\input{labpreamble}
\begin{document}
\parskip=.2in

\subsubsection*{PHSX 222L\hfill Magnetic Field of the Earth\hfill Name:\underline{\hspace*{2in}}}

\underline{Object:}  To measure the magnetic field of the Earth.

\underline{Theory:} The Earth's magnetic field resembles that of a 
huge bar magnet; however, the interior temperature of the Earth is 
above the Curie temperature for ferromagnetic materials and a bar 
magnet would lose its magnetism.  The Earth's magnetic field must be 
related to motion or currents within its core.

Navigators have used the Earth's field for centuries.  Compass needles
are light bar magnets which align themselves with the Earth's field
when they are free to rotate.  By long tradition, the end of the
compass needle which points north is called the north pole.  But since
opposite poles attract, it must be a magnetic south pole that is
closest to the Earth's geographic north pole (the magnetic poles of
the Earth do not coincide exactly with the geographic poles).  The
Earth's field varies from place to place and therefore must be
determined experimentally.  In general, the magnetic field lines
$\boldsymbol{B}_{\rm E}$ enter the Earth's surface at an angle and so
can be resolved into horizontal ($\boldsymbol{B}_{\rm EH}$) and
vertical ($\boldsymbol{B}_{\rm EV}$) components, as shown in the
following figure drawn in a vertical plane.  The dip angle ($\phi$)
can be measured with a compass in such a vertical plane.

\vspace{-.6in}
\begin{center}
\begin{picture}(350,180)(50,0)
\thicklines
\put(180,130){\vector(1,0){60}}     % arrow for B_{H}
\put(180,130){\vector(0,-1){120}}   % arrow for B_{V}
\put(180,130){\vector(1,-2){60}}    % arrow for B_{Earth}
\thinlines
\put(192,122){\makebox(0,0){$\phi$}}
\put(230,95){\makebox(0,0){dip angle}}
\put(219,100){\line(-1,1){20}}
\put(228,122){\makebox(0,0){$\boldsymbol{B}_{\rm EH}$}}
\put(300,130){\makebox(0,0){(magnetic north)}}
\put(196,18){\makebox(0,0){$\boldsymbol{B}_{\rm EV}$}}
\put(118,10){\shortstack{(radially\\downward\\toward\\Earth's\\center)}}
\put(225,18){\makebox(0,0){$\boldsymbol{B}_{E}$}}
\put(280,18){\makebox(0,0){(into earth)}}
\end{picture}
\end{center}

% When wire is wound around an air-core solenoid and 
% a current $I$ is established in the windings, a magnetic field ${\bf 
% B}_{o}$ is produced inside the hollow solenoid which has magnitude
% \begin{equation}
% B_{o}=\mu_{o}n I
% \end{equation}
% where $n$ is the number of windings per unit length. This diagram is 
% in a horizontal plane

A \underline{tangent galvanometer} can be used to measure small uniform magnetic
fields such as the Earth's.  It consists of $N$ turns of wire
around a ring of radius $R$ in a vertical plane.  If a current $I$ is
passed through the coil then a horizontal magnetic field is created at
the center of the ring with magnitude
\begin{equation}\label{coil}
	B_{\rm coil}={\mu_{o} N I\over 2R}.
\end{equation}
If the tangent galvanometer is oriented so the Earth's magnetic field
vector is in the plane of the coil then the field created by the coil
will be perpendicular to the Earth's field vector $\boldsymbol{B}_{\rm
E}$ (and, obviously, its two components $\boldsymbol{B}_{\rm EH}$ and
$\boldsymbol{B}_{\rm EV}$).  Then the compass will point in the
direction of the vector sum of $\boldsymbol{B}_{\rm EH}$ and
$\boldsymbol{B}_{\rm coil}$, as shown in the following diagram which
is drawn in a horizontal plane.
From the diagram it is easy to see how the tangent 
galvanometer got its name.
\begin{equation}\label{tan}
	B_{EH}={B_{\rm coil}\over \tan\theta}.
\end{equation}

\vspace{-.6in}
\begin{center}
\begin{picture}(350,180)(50,0)
\thicklines
\put(240,10){\vector(-1,0){60}}   % arrow for B_{coil}
\put(240,10){\vector(0,1){120}}   % arrow for B_{H}
\put(240,10){\vector(-1,2){60}}   % arrow for resultant B
\thinlines
% \put(192,122){\arc}
\put(230,50){\makebox(0,0){$\theta$}}
\put(300,35){\makebox(0,0){deflection angle}}
\put(250,35){\line(-1,1){15}}
\put(255,115){\makebox(0,0){$\boldsymbol{B}_{\rm EH}$}}
\put(300,130){\makebox(0,0){(magnetic north)}}
\put(191,18){\makebox(0,0){$\boldsymbol{B}_{\rm coil}$}}
\put(160,10){\makebox(0,0){(west)}}
\put(195,120){\makebox(0,0){$\boldsymbol{B}$}}
\put(120,100){\shortstack{(compass\\deflection)}}
\end{picture}
\end{center}

\underline{Experimental Apparatus:} Draw the circuit including some 
details of the tangent galvanometer.
\vspace{.20in}

\underline{Procedure:} 
\vspace{-.20in}
\begin{enumerate}

\item Hook up the tangent galvanometer (and an ammeter in series) to a 
low voltage, high current power supply.  Also adjust the tangent 
galvanometer so the compass platform is level.

\item Orient the tangent galvanometer so the Earth's magnetic field
vector is in the plane of the coil; then rotate the compass dial until
the needle indicates north.

\item Note the compass needle deflection $\theta$ for various values
of $N$ and $I$, including an equal number of readings with the leads
switched so $I$ is negative (but remember not to go above the rated
current of the supply or 6 A, whichever is less, in either direction).

\item Use a dip needle to measure $\phi$.

\end{enumerate}

\underline{Results and Data Plots:} 
\vspace{-.20in}
\begin{enumerate}

\item Make a plot of $\tan\theta$ versus $NI$ and determine $B_{EH}$ 
from the least squares slope.

\item Use $\phi$ and $B_{EH}$ to compute $B_{E}$.

\item How close did you come to accepted values for Ephraim?  See the
web for accepted values; try 
\verb=<http://www.ngdc.noaa.gov/cgi-bin/seg/gmag/igrfpg.pl>= 
for pictures and a FAQ and 
\verb=<http://www.ngdc.noaa.gov/cgi-bin/seg/gmag/fldsnth2.pl>= 
for the value of the field (and its components) at any location.

\end{enumerate}

\underline{Questions:} 
\vspace{-.20in}
\begin{enumerate}

	\item Did the magnitude of the Earth's field surprise you?
	
	\item How would this expermient vary at different latitudes?

	\item What other ways could you measure the Earth's field?
	
\end{enumerate}

\underline{Conclusions:} What clever techniques did you use to reduce 
error?

\end{document}
