\section{E-4 and E-5 Null Measurements}\label{e04_bridge} 
\begin{center}
(Suitable for two lab periods if one includes the optional parts)
\end{center}

\noindent OBJECTIVES: \vspace*{.1in} \\ 
\begin{mp1}
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.\\
\end{mp1} 

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

\noindent APPARATUS: \vspace*{.1in} \\ 
\begin{mp1}
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.\\
\end{mp1}

\framebox{CAUTION: Transport galvanometers only when switch is in the ``short'' 
position.} 
\vspace*{.1in} \\
\noindent INTRODUCTION:  \vspace*{.1in} \\ 
\begin{mp1}
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. 
\end{mp1}

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

\end{center}
\begin{mp1}
\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 \\
%\end{mp1}

%\begin{mp1}
\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}
\end{mp1}

\begin{mp2}
\noindent 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.\\
\end{mp2}

\noindent SUGGESTIONS:  \medskip \\
\begin{mp2}
\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 \hspace{-.2in} PROCEDURE: %\medskip 
%\begin{mp2}
\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}
\end{mp2}

\newpage
\vspace*{3.0in}
 Fig.~3 \\

\noindent OPTIONAL:  \medskip \\
\begin{mp2}
\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}
\end{mp2}

\newpage
\begin{center}
{\bf Part II, the Potentiometer}
\end{center}

\noindent PROCEDURE: \medskip \\
\begin{mp1}
Connect the apparatus as in Fig.~4 but have the TA check your 
hook up before you close any switches. \\
Fig.~4

\vspace*{3.in}
\end{mp1}

\begin{mp1}

\noindent {\em CAUTION:}  
A 10~K$\ ohm $ series resistor with the standard cell 
prevents damage from currents $>$~0.1 mA. \\  {\em Short out this resistor only 
for the final balance on the most sensitive galvanometer setting}. \\

\end{mp1}

\noindent INTRODUCTION:  \vspace*{.1in} \\
\begin{mp1}
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.
\end{mp1} 

\noindent \begin{mp1} 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.\\
\end{mp1}

\newpage
\noindent SUGGESTIONS:  \vspace*{.1in} \\
\begin{mp1}
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.\\
\end{mp1}

\noindent SUGGESTED~MEASUREMENTS: \medskip \\ 
\hspace*{.1in} \begin{minipage}[h]{3.5in} 
\begin{enumerate}
\item Measure the {\em emf of a dry cell}.
\item Measure the {\bf p.d.}~{\em 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.
\end{enumerate}
\end{minipage} 
\hspace*{1in} Fig.~5 \\

\begin{enumerate}
\item[]
\noindent {Note:}  When large currents flow from the cell,
its emf, $\xi $, will fall 
because of a {\bf back} {\em 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 {\em Hence for reproducible results in item 2. above, 
avoid drawing current from the cell except during measurement}.  

\noindent However, 
hydrogen gas is {\em 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}
\medskip 
\noindent OPTIONAL:
\begin{enumerate} \epind{
\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*{1.2in}

 Fig.~6  \hspace*{3in} Fig.~7}
\end{enumerate} 


\begin{enumerate} \epind{
 
\item Calibrate the 0.25~A scale of an ammeter; use the circuit in 
Fig.~7. 
For R {\em 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. \\} 
\hspace*{.1in} \begin{minipage}[h]{2.5in} 
\noindent Let 
 R~=~{\em resistance between} points A and B and R$_{p}$~=~{\em 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
$$ \frac{{\mbox{\it correct~voltmeter~reading,}}~\left( V\right) }
{{\mbox{\it potentiometer~reading,}}~\left( P\right) }~=~\frac{R}{~R_{p}}
$$
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{minipage}
\end{enumerate}
