\input{labpreamble}
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\subsubsection*{PHSX 222L\hfill Magnetic Hysteresis Curve\hfill Name:\underline{\hspace*{2in}}}

\underline{Object:}  To plot a hysteresis curve for a steel ring, and 
determine the coercive force and the retentivity for the sample used.

\underline{Theory:} When wire is wound around an air-core solenoid and 
a current $i$ is established in the windings, a magnetic field $\boldsymbol{
B}_{o}$ 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.

See pages 958--962 in Serway.  If the wire is wound on a core of
ferromagnetic material such as iron or steel, the magnetic field
inside the solenoid becomes much stronger.  This is due to the fact
that, in addition to the magnetic field $\boldsymbol{B}_{o}$ produced by the
current windings, there is the field $\boldsymbol{B}_{M}$ produced by the
ferro-magnetic core material.  $\boldsymbol{B}_{M}$ can be hundreds or
thousands of times stronger than $\boldsymbol{B}_{o}$.  
The total field $\boldsymbol{B}$ within the solenoid is written as
\begin{equation}\label{B}
\boldsymbol{B}=\boldsymbol{B}_{o}+\boldsymbol{B}_{M}
\end{equation}
We define a new quantity, the magnetic field strength (or magnetic 
intensity), $\boldsymbol{H} = (\boldsymbol{B}/\mu_{o})-\boldsymbol{M}$, and equation \ref{B} 
can be written as
\begin{equation}
\boldsymbol{B}=\mu_{o}(\boldsymbol{H}+\boldsymbol{M}).
\end{equation}
where $\boldsymbol{M}$ is the magnetization vector of a substance and is 
related by $\boldsymbol{B}_{M}=\mu_{o}\boldsymbol{M}$.  The magnetic field 
strength $\boldsymbol{H}$ is due solely to the current in the windings and 
has magnitude $H=nI$.  (Both $\boldsymbol{H}$ and $\boldsymbol{M}$ have units of 
amperes per meter.)

Once the magnetic material in the core has been fully magnetized, the 
current in the windings can be turned off, and there will remain a 
certain amount of magnetization in the material.  This remaining 
field is called the \underline{retentivity} (or remanence) of the material.  In 
order to remove this residual magnetism, or de-magnetize the 
material, it is necessary to send a current through the windings in 
the reverse direction.  The reverse magnetic field strength necessary to 
de-magnetize the ferro-magnetic core after having been fully 
magnetized is called the \underline{coercive force} (or coercivity).

\underline{Set-Up:}  The ferro-magnetic sample will be in the form of 
a toroid, or Rowland ring, on which has been wrapped $N_{1}$ primary 
windings.  A second set of windings of $N_{2}$ turns is wrapped on top 
of the primary windings.  We will assume that the ring is large and 
thin enough that the magnetic field produced by the windings is given 
by the same formula as that for a solenoid: $B_{o}=\mu_{o}n I$.

If a current is suddenly established in the primary windings, the
resulting flux change through the secondary windings will induce an
emf according to Faraday's Law.  A current pulse will occur around the
secondary coil circuit, and this pulse can be measured by a ballistic
galvanometer.
% in the same manner as in the earth inductor experiment.

\goodbreak
A mutual inductance can be placed in the secondary circuit 
to calibrate the galvanometer:
\begin{equation}
\Phi = kd
\end{equation}
in which $k$ is to be determined.

The total field change $\Delta B$ occurring within the sample is
proportional to the displacement of the galvanometer.  Thus, as the
current in the primary windings goes through sudden changes around a
cycle, the corresponding magnetization field within the sample can be
plotted against $B_{o}/\mu_{o}$, producing a \underline{hysteresis
curve}, from which the retentivity and the coercive force can be
found.

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

\item Connect the primary winding of the Rowland ring through an 
ammeter to a DC power source.

\item Connect the secondary winding of the Rowland ring in series with 
the ballistic galvanometer and the secondary winding of a mutual 
inductor.

\item Calibrate the ballistic galvanometer by producing a known flux 
change in the mutual inductor and noting the corresponding deflection.
The calibration constant is calibrated from 
\begin{equation}
k={{\rm flux\ change}\over{\rm deflection}} = {\Delta\Phi_{c}\over d_{c}
} 
\end{equation}

\item Leaving the mutual inductor in the circuit, step the current in 
the primary winding of the Rowland ring from zero to maximum forward, 
then back to zero, then to maximum negative, and back to zero.  
Record the deflection for each step.

\item Make a plot of $B$ versus $H$ (called a magnetization curve).

\item From your plot of the hysteresis curve, identify the value of 
the retentivity and the coercive force (put the origin at the center 
of the hysteresis loop).

\end{enumerate}

\underline{Experimental Apparatus:} 
\vspace{.25in}

\underline{Results and Data Plots:} 
\vspace{.25in}

\underline{Questions:} 
\vspace{-.25in}
\begin{enumerate}
\item Show that the magnetic field $\boldsymbol{B}_{o}$ for a toroid reduces 
to that of a solenoid it the toroid is large and thin.
\end{enumerate}

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


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