator.  In part I the parallel plate capacitor consists 
of two circular metal plates held on lucite 
insulators.  Part II uses commercial capacitors consisting of two long 
strips of aluminum foil separated from each other by a thin plastic foil, 
all rolled into a cylinder.
\begin{center}
Part I: THE PARALLEL PLATE CAPACITOR 
\end{center}

\noindent {\bf Experiment A} {\bf Potential~Difference~vs~Separation} (for fixed charge)

\noindent {\bf INTRODUCTION}  

For a fixed charge the potential of a conductor depends on 
what bodies are nearby.  If you momentarily charge a parallel plate 
capacitor and then change the plate spacing, you expect 
the potential difference will change 
since Q = CV remains constant, but C~=~$\frac{\varepsilon A}{d}$ changes.

\noindent {\bf SUGGESTIONS}
\begin{enumerate}
\item[1)] Use the 30~V scale and terminal of the voltage source . ({\bf
Standby switch 
must be in proper position or no voltage results even though meter reads)}.

Connect the electrometer across 
the capacitor, but use the special low capacitance lead plus separate 
ground instead of the shielded coaxial cable. (Why do we 
need a low capacitance lead? Hint: see optional analysis below.) 

Use the 
moveable plate as the grounded one, thereby shielding the other plate 
from charges on your hand or clothing as you move the one plate. 
\item[2)] {\bf Start with a plate separation of 10~cm.  
Momentarily charge the 
capacitor to 30~V by touching the plates with the voltage supply leads.}

Then measure the voltage on the capacitor as a function of plate spacing.  Take 
large steps at first, then smaller steps as the spacing becomes 
small.  (Note that the insulating spacers limit one to $>$ 0.1 cm separation; 
also, zero on the cm scale may not be zero separation).  What do you find?
\item[3)] {\bf OPTIONAL ANALYSIS} 
What should one expect? We measure  V = Q/C where 
C is the capacitance of the parallel plates (=~$\varepsilon $A/d) {\bf plus} the 
capacitance  C' of the leads and the electrometer input since they also 
charge up to the same voltage V. Thus we write
\[C = \frac{\varepsilon A}{d} + C' = \frac{Q}{V} \]  
but we don't know C' and Q.  For an elegant way out, plot the reciprocal 
of the measured V against the reciprocal plate spacing.  What should you get?  
Why does it hold only for {\bf small} plate spacings?  
Can one find C'?
\end{enumerate}

\noindent  OPTIONAL Part I {\bf Experiment B} \\
\hspace*{.5in} Surface Charge Distribution on a Parallel Plate Capacitor \\
\hspace*{1in} (at fixed Potential Difference) 

\noindent {\bf SUGGESTIONS}
\begin{enumerate}
\item[1)] Ground yourself, the electrometer and the moveable plate of the 
capacitor. Turn the parallel plate capacitor so that you are 
behind the moveable plate.  Set up electrometer and cup as in E-1 but not 
close to the capacitor. (Why?)
\item[2)] {\bf Connect 500 volts to the fixed plate}. (Switch must be in the proper 
position or no voltage results!)
\item[3)] Use the aluminum paddle (as in experiment 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 
insulator, touch the paddle to the bottom of the cup and remove the paddle 
before taking the reading (see p. 63).
\item[4)] 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}

\noindent {\bf QUESTIONS} for OPTIONAL Experiment B: 
\begin{enumerate}
\item[1)] Why are measurements for separations $<$ 2.5 cm not very 
meaningful? CHANGE WORDS
\item[2)] 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 {\bf OPTIONAL} Part I Experiment C:\\
\hspace*{.5in} For a fixed spot inside the capacitor, find how $\sigma $ varies with voltage.
\begin{center}
Part II: \\
CAPACITORS IN PARALLEL, IN SERIES, AND CONNECTED TO RESISTORS
\end{center}
\noindent {\bf SUGGESTIONS} \\
{\bf Do~not~ground~the~electrometer~in~Part~II~experiments}. 

\noindent 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.  To 
measure voltages use the electrometer with the probe leads of the E-2 
experiment.  

\noindent {\bf Experiment~II.1} CAPACITORS IN SERIES
\begin{enumerate}
{\rightskip=2.5in
\item[1)] Discharge all capacitors first by 
momentarily shorting with a banana 
plug connector.
\item[2)] Connect 20~V across two capacitors in SERIES.  

Measure the voltage across the individual capacitors.
\item[3)] Calculate the potential difference expected across each capacitor.
\smallskip\par}
\begin{enumerate}
\item[] {\rightskip=2.5in {\bf OPTIONAL}  If that's too easy, try this:  
charge C$_1$ to 30~V; discharge 
C$_2$ and {\bf then} connect 10~V across C$_1$~+~C$_2$.  
Now measure the voltage 
across C$_1$ and across C$_2$.  Calculate the expected voltages. \smallskip\par}
\end{enumerate}
\end{enumerate} 
\begin{center}
-------------------------------------------------------
\end{center}
\noindent {\bf Experiment~II.2} CAPACITORS IN PARALLEL
\begin{enumerate}
\item[1)] {\rightskip=2.5in Discharge C$_2$. 
\item[2)] Next charge C$_1$ to 20~V and 
\item[3)] measure voltage on C$_1$.
\item[4)] {\bf Disconnect~supply}, 
\item[5)] then connect C$_2$ in PARALLEL to C$_1$ and 
\item[6)] measure the final voltage across the capacitors.
\item[7)] Calculate the voltage you would expect. \smallskip\par}
\begin{center}
-------------------------------------------------
\end{center}
\end{enumerate} \noindent {\bf Experiment~II.3}  DISCHARGING OF CAPACITOR THROUGH A RESISTOR
\begin{enumerate}
\item[1)] {\rightskip=2.5in Connect the $10^{7}\ ohm$ 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.
\item[2)] 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.  Switch to lower scales as voltage decays. \smallskip\par}
In other words, repeat the decay several times, with one student watching the second hand 
of a watch and giving the time, and a partner reading the voltage. 
This procedure can be repeated several times 
to make as many measurements as practical. For example, the voltages could be read at times
(in seconds after each recharge);
0, 5, 10, 15, 20, 25, recharge, 
2, 7, 12, 17, 22, 27, recharge,
4, 9, 14, 19, 24, etc
\item[3)] {\bf Turn~off~electrometer} and set zero check to lock position.
\item[4)] Graph how V$_{c}$ depends on time; also try plotting ln V$_{c}$ 
vs time.
\item[5)] {\bf QUESTIONS}
\begin{enumerate}
\item[i)]  What is the functional form of the curve?
\item[ii)] What happens if you use other values of R and C?  Compare the product 
R x C to the time required for the voltage to drop to 1/e of the initial 
value (e~=~2.72....). 
\end{enumerate}
\end{enumerate}
\newpage

\subsection{E4, E5.~~Null~Measurements} 
\begin{center}
(Suitable for two lab periods if one includes the optional parts)
\end{center}

\noindent {\bf OBJECTS}  To study a null method (the Wheatstone bridge) 
for precision measurement of resistances, 
and to use another null method (the potentiometer) 
for accurate measurement of 
emf's and potential differences. 

\subsection{Preliminary Questions}
\begin{enumerate}
\item
Contrast the meaning of the words ``resistor'' and ``resistance''.
\item
?
\item
?
\item
?
\end{enumerate}

\noindent {\bf APPARATUS}  

Slidewire suitable for Wheatstone bridge or potentiometer;
electronic voltage source, resistance boxes, 
unknown resistances, galvanometer, 
reference voltage source (standard cell or Zener diode), cell of unknown 
emf, switches.

\noindent {\bf CAUTION}  

{\bf Transport galvanometers only when switch is in the \underline{``short''} 
position}. 

\noindent {\bf INTRODUCTION}  
We can test whether two potential drops are equal with much more precision 
than measuring their values:  the technique is to use an ultra sensitive 
null detector, e.g.~a sensitive galvanometer. 

\begin{center}
{\bf Part I Wheatstone bridge}

\end{center}

\noindent Since V~=~IR, the voltage drops across 
resistors in series are proportional to the resistances.  Wheatstone's 
bridge makes ingenious use of this fact to compare resistances.  See Fig.~1 
and Fig.~2.

\vspace*{.5in}

\noindent \hspace*{1.8in}R$_{s}$~=~standard R\\
\hspace*{1.8in}R$_{x}$~=~unknown R\\

\vspace*{.5in}

\noindent \hspace*{1.8in}R$_1$~=~variable R\\
\hspace*{1.8in}R$_2$~=~variable R\\

\hspace*{3in} R$_1$~$\propto $~L$_1$ \hspace*{.5in} R$_2$~$\propto $~L$_2$\\

Fig.~1 - Schematic      \hspace*{1.8in} Fig.~2 - Slidewire Bridge

\noindent Adjust the variable resistances R$_1$ and R$_2$ of the slide wire 
until no current 
flows through the galvanometer (i.e., zero deflection): then 
I$_1$~=~I$_2$, I$_{s}$~=~I$_{x}$, and point b has the same potential 
as point c.  
Hence the potential drop from a to b equals that from a to c.  Therefore
\begin{center}
I$_{s}$R$_{s}$~=~I$_1$R$_1$ 
\end{center}
and
\begin{center}
I$_{x}$R$_{x}$~=~I$_2$R$_2$.
\end{center}
Substituting the equal currents and dividing gives

\[\frac{I_{x}R_{x}}{I_{x}R_{s}}~=~\frac{I_2R_2}{I_2R_1} \hspace*{.5in}
 \mbox {or} \hspace{.5in}
        R_{x}~=~\left( ~\frac{R_2}{R_1}\right) R_{s}.\] 
For wire of uniform cross section (Fig.~2) 

\[R~\propto ~L  \hspace*{.5in} \mbox {and} \hspace*{.5in} R_{x}~=~
\left( ~\frac{L_2}{L_1}\right) R_{s} .\]

\noindent Our R$_{s}$, is
a General Radio decade resistance box with a claimed accuracy of 
$\pm $(0.01\%~$\pm ~2 \times 10^{-3}\ ohm $).  The connectors in the bridge must themselves 
be low resistance to achieve accurate resistance comparisons; hence the 
brass strips.

\noindent {\bf SUGGESTIONS}
\begin{enumerate}
\item The connectors to R$_{s}$ and R$_{x}$ must also be of low and constant 
resistance so use 
spade lug connectors.  Banana plug type leads are satisfactory for all 
other connections.
\item If the edge of the slider which rests on the slide wire is reasonably 
sharp, L$_1$ and L$_2$ can be read to a fraction of a mm.  To obtain greatest 
accuracy, adjust the resistance box R$_{s}$ so that L$_1$ and L$_2$ are nearly 
equal.  Why?
\item In very dry weather the sensitive Evro galvanometers may show 
erratic behavior as one moves around.  If so, ground the slider connection.  
Then as you touch the slider to adjust it, you are automatically grounded.
\item With the galvanometer switch on the 0.001 scale, set the mechanical 
zero of the galvanometer to the center of the illuminated 
scale.
\item In balancing the bridge always start on the least sensitive galvanometer 
scale (0.001) and switch to more sensitive scales 
as the bridge becomes balanced.
If the bridge is badly unbalanced, the galvanometer may bang off scale and
not return.  If so tap the galvanometer box or consult instructor.
\end{enumerate}

\noindent {\bf PROCEDURE}
\begin{enumerate}
\item Measure a known resistance\footnote{R$_{s}$ and R$_{x}$ 
must not be too small or the resistance of the connectors 
and leads will be important.}
(in a second resistance box) as your first 
``unknown'' in order to check the equipment and get an estimate of the 
reliability of your measurements.  Check the uniformity of the slidewire and 
connections by interchanging the known and ``unknown'' 
resistances and seeing if you get the same value for the ``unknown.''
\item Check the commercial resistors on the board furnished.  Are they 
within their rated tolerance?  The commercial color codes for resistance 
values and tolerances are given in Fig.~3. 
\end{enumerate}
 Fig.~3
\vspace*{3in}

\noindent {\bf OPTIONAL}
\begin{enumerate}
\item Estimate the reliability of your measurements and justify your estimate.  
\item What limits the accuracy of this 
experiment?  How could the accuracy be improved?
One possible improvement:
Since the slider and slide wire are of different materials, the warming 
of the slider by the fingers may produce a thermoelectric emf which 
will introduce error when the bridge is connected as shown on Fig.~2.  
If one interchanges the connections 
of the battery and the galvanometer, the 
thermal emf then is in series with the battery and will cause no error.
\item Measure each of the unknown resistances furnished.  Then 
measure them in series and in parallel.  Compare with the calculated 
values. Do they fall within your predicted errors?  If not, why not?
\item Set up a more complicated resistor network.  Calculate the equivalent 
resistance and check it with the bridge.
\end{enumerate}
\newpage
\begin{center}
{\bf Part II, the Potentiometer}
\end{center}
\noindent Connect the apparatus as in Fig.~4 but have the TA check your 
hook up before you close any switches.  {\bf CAUTION}  
A 10~K$\ ohm $ series resistor with the standard cell 
prevents damage from currents $>$~0.1 mA.  {\bf Short out this resistor only 
for the final balance on the most sensitive galvanometer setting}.

Fig.~4

\vspace*{3in}

\noindent{\bf INTRODUCTION}  

If in Fig.~4 our $\cong $~2 volt source sends 
current only through the outer circuit, then the 1 meter long high 
resistance slide wire (between A and B) will show a constant rate of
potential drop k ($\cong $~0.02~V/cm) along the wire.  
The (SPDT) switch 
in position S connects the standard 
reference voltage through the galvanometer G
to the slide wire through the slider D. With the slider 
adjusted so that the
galvanometer shows no deflection (with tap switch closed), then no galvanometer 
current means that the reference voltage exactly equals the potential drop 
of the wire from A to D$_1$.  We thus have calibrated accurately k, the voltage 
drop per cm along the wire.  

\noindent To measure an unknown emf between the terminals 
A and X (or potential drop V$_{x}$) set the SPDT switch to the X 
position and readjust the slider to position D$_2$ such that again the 
galvanometer deflection goes to zero.  Then the potential drop between A
and D$_2$ equals the unknown emf or unknown potential drop:


\[ \left.\begin{array}{ll}
V_{x} = \; \mbox {P.D. across} \; A-D_2 = k \; L_2\\
V_s = \; \mbox {P.D. across } \; A-D_1 = k \; L_1 \end{array} \right\}
 \mbox {therefore} \;
V_x = V_s \frac{L_2}{L_1} \]


\noindent Note that the potentiometer will measure any potential drop you connect in 
place of V$_{x}$ provided it is less than that of the constant current source.

\noindent {\bf SUGGESTIONS}  

Use the least sensitive scale of the galvanometer (0.001) at 
the start.  Switch to more sensitive scales as you approach balance.  In 
very dry weather the Evro galvanometers often show erratic effects on the 
most sensitive scale when one moves around.  If so, ground the slider.  In 
this way you automatically ground yourself when you adjust the slider.
When finished with a measurement, recheck the voltage calibration of the 
resistance wire.  If it has changed appreciably (e.g.~from drift in the 
``constant'' voltage supply), then recheck 
the unknown potential difference; finally redo the voltage calibration and 
correct your unknown potential difference for any drift in the calibration.

\noindent {\bf SUGGESTED~MEASUREMENTS}
\begin{enumerate}
{\rightskip=3in \item Measure the {\bf emf of a dry cell}.
\item Measure the {\bf p.d.~between terminals of the same 
dry cell when furnishing current through an external resistor}.  See Fig.~5. 
Use first a 40~$\ ohm $ and then a 20~$\ ohm $ resistor.  Explain the difference. Hint: recall 
that\\ \\
\hspace*{1in} \(V_{t} = \xi  - Ir_{int}\)\\ \\
with V$_{t}$ the terminal voltage and $\xi $ the emf.
 \smallskip\par}

\hspace*{3in} Fig.~5

\noindent {\bf Note}  When large currents flow from the cell,
its emf, $\xi $, will fall 
because of a {\bf back emf} from hydrogen gas which accumulates around the carbon 
electrode.  Upon standing (open circuited) the hydrogen gas is gradually 
oxidized to water by the manganese peroxide present, and the emf 
slowly returns to normal.  

\noindent {\bf Hence for reproducible results in item 2. above, 
avoid drawing current from the cell except during measurement}.  

\noindent However, 
hydrogen gas is {\bf not} the explanation of the difference between using a 40 
or 20 ohm external resistor but is the reason we ask you to measure first 
with the 40 ohm resistor.  
\end{enumerate}
\noindent {\bf OPTIONAL}
\begin{enumerate}
\item Calibrate a low range voltmeter at several points; use 
circuit of Fig.~6.  For R use the resistance box and/or the 4.7~kohm and/or 
the 10~k$\ ohm $ resistors.  Try R~=~0 and large R so that the 
reading drops to intermediate values.

\vspace*{2.2in}

 Fig.~6  \hspace*{3in} Fig.~7
\item Calibrate the 0.25~A scale of an ammeter; use the circuit in 
Fig.~7. 
For R {\bf use only the precision resistance box}.  To avoid blowing fuses on 
the ammeter, first set R to be $\geq$ 6 ohms.  The potentiometer measures the 
voltage V across the known resistance R.  The current then is 
I~=~V/R.  Adjust R to give several currents within the range of the ammeter.
\item Calibrate the voltmeter at a voltage higher than the 
potentiometer can measure directly.  Use the circuit of Fig.~8.  The voltage 
supply must exceed the range of the voltmeter to be calibrated.  

{\rightskip=3in \noindent Let 
{\bf R~=~resistance between points A and B} and {\bf R$_{p}$~=~resistance 
between A and 
X}.  Since the balanced potentiometer does not take current from X, the 
current is the same through R$_{p}$ as through R.  Hence

\(\frac{correct~voltmeter~reading,~\left( V\right) }
{potentiometer~reading,~\left( P\right) }~=~\frac{R}{~R_{p}}
\) \\ \\
or ***WHAT IS P?***

\( V = (\frac{R}{R_{p}}) P.\)

\noindent To change V, adjust r;
R/R$_{p}$ can be any convenient value.
\smallskip\par}
\hspace*{4in}Fig.~8
\end{enumerate}



\section{E6.~~Measurement~of~Charge~to~Mass~Ratio~(e/m)~for~Electrons}
\noindent {\bf OBJECT}  

To measure e/m by magnetic deflection of electrons accelerated 
through a known voltage.

\subsection{Preliminary Questions}
\begin{enumerate}
\item ?
\item ?
\item ?
\item ?
\end{enumerate}

\noindent {\bf APPARATUS}  

Sargent-Welch e/m equipment (Fig.~1); 
power supplies with built-in
digital meters; dip needle and compass; black cardboard and black cloth.

\vspace*{2.5in}

\hspace*{.5in} Fig.~1

\noindent {\bf INTRODUCTION}  

The evacuated e/m tube contains a 
saturated vapor pressure of Hg.
When electrons with E~$\geq $~10.4~eV strike Hg atoms, some atoms 
become ionized and some become 
excited.  Most quickly recombine and/or de-excite to emit a bluish light 
before moving appreciably.  Hence the bluish light marks the 
path of the electron beam.  (Of course the few electrons which strike 
Hg atoms are no longer in the beam).

\noindent The potential difference V between the filament and 
anode (Fig.~2 and 3) 
accelerates electrons thermionically emitted by the filament.  Those 
electrons directed toward the slit S emerge with a velocity v given by
\begin{equation}
Ve = \frac{1}{2}mv^2
\end{equation}
\noindent provided that the thermal energy at emission is small compared to 
Ve.  
\newpage
{\leftskip=1.5in \noindent F is a hot filament emitting electrons\\
C is a cylindrical anode at potential +V with respect to F\\
S is a vertical slit thru which electrons emerge\\
B is electron path in magnetic field\\
A is a rod with cross bars at known distances from the filament\\

\vspace*{1in}
\noindent Fig.~2: The e/m tube viewed along the earth's magnetic field

\vspace*{1in}
\noindent GG are the filament leads and support.\\
E is the anode lead. It supports the anode C and the rod \\
\hspace*{2in} with cross bars.\\
L is an insulating plug in the end of anode cylinder C.\\

\vspace{1.5in}
\noindent Fig.~3: Side view of Fig.~2
\smallskip\par}
\noindent If the tube is properly oriented, the velocity of the emerging electrons is 
perpendicular to the magnetic field.  Hence the magnetic force vector $\vec{F}$, 
which is described as a cross product
$\vec{F}~=~e~(\vec{v}~\times~\vec{B})$,
supplies the centripetal force $\frac{mv^{2}}{r}$ for a circular path of radius r.
Since $\vec{v}$ is perpendicular to $\vec{B}$
\begin{equation}
evB = \frac{mv^{2}}{r}
\end{equation}
\noindent Eliminating v between (1) and (2) gives 
\begin{equation}
e/m = \frac{2V}{(B^2r^2)}
\end{equation}
\noindent Our digital voltmeter measures accurately the accelerating voltage 
V.  The 
radius of curvature, r, is half the distance between the filament F and one 
of the cross bars attached to rod A. See table which follows:
\newpage
\begin{tabular}{ccc}
{\bf Crossbar~No.} & {\bf Distance~to~Filament} & {\bf Radius~of~Beam~Path}\\
    1 & 0.065 meter & 0.0325~m\\
    2 & 0.078 meter & 0.039~~m\\
    3 & 0.090 meter & 0.045~~m\\
    4 & 0.103 meter & 0.0515~m\\
    5 & 0.115 meter & 0.0575~m          
\end{tabular}

\noindent To find e/m we still need the magnetic field strength B.  Instead 
of measuring B, we calculate it from the dimensions of the Helmholtz 
coils and the measured coil current, $I$,  needed to bend the beam so that it hits a 
given cross bar.

\vspace*{1in} 

\noindent \hspace*{.5in}Fig.~4\\
\hspace*{.5in}Helmholtz Coils\\
\vspace*{1in}

\noindent Helmholtz coils consist of two identical coaxial coils which are 
separated as in Fig.~4 by a distance R equal to the radius of either coil.
They are useful because near the center (X~=~R/2) the field 
is nearly uniform over a large volume.  (Most textbooks assign this proof as 
a problem. Hint: Find how dB/dX changes with X for a single coil at 
X~=~R/2, etc.) 

\noindent The field at X~=~R/2 is in fact:\footnote
{The actual B at the maximum electron orbit is only $\sim $~0.5\% less.  See 
Price, ``Electron trajectory in an e/m experiment'', 
{\bf Am.~J.~Phys.}~55,~18,~(1987)}

\begin{equation}
B = \frac{8}{{\sqrt 125}}
\left( \frac{N\mu _0I}{R}\right)  \mbox {teslas} 
\end{equation}

as one easily sees by adding the axial fields B(x) from each of the 
single coils: 
\[B(X) = \frac{N\mu _0IR^2}{2\left[ R^2~+~X^2\right] ^{3/2}}.\]
Thus when X~=~R/2, then~ 
\[B~=~2B(X)~=~\frac{N\mu _0I^2R^2}{\left[ R^2~+~R^2/4\right] ^{3/2}}
~=~\frac{N\mu _0I}{R~\left[ \frac{
5}{4}\right] ^{3/2}}~=~\frac{8}{\sqrt{125}}
\left(\frac{N\mu _0I}{R}\right) .\]
\begin{tabbing}
 In equation (4), \= N~=~\= number of turns on each coil (72 for these coils)\\
\> I~=~ \>current through each coil in amperes\\
\> R~= \>mean radius of the coils in meters (approximately \\
\> \>0.33~m but varies slightly from unit to unit)\\
\>$\mu _0$~=\> 4$\pi \times ~10^{-7}$ tesla meter/ampere.
\end{tabbing}
\noindent Finally if we substitute (4) into (3) we obtain

\begin{equation}
e/m~=~\left( 2.47~\times ~10^{12}~\frac{R^2}{N^2}\right) \left(
\frac{V}{I^2r^2}\right) ~~\mbox {coulombs/kg~.}
\end{equation}


\noindent{\bf CIRCUIT~DIAGRAM}  Fig.~5 should help you hook up the 
components.

\vspace*{3in}
\hspace*{4in}Fig.~5

\noindent {\bf SUGGESTIONS}
\begin{enumerate}
\item Set the axis of the Helmholtz coils along the local direction of 
the earth's field, B$_{e}$, as determined with a compass and dip needle.
The axis should point towards the geographic north but at an angle about 60 degrees
from the horizontal. Thus the axis should point deep under Canada. 
Put the long axis of the e/m tube in the north-south direction and with the cross bars 
up.
\begin{enumerate}  
\item[]CAUTION: Nearby ferromagnetic material (e.g.~steel in the power supply, the table
and in the walls) can alter 
the local direction of B$_{e}$. 
As long as the field 
direction does not change during the experiment, there should be no 
problem.
\end{enumerate}
\item Set the filament supply knob to zero before turning the power on.
A red light comes on when excessive filament current or a too rapid 
increase trips a protective relay. 
Reset by putting the filament control to zero and slowly turning it up. 
\item Since e/m depends on V/I$^2$, one needs high quality meters for V 
and I.
Our digital meters have an accuracy of $\pm $~one digit in the last displayed 
digit.
\item Start with $\sim $~22~V between filament and 
anode.  With the room dark, {\bf gradually} turn up the filament control
until the beam is visible.  To make the beam more visible use 
black cloth and black cardboard to block out stray light:
place the black cardboard {\bf inside} the Helmholtz coils and view from 
the top. If the protection circuit trips and a red light comes on,
reset by zeroing the filament control; then turn it up 
{\bf slowly}.  When the beam appears, adjust the 
filament control to give 5-10~mA of anode current. {\bf Do not exceed 
15~mA!}  With no current through the Helmholtz coils, 
the earth's magnetic field 
should slightly curve the beam (like the dotted line in Fig.~2.)
\item {\bf Correction for the earth's magnetic field}  
Increase I through the Helmholtz coils 
until the beam deflects. If the sense of deflection 
is the same as that from the 
earth's field (i.e.~when I~=~0), reverse the leads to the Helmholtz coils; 
the curvature should then be like 
B in Fig.~2.  The field from the coils then opposes the 
earth's field; hence adjust the coil current until the beam 
path is straight.  Since light travels in straight lines, the beam 
should then hit the glass at the center of the area illuminated by light 
from the filament.  

\noindent This field current which just cancels the earth's field, we call I$_{e}$.  
Clearly the I for equation (5) must have I$_{e}$ subtracted: Let the correct 
field current for crossbar n be I$_{n}$, and let I$_{n}'$ be the measured 
current to bend the beam around so that the outside sharp edge of the beam hits 
the center of the nth
crossbar. Hence I$_{n}$~=~I$_{n}'$~-~I$_{e}$ is the correct 
current to bend the beam to the appropriate radius, r$_{n}$.
\end{enumerate}
\noindent {\bf MORE ACCURATE METHODS TO ELIMINATE THE EFFECT 
OF THE EARTH'S FIELD}\\
Uncertainty in I$_{e}$ can dominate the uncertainty in 
e/m, so consider using the following more accurate method to
eliminate the effect of the earth's field: 
\begin{enumerate}
\item[i.)] Take two readings of coil 
currents, I$_3'$ and I$_3''$:  one with the earth's 
field opposing that 
of the Helmholtz coils, the other with it aiding.  To 
accomplish this, first (with the cross bars up) measure I$_3'$ (as described 
above); then rotate the e/m tube 180$^o $ about its long axis. 
\begin{enumerate}
\item[] CAUTION: 
{\bf always rotate the tube in the sense to REDUCE the TWIST in the FILAMENT \&
ANODE LEADS}. {\bf Otherwise one can twist the leads off}.  
\end{enumerate}
\noindent Next
reverse the current in 
the Helmholtz coil and adjust it so that the beam again hits the same cross 
bar. Call this reading I$_3''$.  Then
\[I_3'~~- ~I_3''~~=~~2I_{e}.\]
\item[ii.)] Or better yet, for each cross bar take an I$_{n}'$ and I$_{n}''$ reading.  The 
average
\[\frac{I_{n}'~~+~~I_{n}''}{2}~~=~~I_{n}\] 
is the current required if the earth's field were zero.
\end{enumerate}

\noindent {\bf MEASUREMENTS~AND~ANALYSIS} (If time is short, measure only two cross bars 
at each voltage.)
\begin{enumerate}
\item Make two determinations of e/m for each cross bar.  
\item Repeat for an 
accelerating voltage of $\sim $~44 volts.  (Note I$_{e}$ should remain the same and 
one can measure it more precisely at the lower accelerating voltage.)

\noindent SUGGESTED DATA TABULATION: \{If you use the alternative method 
(ii.~above), 
replace I$_{e}$ by I$_{n}''$ and  
I$_{n}$~=~I$_{n}'$~-~I$_{e}$ by I$_{n}$~=~$\frac{1}{2}$(I$_{n}'$~+
~I$_{n}''$).\}\\

\begin{tabular}{|p{19mm}| p{19mm}| p{19mm}| p{19mm}| p{19mm}| p{19mm}| p{19mm}| } \hline
        Volts& bar & radius & I$_{e}$ & I$_{n}'$& I$_{n}$~=
~I$_{n}'$~-~I$_{e}$ & e/m \\ \hline
& & & & & & \\ \hline 
& & & & & & \\ \hline 
& & & & & & \\ \hline 
& & & & & & \\ \hline 
\end{tabular}
\item Estimate the reliability of your value of e/m and compare to 
1.76~$\times $~10$^1$$^1$~C/kg.  
\end{enumerate}
\noindent {\bf OPTIONAL} \\
Note that the electron beam 
expands in a fan shape after leaving the slit.  
Interchange the filament leads and note how the fan shape flips to the 
other side of the beam. 

\noindent {\bf Explanation}  The fan shape deflection arises 
from the filament current's magnetic 
field acting on the beam.  Our filament supply purposely uses unfiltered 
half wave rectification of 60 Hz AC so that for half of the cycle 
no current flows thru 
the filament but which is still hot enough to emit sufficient electrons.  The 
electrons emitted in this half cycle constitute the sharp beam edge 
which we use for 
measurement purposes. This trick not only avoids deflection effects for 
this part of the beam, but also avoids the uncertainty in electron energy 
arising from electrons being emitted from points of different 
potential along the filament. (During this half cycle of no filament current 
the filament is $\sim $ an equipotential.) The fan shape deflection 
relates to electrons emitted during the other half cycle when filament 
current flows.

\noindent See: F.C.~Peterson, {\bf Am.~J.~Phys.} 51, 320, (1983)
\newpage

\section{E7U.~~(6) Electromagnetic Induction}

\subsection{Relevant Physics Theory}
*** Place here some text and at least one diagram. 
This text can describe magnetic induction as in the 
208 text books and 208 classwork. 
It should be sufficient for the rare cases in which the experiment is done 
before the classwork.The diagram should clearly show the direction of an induced emf. 
Be careful with notation and defn of emf. ***

\subsection{OBJECT}  To study electromagnetic induction and to 
verify both Lenz's law and Faraday's Law.

\subsection{Preliminary Questions}
\begin{enumerate}
\item
When you are looking along the axis towards a circular coil, 
a current is present moving in a clockwise direction.
(This means that the apparent movement of positive changes is clockwise and
the movement of the negative electrons is anticlockwise.)

In what direction is the magnetic field at the center of the coil? 
(Towards you or away from you?)
\item
Does an unchanging magnetic field cause an induced voltage in any circuit?
\item
Describe two common items in your house or car whose operation relies upon 
``Electromagnetic Induction''.
\item
If the magnetic flux through a circuit changes at twice the rate, 
does it cause the induced voltage to be doubled, halved or multiplied by $\sqrt{2}$?
\item
?
\end{enumerate}
\subsection{Demonstration}
\begin{itemize}
\item A strong permanent magnet with a gap of about 100 mm, a coil of wire and a 
medium impedance moving coil voltmeter. The magnetic poles are labelled N and S. 
The voltmeter inputs are labelled `current in'' and ``current out''.

Use this simple system to verify the direction of the emf ${\cal E}$ which is induced 
when the coil is moved into the magnetic field.
\item
An AC generator which can be turned by hand with a DC voltmeter to see the 
output voltage varying with time. This illustrates the significance of the magnetic induction 
which will be measured in this experiment. Notice that the voltages are increased by 
changing, more quickly, the magnetic field through the coils of wire. In fact, Faraday's law predicts 
that the emf ${\cal E}$, which is induced in coils of contant area, is proportional to 
$\frac{\partial \Phi}{dt}$ where $\Phi$ is the magnetic flux through the circuit. 
The electrical power which is delivered to us all depends upon magnetic induction.

Industrial electrical generators, such as those at Madison Gas and Electric,  have iron
cores to greatly 
increase the magnetic field and the voltages which are induced by its changes. 
However, the iron cannot be everywhere in a system of moving coils since some clearance
must be left for insulation and between the moving parts.
In a system with iron in only some places, the field $\vec{B}$ and rate of change 
$\frac{d\vec{B}}{dt}$ are hard to calculate. 

For that reason, in this experiment, we will 
avoid the use of iron and measure some smaller but predictable voltages.
\end{itemize}

\subsection{Preliminary Questions}
\begin{enumerate}
\item ?
\item ?
\item ?
\item ?
\end{enumerate}

\subsection{APPARATUS}
\begin{itemize}  
\item Three coil Systems;\\
--one primary pair of large Helmholtz Coils, \\
--two similar many-turn secondary coils;\\
\item a computer with an internal plug-in board with Analog-to-Digital-Conversion (ADC) to give
the ability to measure voltages; \\
\item a high impedance digital voltmeter; \\
\item a dry battery with a 100k ohm resistor in series and the  positive terminal colored red 
with a large ``{\Large +}'';\\ 
\item permanent rod magnet with its North Magnetic pole marked in red and with a ``{\Large N}'';\\
\item brass and iron rods; \\
\item Ramping Current supply; \\
\item small compass with the North Magnetic Pole marked with luminous paint; \\
\item the Earth with\\
-- the Geographic North Pole and a South Magnetic Pole near it but deep under 
   upper Canada and also \\
-- the Geographic South Pole with a North Magnetic Pole near it but deep under 
   the Antarctic Plateau.\\
\item (optional:  model generator/motor).
\end{itemize}

\noindent {\bf CAUTION}    If a strong magnet has been placed close above or below the 
compass needle, 
the needle may have remagnetized to the opposite polarity.

In this experiment, we will use a pair of Helmholtz Coils as the ``primary''. 
The Helmholtz Coils of radius $R$ are 
placed a distance $R$ apart on the same axis and the Coils are connected in series so 
that the current flows in the same way (clockwise or anticlockwise) in both coils. 
This particular configuration, of radius = distance apart, is often used
to give a very uniform magnetic field in the central region.

\subsection{Why use a Computer?}
In this experiment, a computer is very helpful for 3 reasons.
\begin{enumerate}
\item  It can record voltages very quickly during the change of the magnetic field. 
It would be impossible for a person to record 10 or more voltages during a change lasting 2 seconds.
\item It can meaure voltages more precisely than a person can read a ordinary meter.
\item It can sum thousands of results to calculate an integral with respect to time.
\end{enumerate}
\subsection{INVESTIGATIONS OF LENZ'S LAW}
Reminder: ``An induced current in a closed conducting loop will appear in such a direction
that it opposes the change that produced it.''

 Procedure:
Verify the direction of the induced emf ${\cal E}$ in the secondary Coil due to several procedures.
\begin{enumerate}
\item {Initial Set-up of Apparatus}\\
Set up the secondary circuit with 2 jumper cables twisted together and loosely connecting 
the 4800 turn winding of the secondary coil (ie n=4800) to the terminals on the base. 
Connect these, in turn, to the computer ADC input terminals. \\
Set the computer to read 2 kSample/second ...\\
Set the Range Configuration to be ``-10 V to + 10 V'' so that both positive and 
negative voltages can be read.\\
Set the gain to be 100.0 so that each Least Count of the ADC corresponds to 
a voltage input of 3.0518 microvolts.


\item Move the magnet so that the magnetic field enters the secondary coil from the right
(ie bring the magnet North pole  closest to the right side of the secondary coil).\\
        --Is the emf induced clockwise or anticlockwise in the secondary as seen along the axis 
        from the right?\\
        --If a small resistor was connected across the terminals of the secondary,
        so that a current surge could flow in the secondary during the movement then, in which direction 
        would the secondary 
        experience a force due to this current surge and the magnetic field of the magnet?
        (In this setup, the force is too small for you to feel but, if you can predict the 
        direction of the force.)\\
        --In which direction would Lenz's Law predict the force?
\item Move the magnet so that its magnetic field $B$ enters the secondary coil from the left
(ie bring the magnet North pole  closest to the left side of the secondary coil).\\
        --Is the emf ${\cal E}$ induced clockwise or anticlockwise in the secondary as seen along the axis 
        from the right?\\
        --If a small resistor was connected across the terminals of the secondary,
        so that a current surge could flow in the scondary during the movement, in which direction 
        would the secondary 
        experience a force due to the current surge and the magnetic field of the magnet?
\item Place the magnet in a position in which its magnetic field enters the secondary 
        coil from the right. Then remove the magnet and its field.
        During the removal, is the emf ${\cal E}$ induced clockwise or anticlockwise 
        (as seen looking along the axis from the RHS) in the secondary?
\item 
\begin{enumerate}
\item Consider the solenoidal sense coil or ``secondary coil'' already mounted within the pair of Helmholtz Coil.
        Connect the Ramping Current Supply to the Hemholtz Coils  so 
        that the current flows clockwise (seen from the right hand side) in both coils
        and causes a magnetic field along the axis from your right to left.
        Use a current of about +10 Amp. 
\item Push the transistion button on the Ramping supply and 
        notice how the current ramps down and up. It can ramp down to either zero current 
        (from +10 A to 0) or to the negative equal current (from +10 A to -10 A).
        The current is restored to +10 A when you release the pushbutton.
\item Use a compass to check the direction of the magnetic field $\vec{B}$in the 
        central region.
        Push the button to reverse the current and notice whether the field $\vec{B}$ reverses.
\item Connect, in series, another simple circuit consisting of only the dry battery, 
        a 100k ohm resistor and the galvanometer in series.
        Determine whether the current must flow from the battery ``+'' terminal
        into the red or the black banana
        socket for the galvanometer to show a positive current.
\item Connect the galvanometer across the secondary so that a clockwise  induced emf 
        ${\cal E}$ (as seen looking along the axis from the RHS)in the secondary will 
        cause the galvanometer to indicate a positive current. 
        You can do this by following the direction of the wires from the secondary's wire turns 
        to the galvanometer.
\item Push the transistion button on the Ramping Supply and notice the induced emf 
        in the secondary.
        When the primary current changes from 0 to +10 A, (in a clockwise direction seen 
        from the RHS), record the direction (Clockwise or anticlockwise seen from the RHS)
        of the induced current flow and induced emf in the secondary.\\
        In which direction does Lenz's Law predict the direction of the emf ${\cal E}$ 
        in the secondary?
        
        *** At what point should the student start using the computer to measure voltages?***

\end{enumerate}
\item Connect the dry battery and 100k ohm resistor in series with the computer's 
        voltage measuring terminals. 
        As before, determine whether the current must flow into the red or the black banana
        socket for the computer's terminals from the computer to show a positive current.
        
\item Connect the computer terminals across the secondary so that a clockwise 
        (around the axis of the secondary solenoid as seen from the RHS) 
        induced emf ${\cal E}$ in the secondary will cause the computer to indicate a positive 
        current. 
        You can do this by following the direction of the wires from the secondary's wire turns 
        to the computer.
        
\item Push the transistion button on the Ramping Supply and notice the induced emf.
        When the primary current changes from 0 to +10 A, (in a clockwise direction seen 
        from the RHS), record the direction (clockwise or anticlockwise seen from the RHS)
        of the induced current flow and induced emf in the secondary.\\
        In which direction does Lenz's Law predict the direction in the secondary?
\item ?
\item ? Put a lamp or LED across the secondary? (It can receive 5.5 mA from an 
impedance of 1.7 kohm.)
\item ?
\end{enumerate}
Write your conclusions about the directions of the induced emfs.

\subsection{INVESTIGATIONS OF FARADAY'S LAW}
First Reminder: The ``Magnetic Flux $\Phi$'' which passes through a surface, bounded by
a closed loop, measures the number of magnetic lines of force through that surface.''\\
More precisely, the ``Magnetic Flux $\Phi$'', through a surface $\vec{A}$, is the integral 
of the scalar product

$\Phi=\int \vec{B}.\vec{dA}$   ''

\vspace{10mm}

Second Reminder:``If a circuit is temporarily broken and a voltmeter is placed across 
the break, the voltmeter is said to measure the ``Electromotive Force'' or 
emf ${\cal E}$ of the rest of the circuit''.

\vspace{10mm}

Third Reminder: ``The Electromotive Force (emf) ${\cal E}$ induced in a conducting loop is equal to the 
negative of the rate at which the magnetic flux $\Phi$ through that loop is changing with time.''

${\cal E}=-\frac{d\Phi}{dt}$

Notes 
\begin{itemize}
\item If we make a conducting loop with the wire having n turns, then the magnetic field
induces an emf in each turn and so the total emf ${\cal E}$ of the n turns is n times that of each turn. 
The result is the same as if n times the flux was through a single turn.

\item We measure the voltage with a very high impedance ``voltmeter'' in the computer and so we 
can directly measure the emf ${\cal E}$ as a simple voltage.

\item The magnetic flux $\Phi=\int \vec{B}.\vec{dA}$ can be changed in several ways.\\ 
The current in a primary coil can be changed or\\
the position of a magnet or primary coil can be changed or\\
the position of the secondary coil can be changed or\\
the shape and size of the secondary coil can be changed or\\
the orientation of the secondary coil to the magnetic field can be changed.

\item We will neglect the Earth's magnetic field for 2 reasons
\begin{itemize}
\item If I is near 18 A, the field of the coil is near 2.3 millitesla while the Earth's field 
is only about 60 microtesla.
\item We are concerned with only changes in the flux through the coil and, unless the secondary 
coil is rotated, the flux due to the Earth is constant.
\end{itemize}
\end{itemize}
 
 Faraday's Law agrees with Lenz's Law but Faraday's Law gives us the \underline{magnitude} 
 of the induced 
 emf while Lenz's Law gives us \underline{direction} of the induced emf.\\
 In our experiment, we will use two alterations to make the apparaus more simple.\\
 \begin{itemize}
 \item We will use the Helmholtz Coils, instead of a magnet, to create or destroy a Magnetic 
 field because 
 \begin{itemize}
 \item the field B of a Helmholtz system is uniform and can be readily calculated whereas
 \item the field B of a magnet is nonuniform and is not easily measured.
 \end{itemize}
 \item Although in the next procedure we will test Faraday's Law 
 
 ${\cal E}=-\frac{d\Phi}{dt}$, 
 
 \noindent
 in the later procedures, we will test the integral of both sides of the equation
 
 $\int{\cal E}dt=-\Phi$
 
 We use the integral method later because the computer can easily add many measurements to integrate ${\cal E}$
 whereas it is hard to cause a precise $\frac{d\Phi}{dt}$ since it is hard to
 cause a current ramp with a precise $\frac{dI}{dt}$.
 \end{itemize}
 
 \subsection{Procedure:}
Verify the magnitude of the induced emf ${\cal E}$ in the secondary Coil due to several 
procedures.

*** This procedure assumes the use of the
{\bf National Instruments Enhanced Multifunction I/O board PCI-MIO-16XE-50 for PCI
(Natl Inst 1996 Catalog pg 3-104)}. ***
\begin{enumerate}
\item {Initial Set-up of Apparatus}\\
Retain the set up you already have for the secondary circuit.\\
Set up the primary circuit by connecting the Ramping Supply so that the current output
will cause a current to flow clockwise in the Helmholtz Coils (as seen from the RHS).
Remember that the current must flow from the supply to one big Coil then to the next
big Coil and then back to the supply.\\
Set the maximum current to 18.0 Amp by adjusting the helipot while watching the digital current scale.
(The helipot numbers have no meaning except as a fraction of the maximum current.)

\item 
To keep your notes tidy, we suggest making a short table of your results; with the headings

\noindent
\begin{tabular}{|p{12mm}|p{17mm}|p{17mm}|p{17mm}|p{17mm}|p{17mm}|p{17mm}|p{17mm}|}
\hline
item \# & $\Delta I$&$\frac{dI}{dt}$&$n$ & $N$ & Calculated $-\frac{d\Phi_{1}}{dt}$ 
                                        & Measured ${\cal E}_{1}$ 
                    & ratio $\frac{{\cal E}_{1}}{-\frac{d\Phi_{1}}{dt}}$\\
\hline
\hline
1 & & & & & & &\\
\hline
\end{tabular}

\begin{enumerate}
\item Set the Ramping Current Supply to have transitions from +18 A to -18 A 
($\Delta I_{1}=36\ A$) and back.
Set the transistion rate to $\pm 20\ A/s$ meaning it will take a time 
$\Delta t=\frac{\Delta I_{1}}{(\frac{\Delta I}{dt})}=\frac{36\ A}{20\ A/s}=1.8\ seconds$
 for each transition.
 \item Set the computer menus to have a full scale voltage of ?? volts

\item Calculate the rate of change of the magnetic flux $\frac{d\Phi}{dt}$ that will occur 
if the primary current is changed at a rate near $\frac{dI}{dt}=20\ A/s$.
\item
Since the magnetic field at the center of a Helmholtz Coil carrying current I 
in each of two coils with $N$ turns is 
                
                $B=\frac{\mu_{0}8}{\sqrt{125}}\frac{NI}{R}$ 
                
\noindent
The effective area of a solenoid of $n$ turns when pointing at angle $\theta$
from the magnetic field is $A=n\pi r^{2}.Cos\theta$.
Thus a change in current $\Delta I$ will create an induced emf

$\Phi=\frac{\mu_{0}8}{\sqrt{125}}.\frac{\pi r^{2}Cos\theta }{R}. Nn\delta I$ 
                

Since you will need to calculate $B$ for several cases, you might as well now calculate a
temporary useful factor $F$;

        $F=\frac{BA}{NI}=\frac{\mu_{0}8}{\sqrt{125}}.\frac{\pi r^{2}Cos\theta }{R}$
                
\noindent
Write the value of $F$ and be careful of powers of 10 and of the units.
Now you will be able to quickly calculate the change in magnetic flux due to a 
change in current

$\Delta \Phi=F. Nn\delta I$ 

\noindent
and

$\frac{d\Phi}{dt} = FNn\frac{dI}{dt}$.
                
\item Use the factor $F$ to calculate the rate of change of the magnetic flux $\frac{d\Phi}{dt}$ that will occur 
if the primary current is changed at a rate near $\frac{dI}{dt}=20\ A/s$.

\item Push the Transition button to cause the current in the primary and its magnetic field
to reverse over a time of 1.8 seconds. Note the value of the induced emf ${\cal E}_{1}$ which 
the computer records during the change. The list in the computer of values of ${\cal E}_{1}$
should start at zero, become non-zero during the flux change, then become zero after the 
flux change.
\item
Check the agreement between ${\cal E}_{1}$ and $-\frac{d\Phi_{1}}{dt}$. 
\item Release the pushbutton so that the reverse transition occurs. 
Again, check ...
\end{enumerate}
\item 
Make a larger table with at least 4 more lines and record the results of the following 
measurements.

\noindent
\begin{tabular}{|p{12mm}|p{17mm}|p{17mm}|p{17mm}|p{17mm}|p{17mm}|p{17mm}|p{17mm}|}
\hline
item \# &$\Delta I$ &$\frac{dI}{dt}$&$n$ & $N$ & Calculated $-\Phi$ 
                                        & Measured $\int {\cal E}dt$ 
                    & ratio $\frac{\int {\cal E}dt}{-\Phi}$\\
\hline
\hline
2 & & & & & & & \\
\hline
3 & & & & & & & \\
\hline
4 & & & & & & & \\
\hline
etc & & & & & & & \\
\hline
\end{tabular}
Keep the same Current and Current Transition Rate as in item 1, 
(ie $\Delta I_{2}=\Delta I_{1}$, $n_{2}=n_{1}$, $N_{2}=N_{1}$ and 
$(\frac{dI}{dt})_{2}=(\frac{dI}{dt})_{1})$.
\begin{enumerate}
\item Calculate the magnetic flux change that will occur if the primary current is reversed.
\item Push the Transition button to cause the current in the primary and its magnetic field
to reverse over a time of 1.8 seconds. Note the values of the induced emf ${\cal E}_{2}$ which 
the computer measures.
\item Use the computer to integrate the $\int {\cal E}_{2}dt$.
\item Check the agreement between $\int {\cal E}_{2}dt$ and $-\Phi_{2}$. 
\end{enumerate}

\item Take another variation of 4 variables 
(the current change $\Delta I_{3}$, 
the number of turns $n_{3}$ in the secondary  
the number of turns $N_{3}$ in the primary) and
the rate of change of current $(\frac{dI}{dt})_{3}$. 
\begin{enumerate}
\item Calculate the expected flux change and 
\item Push the Transition button again to cause the primary's magnetic field
to reverse over a time of 1.8 seconds. Again the computer can measure the induced 
emf ${\cal E}_{3}$.
\item Integrate $\int {\cal E}_{3}dt$.
\item
Check the agreement between the third values of $\int {\cal E}_{3}dt$ and $-\Phi_{3}$. 
\end{enumerate}

\item Repeat the procedure for another variation of the 4 variables 
($\delta I$, $n$, $N$ and $(\frac{dI}{dt})$
and check the agreement between the fourth values
of $\Phi_{4}$ and $\int {\cal E}_{4}dt$

\item Reduce the Ramping Supply's $\frac{dI}{dt}$ to, say $\frac{dI}{dt}=2\ A/s$ and, 
keeping the same values of the other 3 variables in the primary as in the previous item
(ie $I_{5}=I_{4}$, $n_{5}=n_{4}$, and $N_{5}=N_{4}$), repeat the procedure 
and check the agreement between the fifth values
of $\int {\cal E}_{5}dt$ and $-\Phi_{5}$. Notice that although the values of $\int {\cal E}_{5}dt$
were smaller, the change and the integral lasted for a longer time and so $\Phi_{5}$ 
and $\int {\cal E}_{5}dt$ were not changed.

\item \begin{enumerate}
\item Set $I_{6}=18\ A$, 
\item Rotate the Secondary Coil and let the computer measure the emf ${\cal E}_{6}$
for about a rotation of 90 degrees.
\item Study the nature of the measured values of ${\cal E}_{6}$. 
\item Why would you expect the voltage to have a sinusoidal nature?
\end{enumerate}

\item \begin{enumerate}
\item Set $I_{7}=18\ A$, 
\item Rotate the Secondary Coil and let the computer measure the emf ${\cal E}_{7}$
for about 3 turns.
\item Study the nature of the measured values of ${\cal E}_{7}$. 
\item Why would you expect the voltage to have a sinusoidal nature?
\end{enumerate}

\item If you have time, try a few more choices of the 4 variables
($\Delta I$, $n$, $N$ and $(\frac{dI}{dt})$). 
In each case, calculate the expected $\Phi$, measure $\int {\cal E}dt$  compare the
expected and measured values. 
\item ?
\end{enumerate}
In all of the above procedures, did your measurements indicate that  $\int{\cal E}dt=-\Phi$
or ${\cal E}=-\frac{d\Phi}{dt}$ as predicted by Faraday's Law?

\subsection{Observe Useful Magnetic Induction Systems}
\begin{enumerate}
\item repeat one of the above with a iron core
\item repeat one of the above with a brass core
\item transformer
\item Electric generator
\item Inductance
\end{enumerate}

\subsection{Questions}
draft questions
\begin{itemize}
\item If the solenoid had a resistance of 1 kilohm and a resistor of 2 kilohm were placed across
the solenoid, and an emf of ${\cal E}=10.0$ volts was induced in the solenoid,
what voltage might be measured across the resistor?\\
\item If no resistor is added and the angle $\theta$, between the solenoid and the magnetic field, were changed at a
rate $\omega=\frac{d\omega}{dt}$, and the maximum emf found across the solenoid is
${\cal E}=10.0$ volts, what would be the equation for the voltage at any time $t$?\\
\item Describe three common applications of magnetic induction\\
in most homes,\\
near but outside most homes in Madison,\\
in all cars.
\item ?
\end{itemize}
\newpage
\section{E7.~~Electromagnetic~induction (BC)}
\noindent {\bf OBJECT}  

To study electromagnetic induction and to 
verify Lenz's law.

\subsection{Preliminary Questions}
\begin{enumerate}
\item ?
\item ?
\item ?
\item ?
\end{enumerate}

\noindent {\bf APPARATUS}  

Two coils --one a
primary, the other a many-turn secondary; 
galvanometer; rod magnet; 
brass and iron rods; electronic power supply; 
switches; small compass; (optional:  model generator/motor).

\noindent {\bf CAUTION}  

If a magnet is placed or stored close above or below the compass needle, 
the needle may remagnetize to opposite polarity.

\noindent {\bf INVESTIGATIONS}
\begin{enumerate}
\item With the compass explore qualitatively the magnetic field near 
the bar magnet: first check that the compass 
needle points north, (see CAUTION above).  Sketch the field pattern, and 
put arrows on the field line directions. 
\item Similarly explore the field of the primary\\ coil after closing the 
switch of the\\ Fig.~1 set-up. Sketch the results.

\vspace*{.5in}

\hspace*{4in} Fig.~1

\item To test which way the galvanometer deflects for a given current 
direction, connect the dry cell's (-) terminal to the left terminal of the 
galvanometer; then connect the dry cell's (+) terminal {\bf through your body} 
to the galvanometer.  Unless your skin is very dry, enough current should 
flow to give a small deflection.
\end{enumerate}
\noindent {\bf SUGGESTIONS}   

With coil axes aligned horizontally, place the two coils end to end.
Connect the primary as in Fig.~1, but connect the secondary to the 
galvanometer.  Use long (about 1 m) leads for both coils.  
Note the direction of the current in the primary.

\noindent Specify currents as clockwise or counterclockwise as 
viewed always from the right. Tabulate the results of the experiments 
as follows:
\begin{tabbing}
\hspace*{.2in} Action \hspace*{.8in} \=  
~~~~~Direction of current \hspace*{.8in}\=  Size of deflection\\
\>Primary \hspace{.5in} Secondary \> ~(if appropriate)\\
Closed primary \\
~~~switch\\
\end{tabbing}
\newpage
\noindent {\bf EXPERIMENTS ON INDUCTION}
\begin{enumerate}
\item Note the current direction in the secondary when primary 
switch is closed, and then opened.  Explain in terms of Lenz's law.
\item With primary switch closed move secondary away from primary;  then toward 
primary.  Note directions of secondary currents.
Now repeat but move the primary instead of the secondary.  Explain in 
terms of Lenz's law.
\item With the coil axes aligned horizontally (and with 
switch closed), rotate the secondary coil in 90$^o $ steps about a vertical 
axis through the secondary coil.  Record the current directions in 
each step.  Explain in terms of Lenz's law.
\item With the coil axes aligned but separated by several 
centimeters, 
\begin{enumerate}
\item[a)] note deflection size as one closes and opens the primary 
circuit. 
\item[b)] Repeat with the soft iron rod inserted along 
the axes of the coils.  
\item[c)] Repeat with a brass rod instead of the iron rod.  
\item[d)] Explain the results.
\end{enumerate}
\item For the Fig.~2 set-up, 
keep S$_2$ open, but close S$_1$.  Wait several seconds, and 
then close S$_2$.  Does current flow in the galvanometer?  Explain.

\vspace{.5in}
Fig.~2

\vspace{.5in}

\item Remove the iron rod of Fig.~2 and reconnect secondary directly to 
galvanometer.
With primary circuit open, note the deflections as you move a magnet's
{\bf north} pole 
toward and then away from the {\bf right} end of the secondary; then 
toward and away from the {\bf left} end of the secondary.  Repeat the first 
observation but with the {\bf south} pole of the magnet.
\item Twist the leads to the secondary so area enclosed by leads is small and 
remains constant.  With S$_1$ open and with
all magnets removed from the vicinity of the secondary coil, place 
the secondary coil on the table with its plane horizontal.  
Keeping its plane horizontal, raise it quickly (vertically) 
through 10 or 20 cm.  Lower 
it to the table top.  Rotate it though 180$^o $ about a horizontal axis.  
Explain any deflections which you observe.
\end{enumerate}
\noindent {\bf OPTIONAL}

Experiment with the model generator/motor: e.g.~connect
the output of the generator to the galvanometer and 
turn the generator slowly.  Try both slip ring and commutator connections.
\newpage

\section{E8~~Cathode~Ray~Oscilloscope~and~Differential~Amplifiers}
\noindent {\bf OBJECT}  To operate an oscilloscope and auxiliary equipment.

\subsection{Preliminary Questions}
\begin{enumerate}
\item
What cathode Ray systems do you have in your house?
\item
?
\item
?
\item
?
\end{enumerate}

\noindent {\bf APPARATUS}  Dual-trace oscilloscope plus manual, 
signal generator plus 
frequency counter, digital multimeter, (DMM);
circuit plug board \& component kit; differential amplifier.
\begin{center}
{\bf Part~A -- Operation~of~the~Oscilloscope}\\
(Leader Model 1021, or Hitachi V-202F)
\end{center}
\noindent {\bf INTRODUCTION}  

Our dual trace oscilloscopes have circuits which 
alternately 
display sweep signals from two different input channels.  However, because of 
the finite time resolution of the eye, one sees simultaneous dual traces.
\begin{enumerate}
\item[] {\bf CAUTION: 

Avoid placing signal generator on top of scope.  Poor CRT 
shielding 
results in a noisy (fuzzy) trace.}
\end{enumerate}
\noindent {\bf SUGGESTIONS}                                             
\begin{enumerate}
{\rightskip=3in 
\item Connect the sine wave output of the signal (function) 
generator to one of 
the two scope inputs.  Note each input has a lever for 
grounding the input (GND), coupling it through a capacitor (AC), or a 
direct connection (DC).  Use the AC input. \smallskip\par}

\vspace*{.5in}

\item Turn on the scope and note the three main groups of 
controls:  
\begin{enumerate}
\item[I,] those associated with the intensity and focus of the trace; 
\item[II,] those controlling the vertical deflection; and 
\item[III,] those associated with the horizontal deflection.  
\end{enumerate}
\item If you have trouble finding the 
beam, preset the controls as described in the scope instruction manual or 
consult instructor.
\end{enumerate}
\noindent {\bf Discussion of each control group follows} (also see Fig.~1)
\begin{enumerate}
\item[I.] The {\bf INTENSITY} knob controls the trace 
brightness\footnote{CAUTION: a stationary intense spot may damage the screen}
by varying the 
potential of a control grid or an aperture near the electron emitting 
cathode.  The {\bf FOCUS} adjusts the trace sharpness by varying the 
potential on the focusing anode.  {\bf POSITION} knobs 
(see below) move the traces up 
and down, also right and left, by varying the potentials on a pair of
horizontal plates (see Fig.~1) and on a pair of vertical plates (also 
Fig.~1).
\item[II.] The {\bf VOLTS/DIV} and {\bf red VARIABLE} knobs provide coarse and 
fine adjustment of the pattern height by varying the amplification of the 
signals from the input terminals.  To read the correct volts/division 
shown on the scales, the red variable knobs must be in the {\bf CAL}
(calibrated) 
position.

\noindent The vertical {\bf MODE} lever permits display of either or both channels; for 
both use either {\bf ALT} for alternate sweeps, or {\bf CHOP} which switches 
channels at a 250~kHz rate.
(Hitachi's {\bf DUAL}
automatically switches {\bf ALT} to {\bf CHOP} at 
{\bf TIME/DIV} setting =~1~ms/DIV).

\noindent Except for the Hitachi, pulling out {\bf CH~1's} position control ({\bf 
PULL ADD}) 
adds both channels, and
pulling out {\bf CH~2's} position control ({\bf PULL INV}) inverts its 
signal. Hence using 
both {\bf PULL ADD and PULL INV} gives the difference of the two channels. 
(For the Hitachi use {\bf ADD or DIFF} on
the vertical {\bf MODE} lever).
\item[III] The horizontal deflection normally results from an internal sweep 
generator which provides ``saw tooth'' signals

\vspace*{.5in}

Fig.~2

\vspace*{.5in}

\noindent to the vertical plates.  The {\bf TIME/DIV and VARIABLE or SWP/VAR}
knobs give coarse and fine control of this sweep generator.  Again for the 
{\bf TIME/DIV} scales to be correct, set the {\bf VARIABLE}
(or {\bf SWP/VAR}) knob at the {\bf CAL} position.

\noindent The linear rise of the saw tooth voltage sweeps the beam across the screen 
at a uniform rate, and the near vertical drop quickly returns the beam to 
the start. By adjusting
sweep controls one can often synchronize 
a nearly constant frequency vertical signal to an {\bf automatic} horizontal 
sweep and thus produce a stationary trace.  Good scopes blank the 
return trace and have trigger circuits for
synchronizing the sweep start with some feature of the input signal.
See Fig.~3 and below.

\vspace*{.5in}
Fig.~3

\vspace*{1in}

\noindent {\bf To~sweep~automatically}  Set {\bf TRIGGER MODE}
lever to {\bf AUTO}.

\noindent {\bf To~trigger~and~sweep~only~on~a~large~signal}  Set trigger 
{\bf MODE} lever (or 
Hitachi horizontal {\bf MODE}) to {\bf NORM} and set {\bf SOURCE} lever to the appropriate 
{\bf CH~1 or CH~2}; then adjust {\bf TRIGGER LEVEL} knob to desired amplitude.

\noindent The trigger {\bf COUPLING} lever can couple input 
trigger signals 
directly (DC); or through a large capacitance, (AC); or (except 
Hitachi) through a filter
({\bf HF REJ or LF REJ}) which can reject high or low 
frequencies: $>$~4~kHz or $<$~4~kHz.  

\noindent The trigger {\bf SOURCE} lever also has options of triggering on 60 Hz 
({\bf LINE}) or on an external trigger ({\bf EXT}).

\noindent The {\bf TIME/DIV} knob if set at {\bf X-Y},
disconnects the internal sweep generator and connects {\bf CH~1~X} to the 
horizontal deflection plates; {\bf CH~2}, still connected to the vertical 
deflection plates, becomes {\bf Y}. (For Hitachi also set {\bf MODE} 
lever to {\bf CH 2
X-Y}).  The resultant {\bf X-Y} pattern is a Lissajous figure 
(see Part B OPTIONAL).
\begin{enumerate}
\item[] {\bf Caution, if the spot is undeflected, turn down 
the intensity to avoid damage to the screen's phosphor}.
\end{enumerate}
\end{enumerate}
\begin{center}
{\bf Part~B.~~Suggested~Oscilloscope~Experiments}
\end{center}
\begin{enumerate}
\item {\bf Observe both sine and square wave signals of various frequencies}  
Experiment to learn the effect of different controls, (e.g. automatic {\bf vs}
triggered sweeps).  For the square waves look both at low frequencies and 
at maximum frequencies.  Note effect of changing the input coupling 
(switch back and forth from AC to DC).  Sketch the appearance of ``square'' 
waves at high and low frequencies.  Explain the differences.
\item (i) {\bf Check calibration of scope's {\bf Y} deflection} by using the 0.5 volt peak 
to peak square wave signal from the calibration terminal {\bf (CAL)}. 
To connect this signal to the scope, just 
touch the exposed calibration terminal with the center of the 
{\bf b}ayo{\bf n}et {\bf c}oaxial (BNC) connector 
on the input coaxial cable.  

\noindent (ii) Use a digital multimeter (DMM) to {\bf measure} both the {\bf sine}
wave and {\bf square} wave voltage output of the signal generator.  
Avoid very high frequencies since
capacitative loading may then be a problem when the signal generator is 
also connected to the scope.

\noindent (iii) {\bf Compare the DMM readings} (which are r.m.s.~or 
``effective'' voltages 
{\bf with the peak to peak voltages} read off 
the scope screen.  Are they consistent?  (Remember the voltage amplitude is 
one-half the peak to peak voltage and the rms voltage is $\frac{peak\ voltage}{\sqrt{2}}=\frac{peak\ to\ peak\ voltage}{2.818}$!)
\item OPTIONAL: {\bf Check the calibration of the 
scope's {\bf X} (horizontal) deflection} by 
using the {\bf TIME/DIV} reading to measure the period of one wave length of the 
signal.
Compare the deduced frequency with signal generator scale setting and/or 
that of a frequency meter.
\end{enumerate}
\noindent OPTIONAL: 
{\bf Lissajous~Figures}  When two periodic motions at right angles 
combine, the resulting pattern is generally very complex.  However, sine 
waves whose frequencies are in the ratio of small integers give simple 
stationary patterns.  For example see Fig.~4.

\noindent To produce {\bf Lissajous} figures on the scope, set the {\bf TIME/DIV}
control to {\bf X-Y}.  Connect the 60 Hz terminal to the 
{\bf CH~1~X} input.  To {\bf CH~2~Y} apply a sine wave input
whose frequency bears a simple ratio to 60 Hz. (For the Hitachi,             
also set vertical {\bf MODE} lever to Ch 2, {\bf X-Y}.) 
A Lissajous pattern should 
result.  Adjust amplitudes and positions to locate the figure properly
on the screen.  
The frequency scale on the signal generator may not be accurate,
but the 60 Hz power line is an excellent frequency standard.  
{\bf Use Lissajous figures to calibrate your signal generator at 120~Hz and 
180~Hz}. 

\noindent Observe and explain patterns with the frequency ratios of Fig.~4.  Also try 
a ratio 2:3.

\vspace*{1in}

Fig.~4

\vspace*{1.5in}

\begin{center}
{\bf Part~C~--~Differential~Amplifiers} 
\end{center}
\noindent {\bf INTRODUCTION} Amplifiers are devices which usually increase the amplitude 
of the output signal compared to the input signal.  The customary symbol 
for an amplifier is a triangular shape:

\vspace*{.5in}

Fig.~5

\vspace*{.5in}

\noindent In the sine wave example above, the gain is 10 and the input signal is 
between an input terminal and ground.  The internal oscilloscope amplifiers for
channels Y$_1$ and Y$_2$ are of this type and have a common ground.  Because of 
this common ground, one has problems 
in using a scope to examine 
{\bf simultaneously} voltages across individual circuit elements that are in 
series (see below, also E9 lab).

\noindent We can avoid these problems by interposing ``differential 
amplifiers'' which 
have two inputs V$'$ and V$''$, (neither at ground), and which 
amplify only the voltage 
difference (V$'$~-~V$''$).  Symbolically:

\vspace*{.5in}

Fig. 6

\vspace*{1in}

Note now that the ground of the output signal is independent of any input 
ground.

\noindent{\bf SUGGESTED~EXPERIMENTS}
\begin{enumerate}
\item Connect the outputs of the differential amplifiers to 
the scope Y$_1$ and Y$_2$ inputs, and set gains to 1.  Try various 
inputs to the amplifiers; e.g. apply the same signal to both 
inputs of a single amplifier and note V$_{out}$~=~0. 
\item  {\bf Become familiar with the circuit plug board (Fig.~7) and the banana type 
plug-in components}.  Note there are 24 groups with 9 socket plug-in holes in each group.  
While the 9 plug-in holes in each group are electrically connected together, each of the 24 groups 
is isolated unless connected by plug-in components.  Two isolated metal 
bus bars (with banana plug-in holes) on the sides may facilitate common 
connections for some circuits.

\vspace*{1in}

Fig.~7

\vspace*{1in}

\noindent The plug-in components (metal bridges, resistors, 
capacitors, inductors, or 
other circuit elements) will only connect between adjacent groups of the 9 
(already connected) plug-in holes or from the edge metal bus bars to an 
adjacent 9 hole group.  We also provide coaxial cable to 
banana plug-in connectors:
these facilitate using signal generators, scopes, etc.~with the 
circuit plug-in board.
\item {\bf Use the differential amplifiers to observe the action of a 
silicon diode on an A.C.~signal}. 

{\bf NOTE:}
Silicon diodes (Fig.~8) are junctions between silicon semiconductors 
with different type doping: n-type with an impurity which easily donates 
electrons, e.g.~phosphorus (P) in silicon;\footnote{As one might 
expect since silicon has valence 4, phosphorus valence 5, 
and aluminum valence 3} 
and p-type with an ``acceptor'' impurity, 
e.g.~aluminium (Al) in silicon$^5$. 
The junction between the two types
becomes a very non-linear device which if subjected to an electric field
acts almost as a one way valve to current flow.  The diode symbol 
has the arrow pointed in the direction of conventional easy current flow 
(from p-type to n-type) though actual electron flow may be in the opposite 
direction.

\vspace*{.5in}

Fig.~8

\vspace*{.5in}

\noindent {\bf Suggested experiment}
Set up the circuit plug board as shown in Fig.~9. Connect Y$_1$ and Y$_2$ to 
the two scope inputs.  

\vspace*{.5in}

Fig.~9

\vspace{.5in}

Vary the amplitude of the signal generator input 
and sketch the resultant waveforms across the resistor and across the 
diode.
\end{enumerate}
\newpage

\section{E9~~A.C.~Circuits}
\subsection{OBJECT}  To study voltage and phase relations in A.C. circuits

\subsection{Preliminary Questions}
\begin{enumerate}
\item
?
\item
?
\item
?
\item
?
\end{enumerate}
\subsection{APPARATUS} 

Dual trace oscilloscope \& manual; differential amplifiers,
circuit plug board \& component kit; 
signal generator \& frequency counter; digital multimeter (DMM).
\begin{center}
Part A {\bf SERIES~CIRCUIT}
\end{center}
\subsection{INTRODUCTION} 

Remember that ``rms'' value of the AC voltage or current 
means the ``root mean square'' value of the AC voltage or current and is useful
even when the voltage or current is changing but not as perfect sine waves.
For example the voltage used to provide power for most lights and appliances is 
$V_{rms}=110\ Volts$.
Some higher power appliances such as stoves use $V_{rms}=220\ Volts$.
Although the voltage and current may often be negative, the rms values are always positive. 
Similarly, the impedance $Z$ is always positive. Sometimes the word ``effective'' is used
instead of ``rms'' but means the same thing.
For a pure sine wave voltage, the peak voltage and rms voltage are related 
$V_{rms}=\frac{V_{peak}}{\sqrt{2}}$. Thus the peak voltage of a 110 volt AC wire is
$V_{peak}=110\times \sqrt{2}\ volts\approx155\ volts$

The rms concept allows us to describe any AC voltage as having a particular rms voltage and
a particular phase.

In this experiment we will use the subscript ``rms'' for some measurements and will use
``$V$'' or ``$v$'' or ``$I$'' or ``$i$'' without this subscript when referring to instantaneous 
values of the voltage or current. Thus, even if two rms voltages are equal, ($V_{rms}=v_{rms}$), 
$V$ may not be equal to $v$ since $V$ and $v$ may have different phases.

The impedance $Z$ or $X$ of any part of a circuit is the ratio of the rms voltage across that
part and the rms current though that part;

$Z\ or X\ =\frac{V_{rms}}{I_{rms}}$

\noindent
Because impedance is defined as a ratio of voltage/current, impedance is measured in ohms.
If $\omega=2\pi f$ and we measure L and C in henries and farads, then it can be shown   
that the impedance $X_{R}$ of an resistance $R$ is $R$ 
and the impedance $X_{L}$ (in ohms) of an inductance $L$ is  
                $X_{L}~=\omega L=~2\pi fL$.
and the impedance $X_{C}$ (in ohms) of a capacitance $C$ is 
                $X_{C}~=~\frac{1}{\omega C}=~\frac{1}{2\pi fC}$.
It can also be shown that the impedance Z (in ohms) of 
an RLC series circuit (Fig.1) is 
\[Z~=~ \sqrt{R^2 + (X_{L} - X _{C})^2} \]
where, $X_{L}$ and $X_{C}$ are the impedances of the individual inductor and capacitor.

The impedance Z of the RLC series circuit is a minimum for X$_{L}$~=~X$_{C}$.  
The frequency for which this occurs is the {\bf resonant~frequency}.  At this frequency, 
f$_{r}$, the current thru R is maximum, but the 
voltage V$_{rms\ LC}$ across the LC series combination is a minimum and in fact would be zero 
if the inductor had no resistance.  Hence one can search for f$_{r}$ by varying 
the frequency and looking either 1) for a {\bf maximum} V$_{rms\ R}$ signal,  or 2) a 
{\bf minimum} V$_{rms\ LC}$ signal.  A search for the minimum has a
practical advantage that near the resonant frequency, f$_{r}$, one can 
increase enormously the detection sensitivity by going to maximum signal 
generator amplitude and also by going to higher scope gain.

\vspace*{.5in}

Fig.~1

\vspace{1in}

\noindent Our dual trace scope allows 
the simultaneous observation of two voltages.  If we use differential 
amplifiers (E-8) to avoid ground problems,
then we can compare the 
signal across the resistor (displayed on one trace) with the 
signal across other circuit elements (displayed on the other trace).  Since 
across a resistor the voltage and current are in phase, and since the 
current throughout a series circuit is the same, we can observe the
relative phase relationship between the current and any other measured voltage.

\noindent SUGGESTIONS:
\begin{enumerate}
\item Hook up the circuit plug board as shown in Fig.~1. (The signal 
generator supplies $V_{rms}$.)
\item Use the digital multimeter (DMM) to measure $V_{rms}$, V$_{rms\ R}$, V$_{rms\ L}$, 
and V$_{rms\ C}$ at 
frequencies about 400, 700, and 1000 Hz.  Is $V_{rms}$~=~V$_{rms\ R}$~+~V$_{rms\ L}$~+~V$_{rms\ C}$? ~Explain.
\item Calculate the frequency for which resonance should occur.
\item Use the scope to
search for each of the  {\bf resonant frequency} f$_{r}$ by the following 
techniques and also estimate the uncertainty $\Delta $f$_{r}$ in each 
measurement.  Use the trigger output of the signal generator to trigger the 
scope. (For the Krohn-Hite signal generator , use the ``TTL'' output. 
This output gives a ``rectangular'' signal, snapping between zero and +3 volts, which is suitable for 
digital circuitry such as computers and counters.)
\begin{enumerate}
\item[(i)] As one varies f, observe the {\bf maximum} in the {\bf current
$I_{rms}$ thru R}.  
(Since $I_{rms}$~$\propto $~V$_{rms\ R}$, use a differential 
amplifier with scope channel Y$_2$ to look at V$_{rms\ R}$).
\item[(ii)] Observe the {\bf minimum} in the {\bf voltage $V_{rms\ LC}$ across the LC combination}. 
(Use the other differential amplifier to input V$_{rms\ LC}$ to scope channel Y$_1$.
To improve sensitivity, 
turn up signal amplitude and/or scope gain). 
\item[(iii)] {\bf Produce a Lissajous figure} using V$_{rms\ R}$ and V$_{rms\ LC}$ as the X and Y 
inputs. (Use the X-Y setting of the TIME/CM or TIME/DIV knob and set mode 
lever to X-Y).  
At resonance the usual elliptical pattern 
becomes a straight line as V$_{rms\ LC}$ goes to zero.  Again turn up 
the gains to improve sensitivity.
\end{enumerate}
\item Compare the calculated f$_{r}$ with your observed values. 
Is there agreement? 
\item {\rightskip=2.2in {\bf Phase relations} Use scope channel 1 (plus a differential amplifier) 
to observe the total 
voltage $V_{rms\ RLC}$ across the RLC combination. Use scope channel 2 (plus 
differential amplifier) to observe V$_{rms\ R}$.  Fig.~2 shows the phasor relations.

\vspace*{.5in}

\hspace*{4in}
Fig.~2 \smallskip\par}
\begin{enumerate}
\item[(i)] Set the signal generator near f$_{r}$. At 
resonance the two signals should be in phase.  Why?  (If at f~=~f$_{r}$ the 
signals are 180$^o $ out of phase, the input polarity is wrong on one 
differential amplifier; hence interchange
the input leads on {\bf one} of the differential amplifiers.)

\noindent {\bf Find the resonant 
frequency by this method}.  Adjusting gain and position controls so the 
signals nearly overlap will help.
\item[(ii)] With f~=~f$_{r}$, short out the capacitor by a jumper, (e.g.~with metal 
bridge connector).
Note what happens.  Unshort capacitor and then short 
the inductor.  Explain results in terms of Fig.~2.
Does current lead or lag the applied 
voltage in each case?
\item[(iii)] Remove the shorts and try f~$<$~f$_{r}$ and f~$>$~f$_{r}$. 
Explain phase behavior. 
\end{enumerate}
\end{enumerate}
\noindent {\bf OPTIONAL}  

A quality factor Q (so named because the energy loss $\Delta $U per 
cycle of energy U stored in oscillating LC elements is inverse to the 
circuit's Q) is
\[Q~=~\frac{1}{R} \sqrt{\frac{L}{C}}.\]
Also Q measures the sharpness of a resonance.  Rewrite Z in terms of 
Q:
\[ Z~=~R\left[ 1~+~Q^2\left( \frac{f}{f_{r}}~-~\frac{f_r}{f}\right) ^2\right] 
^{1/2}.\]
Note that large Q means Z will assume large values when f differs slightly 
from f$_{r}$. Hence sharp resonant effects occur.
Calculate Q for your circuit.  What $\Delta $Z/Z results for 
$\Delta $f/f$_{r}$~=~0.1?  
(Large Q occurs for R small, but the resulting resonant 
current may be excessive. To achieve sharp tuning, 
radio circuits may have Q's $\sim $~100).
\begin{center}
Part B: {\bf PARALLEL~A.C.~Circuits}
\end{center}

\vspace*{2in}

Fig.~3a \hspace*{3in}   Fig.~3b

\subsection{INTRODUCTION} 

The parallel RLC circuit is very common in radio and TV 
circuits.  The circuit analysis follows from Kirchhoff's laws, but is not 
in many beginning texts.  The common terminal voltage V is across 
each element (see Fig.~3a), and the instantaneous current i is the algebraic 
sum of the instantaneous i$_{R}$, i$_{L}$, and i$_{C}$.  Since i$_{R}$ is in phase with 
V, but i$_{L}$ lags V by 90$^o $, and i$_{C}$ leads V by 90$^o $, we can describe the 
situation by the rotating vectors in Fig.~3b where I is the 
vector sum of I$_{R}$, I$_{L}$, and I$_{C}$.  Hence from Fig.~3b
\[I_{rms}~=~\sqrt{I^2_{rms\ R}~+~(I_{rms\ C}~-~I_{rms\ L})^2}\]

\noindent
where I$_{rms\ R}$ = $\frac{V_{rms}}{R}$, I$_{rms\ C}$ = $\frac{V_{rms}}{X_{C}}$, and 
I$_{rms\ L}$ = $\frac{V_{rms}}{X_{L}}$.  
Substituting these gives 
\[ I_{rms} = V_{rms}. \sqrt{ \frac{1}{R^2} + \left( \frac{1}{X_C} - \frac{1}{X_L}
\right)^2}.\]
\noindent Resonance still occurs for X$_{L}$~=~X$_{C}$ but the total current, $I_{rms}$, is then just 
the current thru R and thus is a MINIMUM (in contrast to the series resonant 
case).   

\noindent Since V is the same for all the parallel elements, the {\bf relevant phase 
differences are between the currents}.
To measure the total current $I_{rms}$ and 
the phase between it and the voltage $V_{rms}$, we will insert a 22 k$\Omega $ 
sampling resistor R$_{S}$ in series with the signal generator.  See Figure 4.  


\vspace*{3in}

Fig. 4

\noindent 
Measure the voltage V$_{rms\ S}$ across the sampling resistor R$_{S}$ by connecting 
the resistor ends to scope 
channel 2 via a differential amplifier.  Since the voltage and current are 
in phase across a resistor, this signal V$_{rms\ S}$ is proportional to the total 
current $I_{rms}$.  

\noindent We use channel 1 and the other differential amplifier to view 
the common voltage V across all the parallel elements.
At resonant frequency f$_{r}$ the currents from L and C 
will cancel since they 
are of equal magnitude but (always) 180$^o $ out of phase.  Hence at f$_{r}$ the 
total current I will be just that thru R, i.e. I~=~I$_{R}$, and V 
and V$_{S}$ (or I) will be in phase.
\begin{enumerate}
\item[] The smaller the sampling resistor R$_{S}$, the less disturbance its voltage 
drop V$_{rms\ S}$ will have on the net voltage $V_{rms}$ applied across the parallel 
circuit.  However distortion in the generated sine 
wave may bother when R$_{S}$ is small and large currents flow.
Our suggested R$_{S}$~=~22~k$\Omega $ is a 
compromise.  Even so you may see a factor 2 change in $V_{rms}$ as f 
varies; however we can compensate by adjusting the signal generator 
amplitude to keep $V_{rms}$ constant as the frequency changes.
\end{enumerate}

\subsection{SUGGESTIONS}
\begin{enumerate}
\item Hook up the circuit plug board as in Fig.~4 and again trigger the scope 
externally from the signal generator.  (We chose a large 
R~=~100~k$\Omega $ circuit resistance to limit the resonant current and thus 
enhance sensitivity.) 
\item Set the frequency near f$_{r}$ found in Part A SERIES RESONANCE.  Adjust 
the signal generator amplitude to give a 1 volt 
scope signal (p to p) for V in channel 1.  
\item Vary the frequency f on both sides of $f_{r}$ by about 300~Hz.  Adjust the 
signal generator amplitude to keep 
$V_{rms}$ (i.e.~$Y_1$) constant as f and 
$V_{rms\ S}$ change.  Since 
$I_{rms}~\propto~V_{rms\ S}$, plot 
$V_{rms\ S}$ versus f. ~Note the resonant minimum in $I_{rms}$ at f$_{r}$.

{\bf OPTIONAL} 
Replace the 22~k$\Omega $ R$_{S}$ with a 1~k$\Omega $ resistor.  Repeat \#3. and 
note the increased distortion of V$_{S}$ at f$_{r}$, but the relative 
constancy now of the voltage V, (channel 1), across the circuit elements.
\item Starting with f~=~f$_{r}$ observe how the {\bf phase relations} between I 
(which is in phase with V$_{S}$) and V 
change as one goes to f~$>$~f$_{r}$ and f~$<$~f$_{r}$.  (Again if at f~=~f$_{r}$, 
the two signals are {\bf not} in phase but are 180$^o $ out of phase, interchange 
the input leads to one of the differential amplifiers).
\item {\bf Use the Lissajous figure technique} to find f$_{r}$: namely, with the 
horizontal deflection (TIME/CM or TIME/DIV) knob at X-Y (and the mode lever 
at X-Y) use V and V$_{S}$ as the X-Y inputs.  As V$_{S}$ (or I) goes 
thru its minimum, the 
elliptical pattern collapses to a straight line.  Again one can increase 
gains to improve sensitivity.
\end{enumerate}

INSERT
\begin{enumerate}
\item Move the scope trace for $V$ near to the top of the scope screen. 
\item Move the scope trace for $I$ near to the bottom of the scope screen.
\item The the ratio of    $\frac{Magnitude\ of\ upper\ trace}
                                                                {Magnitude\ of\ lower\ trace}$ is proportional to the
                                                                impedance $Z$.
\item Notice that, as you change the frequency, $Z$ given by this ratio of traces,
goes through a maximum at the resonant frequency.
\end{enumerate}
\newpage

\section{E10~Transistors}
\noindent {\bf OBJECT} To experiment with a transistor.

\subsection{Preliminary Questions}
\begin{enumerate}
\item
?
\item
?
\item
?
\item
?
\end{enumerate}


\noindent {\bf APPARATUS} 

An {\bf npn} power transistor; dual trace oscilloscope 
\& manual;  signal generator \& frequency counter; 
power supply ($\pm $15 V fixed/$\pm $9 V variable); 
circuit plug board \& component kit; 
two digital multimeters (DMM); differential 
amplifiers.

\noindent {\bf INTRODUCTION}  

Our junction transistor (Fig.~1) is like two back to back 
{\bf np} diodes (see E-8 Part C,~\#3).
Hence there are two possibilities, an {\bf npn} 
transistor and a {\bf pnp}
transistor.  Our {\bf npn} transistor has a central p-type layer (the 
{\bf Base}) 
between two n-type layers 
(the {\bf Emitter} and {\bf Collector}).  There are other type transistors 
which we will not discuss,
(e.g.~a {\bf MOSFET} Metal-Oxide 
Semiconductor-Field-Effect-Transistor).

\vspace*{.5in}

Fig.~1

\vspace{2in}

\noindent For an {\bf npn} transistor the {\bf collector is positive 
relative to the emitter}.  
The base-emitter circuit acts like a diode and is normally conducting (i.e. 
forward-biased).  The base-collector circuit also acts as a diode but is 
normally non-conducting (reverse biased) if no current flows in the 
base-emitter circuit.  However when current 
flows in the base-emitter circuit, the high concentration gradient of 
carriers in the very thin base gives an appreciable diffusion current to the 
reversed biased collector.
The resulting 
collector current I$_{c}$ depends on the base current I$_{b}$.  We write 
I$_{c}$~=~$\beta $I$_{b}$ where $\beta $ is the current amplification factor.

\noindent {\bf PRECAUTIONS} 

Although our power transistor is fairly indestructible, its 
characteristics are temperature sensitive.  Hence avoid exceeding the 
voltages suggested; also leave the power supply off when not making 
measurements; read currents and voltages to two significant figures only.

\noindent {\bf SUGGESTED~EXPERIMENTS}
\begin{enumerate}
\item {\bf Current~Amplification}  Measure $\beta $ for various 
emitter to collector voltages, V$_{ec}$.
Set up the circuit as in Fig.~2.  

\vspace{1in}

Fig.~2

\vspace{1in}

\noindent Start with the voltage divider 
completely counter-clockwise so that the emitter-base voltage V$_{eb}$ is a 
minimum.  With the variable power supply set to 3 V, record the I$_{b}$ 
measured on DMM$_1$ and the I$_{c}$ on DMM$_2$

\noindent Repeat the readings for ten reasonably spaced (higher) 
settings of the voltage divider (i.e.~higher V$_{eb}$ and hence higher I$_{b}$.

\noindent Calculate $\beta $ (=~I$_{c}$/I$_{b}$) and plot it against I$_{b}$.  Over what range of 
I$_{b}$ is $\beta $ reasonably constant?
Repeat the above but with emitter to collector voltage V$_{ec}$ now at 7 V.
\item (Optional) {\bf MEASUREMENT~OF~I$_{c}$~vs~V$_{eb}$:}
Remove the multimeter DMM$_1$ (used as a microammeter) from the 
circuit of Fig.~2 and use it instead as a voltmeter to measure
the emitter to base voltage V$_{eb}$.  With the variable power supply set to 
give 7 V for V$_{ec}$, use the voltage divider to vary V$_{eb}$. 
Read and record both V$_{eb}$ and I$_{c}$ for 
10 reasonably spaced values of V$_{eb}$.
Plot I$_{c}$ vs V$_{eb}$.  The results are very similar to a diode curve (as 
expected since I$_{c}$ is proportional to I$_{b}$ over the region where 
$\beta $ is a constant).
\item {\bf Voltage~Amplification}  By adding a large load resistor 
in series with the collector, one can convert the current 
amplification $\beta $ observed earlier into a voltage amplification.
Hook up the circuit plug board as in Fig.~3.

\vspace*{.5in}

Fig.~3

\vspace*{1in}

\noindent The input voltage to the transistor V$_{in}$ includes the drop across the 4.7~k$\Omega $ 
protective resistor.  The voltage gain is G~=~$\Delta $V$_{out}$/$\Delta $V$_{in}$.
Record input voltages V$_{in}$ and output voltages V$_{out}$~(=~V$_{ec}$) for ten 
reasonably spaced values of V$_{in}$ from 0 to 5 volts.

\noindent Graph V$_{out}$ vs V$_{in}$ and calculate the voltage gain G at the steeply changing 
part of your graph.  The value of V$_{in}$ at the center of this region we 
call the ``operating voltage'' of the amplifier.
How would the gain change if the load resistance was 2.2~k$\Omega $ 
instead of 
10~k$\Omega $?
\item {\bf Distortion~effects~when~amplifier~is~overdriven}  Set 
up the circuit 
plug board as in Fig.~4. Connect Y$_1$ and Y$_2$ to the scope.  To 
connect the signal generator to the board, 
use the BNC to banana plug adapter and 
remember that the side with the bump goes to ground.

\vspace*{1in}

Fig.~4

\vspace*{1in}

\noindent Adjust the voltage divider until the V$_{in}$ is close to the ``operating 
voltage'' found in \#3.

\noindent Vary (and record) the amplitude of the input signal 
from small values to those which overdrive the amplifier
and produce considerable distortion in the output Y$_2$.

\noindent Observe and record the effect of changing V$_{in}$ 
to values outside of the 
operating range (where $\beta $ is constant).
\end{enumerate}
\newpage


\section{E11.~~Electromagnetic~Radiation}

\noindent {\bf OBJECT}  

To study microwave electromagnetic radiation; to
measure the wavelength, $\lambda $, in air and in a wave guide; to
observe reflection, refraction, polarization, interference and diffraction.
\begin{enumerate}
\item[] Note: If this experiment precedes L-1 and L-8, first read 
appropriate material on diffraction, interference, 
polarization, etc.
\end{enumerate}
\subsection{Preliminary Questions}
\begin{enumerate}
\item ?
\item ?
\item ?
\item ?
\end{enumerate}

\noindent {\bf APPARATUS}  

Klystron oscillator kit with wave guide and manual; horns, 
detector and various auxiliary equipment.  

\noindent {\bf INTRODUCTION} 

Our source of microwaves is a klystron amplifier and 
oscillator which is a 
device where electrons can produce oscillations 
at frequencies large compared to those possible from  
lumped LC components.  High velocity electrons pass through a gap 
in an electromagnetic 
cavity.  Their electric fields excite resonate cavity oscillations whose 
electric fields then tend to bunch the electrons at the 
resonate frequency.  (See Fig.~1). 
These velocity modulated bunches in passing through 
gaps in a similar resonator excite the second cavity.  

\vspace*{.5in}

Fig.~1

\vspace*{2in}

{\rightskip=3in \noindent If part of this generated 
power feeds back into the first cavity,  microwave oscillations can result.  
A reflex klystron oscillator (Fig.~2) uses a 
negative repeller to return the original 
velocity modulated electrons back through the first resonating cavity.


\hspace*{3in} Fig.~2

\smallskip\par}
\noindent {\bf SUGGESTED~POSSIBILITIES}
\begin{enumerate}
\item Follow instruction manual to assemble and operate the equipment.
\item To measure $\lambda $ in air and in the wave guide, set up standing 
waves by placing a metal plate reflector about 30-40 cm in front of 
the transmitting horn.
\item Note reflection by a sheet of metal at various angles.
\item Place the metal plate with a 5 cm slot in front of the 
transmitting horn. Measure the diffraction pattern at a distance of about 
two meters from the slot.  Compare with theory.
\item Make a pair of slots one cm wide and 6 cm apart in front of 
the transmitter. Measure the interference pattern two meters away.
\item Use the wire grill to study the polarization of
the radiation.  

\vspace*{2in}

\noindent Describe and discuss what you observe.
\item Check the refraction of the radiation by a piece of plastic.
\item Study the Bragg reflection from arrays of bearing balls imbedded in 
plastic foam.
\item Study the transmission of the radiation through wood, plastic, metal, 
metal screen, sponge, wet sponge, etc.
\end{enumerate}

\vspace{5mm}

\begin{center} ===== End of E \& M, start of Optics ======== \end{center}


\end{document}


