\input{labpreamble}
\begin{document}

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


\underline{Object:} Using a current balance, measure the magnetic 
force between two parallel current-carrying wires, and thereby 
determine the magnetic permeability constant, $\mu_{o}$.

\underline{Theory:} A magnetic field $\boldsymbol{B}$ will exert a force on a 
wire carrying a current $I$ according to the relation
\begin{equation}\label{F}
\boldsymbol{F} = I\boldsymbol{L}\times \boldsymbol{B}
\end{equation}
\noindent
where $\boldsymbol{L}$ is the vector in the direction of $I$ and the magnitude 
$L$ is the length of the wire segment.

Also, according to Ampere's Law, an electric current produces a
magnetic field at point $P$ with a magnitude of
\begin{equation}\label{B}
B = {\mu_{o}I\over 2\pi r}
\end{equation}
where $\mu_{o}$ is the magnetic permeability constant and $r$ is the 
perpendicular
distance from the current to $P$.

Combining these two relations (and changing variables from $r$ to $d$)
results in the force between two parallel current-carrying wires in
the form
\begin{equation}\label{force}
F={\mu_{o}LI_{1}I_{2}\over 2\pi d}
\end{equation}
where $d$ is the distance between the currents.  We will route the 
same current through both wires so that $I_{1} = I_{2} = I$.  A plot 
of $F$ vs.~$I^{2}$ should be a straight line whose slope involves $\mu_{o}$.

If the currents flow in the same directions the force is attractive; 
if in opposite directions, repulsive.  (Use this to your advantage to 
create a stable equilibrium with the current balance.)

There is a second method of obtaining $\mu_{o}$.  When wire is wound
around an air-core solenoid and a current $I$ is established in the
windings, a magnetic field $\boldsymbol{B}$ is produced inside the hollow
solenoid which can be measured with a gaussmeter and has magnitude
\begin{equation}
B=\mu_{o}n I
\end{equation}
where $n$ is the number of windings per unit length.  

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

\item Set up the Coulomb Balance with the straight rods (wires) rather
than the plates used to obtain $\epsilon_{0}$.  Examine the rods for 
straightness and align them by placing a coin on the scale pan to 
bring the rods into contact.  Check alignment from various 
perspectives.  Remove the coin.

\item This experiment may be done in either DC or AC modes of 
operation.  When done with direct current the Earth's field will cause 
erroneous results.  This may be compensated for by switching the 
direction of the current and averaging the values obtained, or by 
orienting the instrument at right angles to the Earth's field.

\item First measure the center-to-center distance $d$ between the
wires in a zero-current equilibrium position.  Note the position of 
your laser beam on the wall.

\item With the current routed to go one way in one wire and then the 
opposite way in the other wire so the force is repulsive, turn up the 
power supply so the rods separate.  (Always keep the current below 
the rated value of the power supply, usually about 4--7 A.)

\item Add small weights to the scale pan to bring the rods back to 
their original equilibrium separation ({\it i.e.,} the laser beam 
returns to its original position on the wall).

\item Starting each time with the same zero-current $d$, vary $I$ and
note the the amount of weight needed in each case to restore the
balance to the original separation.  Plot $F$~vs.~$I^{2}$ and compute
$\mu_{0}$ from the least squares slope (see equation \ref{force}). 
Compare with the exact defined value of $\mu_{0} = 4\pi \times
10^{-7}$~T$\cdot$m/A.

\item Repeat the whole experiment with a different starting 
separation $d$ by adjusting the counterpoise.

\item Use the second method of determining $\mu_{0}$ by plotting the
magnetic field inside a solenoid (as measured by a guassmeter)~vs.~the
current in the windings for various values of the current (don't use 
as much current as in the previous procedures).  Subtract
the contribution from the Earth's field.  Calculate $\mu_{0}$ from the
least squares slope of $B$~vs.~$I$.  Report your error.

\end{enumerate}

\underline{Questions:} 
\vspace{-.25in}
\begin{enumerate}
	\item Why is the force between currents method of determining $\mu_{0}$ 
	more fundamental than the solenoid method?
\end{enumerate}

\underline{Conclusions:} 
\vspace{-.25in}

\end{document}
