\input{labpreamble}
\begin{document}
\parskip=.19in

\subsubsection*{PHSX 222L\hfill  Electrostatic Force\hfill Name:\underline{\hspace*{2in}}}

% \underline{Title:} Electrostatic Force between Two Parallel Plates

% Materials needed:
% Coulomb balances
% High voltage power supplies
% Megohm resistors
% Patch cords
% Lasers and Lab Jacks, Meter sticks, Masking tape
% Milligram weights
% Clay
% Rulers
% Levels

\underline{Object:} To measure the electrostatic force between two 
parallel plates and thereby determine the electric permittivity 
constant (the permittivity of free space $\epsilon_{o}$.)

\underline{Theory:} Consider two parallel metal plates of area $A$ and 
separation $d$.  If these plates carry equal and opposite charges of 
magnitude $q$, then it can be shown that the electric field between 
the plates is
\begin{equation}\label{E}
E_{\rm tot}={q\over A\epsilon_{o}}.
\end{equation}

When an electric charge is brought into an electric field produced by 
another charge it experiences a force given by
\begin{equation}\label{F}
F=qE_{\rm one}.
\end{equation}

Thus, the force of attraction between two parallel plates is the 
charge on one plate multiplied by the field due to the other plate 
(which is one-half the value given by the above formula).  The 
force attracting the plates together can be found by solving for $q$ 
in equation \ref{E} and substituting in equation \ref{F}. 
\begin{equation}
F={E^{2}A\epsilon_{o}\over 2}.
\end{equation}

The electric field $E$ can be calculated from the potential 
difference $\Delta V$:
\begin{equation}
E={\Delta V\over d}.
\end{equation}

The force attracting the plates together can thus be written as
\begin{equation}\label{eqn:f}
F={A(\Delta V)^{2}\epsilon_{o}\over 2d^{2}}.
\end{equation}
All quantities in this expression except $\epsilon_{o}$ are 
susceptible to measurement.  Thus $\epsilon_{o}$ is determined.

There are two ways to obtain the plate separation $d$: direct
measurement with a ruler, and using the formula
\begin{equation}\label{eqn:d}
	d=\frac{Da}{2b}
\end{equation}
where $D$ is the distance between the laser beam spots for the contact
and equilibrium positions of the plates, $a$ is the perpendicular
distance from the line of the knife edges to the center of the plates,
and $b$ is the distance from the mirror to the laser beam spot on the
wall.

The value of $d$ should be small; if it becomes large relative to the 
area of the plates then edge effects become significant and lack of 
paralellism increases (and then equation \ref{eqn:f} isn't valid).

\newpage
\underline{Apparatus:} We'll use a Coulomb balance which has two 
aluminum plates, one of which balances on a knife edge.
Draw a diagram of the apparatus and label appropriate quantities.


\underline{Procedure:} 
\vspace{-.25in}
\begin{enumerate}

\item Measure the area of the plates $A$, and the values of $a$ and
$b$.  Determine the exact center of the movable plate and gently,
nondestructively mark it.
	
\item Mount the plates in the Coulomb balance and take great care to
level the apparatus and align the plates (without bending them) so 
they are parallel.

\item Wire a high-voltage power supply with the positive side to one
plate and the negative side to the other plate, but so that no current
will flow.  Also, in case the two plates accidentally do touch when
the power supply is on, place a megohm resistor in one branch of the
circuit to limit the current to prevent possible damage to the
power supply.  Do not yet turn it on.

\item Place a coin on the upper (movable) plate to bring it into
contact with the lower plate (this is OK when the power supply isn't
on).  Note the location of the laser beam on the wall; this is the
contact position.  Remove the coin.

\item\label{proc:wt} Place a milligram weight (50 mg is a good first
weight) on the very center of the movable plate and note the position
of the laser beam spot; this is the equilibrium position.  Use
equation \ref{eqn:d} to compute $d$ and verify with a ruler.  Remove
the weight.

\item\label{proc:volt} Turn on the high-voltage power supply and
gradually increase the voltage to bring the plates back to the
equilibrium separation distance.  Be careful as this is an unstable
equilibrium (the plates will tend to rush together), and you don't
want the charged plates to touch each other---you should take great
care to insure that they don't.  Now the electrostatic force should be
equivalent to the milligram weight used to balance the plates.  Record
your data and turn off the high voltage.

\item Repeat procedures \ref{proc:wt}--\ref{proc:volt} for various amounts
of weight and corresponding voltage settings, and make a data table 
containing columns for $F$ (the weight), $d$, and $\Delta V$.

\end{enumerate}

\underline{Analysis:} Plot $F$ vs.  $(\Delta V)^{2}/d^{2}$ and find the
least squares slope.  Use equation \ref{eqn:f} to solve for
$\epsilon_{o}$.  Compare with the book value, $8.854\,187\,817
\times 10^{-12}$ C$^{2}$/N$\cdot$m$^{2}$ (which is exact).


\underline{Conclusions:} If you were a grad student at a rich
university and had a whole semester to do nothing but this experiment
how would you reduce errors?  Go into some detail.

\end{document} 
