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\documentclass[a4paper, 12pt,oneside]{article}
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%Début du document
\begin{document}
\title{\normalsize{Lab Work Report - Group N$^\circ$\\ XX - Experiment}}
\date{\normalsize{\today}}
\author{\normalsize{Name} 1\and \normalsize{Name 2}}
%Crée la page de titre
%\maketitle
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%\tableofcontents
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\begin{center}
\large\textbf{\sffamily Experiment N$^\circ$ G1: Magnetic Histeresis Cycle}\\%
\large\sffamily Group N$^\circ$16: Ancarola Raffaele, Cincotti Armando\\%
\large\sffamily \today\qquad Vasiljevic Milica\\%
\end{center}
% Introduction
\section{Introduction}
The main goal of this experiment is to study the magnetization of different materials, and how they behave under the influence of a magnetic field.
Moreover, the phenomenon of Magnetic Histeresis would be studied in this document.
The study of local magnetic properties of matter permit to obtain results that can be applied to a lot of differents domains of technical science.
For exemple it could be usefull for the production of permanent high qualiy magnets so that they can be used to focus charged particles in straight or cicular
accellerators like in The Large Hadron Collider. Furthermore, the results of these kind of studies can be used to obtain more knowledge about microscopic
structure of materials.
\section{A brief Therotical basis}\label{th}
In general, as a first approximation, it exists a linear relation betweent a magnetic field $\vec{H}$ and the magnetic induction $\vec{B}$, given by :
\begin{equation}\label{linearity}
\vec{B} = \mu_0\mu_r \vec{H}
\end{equation}
where $\mu_0 = 4\pi \cdot 10^{-1}$ \si{\volt\second\per\ampere\per\metre} is the absolute permeability of free space and $\mu_r = \chi_m + 1$ is the relative permeability of the material under the influence
of $\vec{H}$. In fact, $\vec{B}$ also depends on the magnetization $ \vec{M} = \mu_0 \chi_m \vec{H}$ of the material,
and $\vec{B} = \mu_0\vec{H} + \vec{M}$ implies (\ref{linearity}). For free space $\mu_r = 1$ by definition.
After the linear domain, the relation is no more linear and the magnetic induction tend to a saturation value $B_r$, but the magnetic permeability $\mu_r^*$ can be locally defined as in the following relations :
\begin{align}\label{relative_permeability}
\mathrm d|\vec{B}| = \mu_0\mu_r^* \mathrm d |\vec{H}| &&\mu_r^* = \frac{1}{\mu_0}\frac{\mathrm d|\vec{B}|}{\mathrm d|\vec{H}|}
\end{align}
Each material acts differently under the influence of the same $\vec{H}$, as their magnetic susceptibility $\chi_m$ value changes.
In fact, its behaviour can be classified with respect to its value of $\chi_m$. A material is said to be diamagnetic if $\chi_m < 0$
and its magnetization $\vec{M}$ is linear in $\vec{H}$ with no relation to the temperature $T$. It is said to be paramagnetic if $\chi_m > 0$
and a non-zero $\vec{H}$ induces a macroscopic $\vec{M}$ in it related to its temperature $T$. If $\chi_m \gg 1$, then the material is said to be
ferromagnetic. This kind of materials, once they've been magnetized, present a residual magnetization even though $\vec{H} = 0$ and the relation between
$\vec{B}$ and $\vec{H}$ is no longer linear. This effect is the one that creates a magnetic hysteresis when $\vec{H}$ changes direction and intensity.
This hyseresis is illustrated on the graph (\ref{hysteresis}), here $H_s$ is the saturation field where $\vec{B}$ gets to its saturation value $B_s$,
$B_r$ is the residual induction and $H_c$ is the coercitif field where the induction $\vec{B} = 0$.
\begin{figure}[H]
\begin{center}
\includegraphics[scale=0.8]{Cycle_hysteresis.PNG}
\caption{Magnetic Hyteresis for a ferromagnetic material, on a graph (B,H) \cite{ref1}}
\label{hysteresis}
\end{center}
\end{figure}
\section{Experimental procedure}
\paragraph{The exeperimental setup} it's shown on Figure (\ref{setup}).
The scheme shows where the plotter x-y is attached in parallel, in order to get the tension values from the circuits and plot them on a graph
($U_x,U_y$). At this point, it is needed a relation between $U_x , U_y$ and $H , B$ so that theese values can be studied.
\subsection{Relations for H and B}\label{calibration}
As it can be seen on figure (\ref{setup}) The voltage plotted on axis x is the one at the bounds of resistance $R_x$.
So, by the Ohm Law $U_x = R_x I_x$ and as $R_x=1\Omega$, then value of $U_x$ is the same as the one of $I_x$.
To finsh with, considering the primary coil as a perfect solenoid the magnitude of H field is given by:
\begin{equation}\label{h}
H = \frac{N_1}{l}I_x = \frac{N_1}{l R_x}U_x = \alpha_H U_x
\end{equation}
where $N_1$ is the number of turns of the solenoid, $l$is its length , and $I_x$ the current passing through it. Since constants $N_1 , l$ and $R_x$ are known
then $\alpha_x$ can be easily measured to obtain $H$ from $U_x$.
Furthermore the $B$ field is linked to the current $I_y$ passing trough the secondary coil (considering it a perfect solenoid) by the following relation \cite{}:
\begin{equation}
\frac{\mathrm d B}{\mathrm d t} =\mu_0 \frac{N_2}{l} \frac{\mathrm d I_y}{\mathrm dt} \; \Longleftrightarrow \; \int_{t_0}^t I_y \, \mathrm dt = \frac{l}{\mu_0 N_2}(B(t) - B(t_0))
\end{equation}
where $N_2$ is the number of turns of the secondary coil and $l$ its length. Setting $t_0$ such that $B(t_0) = 0$ it can be observed that $B$ is
proportionnal to the integral of $I_y$ over time.
It is also known that $U_x$ is proportional to the integral of the input voltage $U_{int}$ \cite{ref1} over time. Thus, as $U_{int} = R_{int} I_y$
(where $R_int$ is the main resistance of the integrator), it can be found that $B \propto \int_{t_0}^t I_y \, \mathrm dt \propto U_y$.
Theese assumptions shows that $B = \alpha_B U_y$.
Now, $\alpha_B$ can be found knowing that in free space $B = \mu_0 H$ and using the relation (\ref{h}) as:
\begin{equation}\label{b}
\alpha_B U_{y,v} = \mu_0 \alpha_H U_{x,v} \; \Longleftrightarrow \; \alpha_B = \mu_0 \alpha_H \frac{U_{x,v}}{U_{y,v}}
\end{equation}
Where $U_{x,v}$ and $U_{y,v}$ are plotted whit no sample between first and second coil.
Even though there's air between the coils, as Table (\ref{mureal}) shows that $\mu_r \approx 1$ for air.
The relation $U_x/U_y$ can be obtained by a linear regression over the plotted points.
To finish with, knowing $\alpha_H$ and $\alpha_B$ permit to define the following relation :
\begin{equation}\label{mu}
U_y = \mu_0\mu_r \frac{\alpha_H}{\alpha_B} U_x \; \Longleftrightarrow \; \mu_r = \frac{1}{\mu_0}\frac{\alpha_B U_y}{\alpha_H U_x}
\end{equation}
{\bf N.B.} The plotted measurements will be kept in Volt because a conversion would bring too much uncertainty to be shown on a clear graph.
\begin{figure}[H]
\begin{center}
\includegraphics[scale=0.8]{montage.PNG}
\caption{Electronical scheme of the experimental setup \cite{ref1}}
\label{setup}
\end{center}
\end{figure}
\subsection{Drift of collected values along y-axis}
As the integrator used in the circuits integrates electronically even though no change in H is recorded, before any calibration a drift along y-axis can
be noticed. To avoid such a behaviour, a potentiometer on the integrator helps containing the drift increasing the resistance of the integrator.
Moreover, to obtain an easy manipulation it is helpfull to increase the gain of the amplifier that also digitize the values. Then it is important to consider
this gain if the recorded values will be used in any measurement.
\subsection{Determination of $\mu_r$}\label{mucomputation}
As mentioned in section \ref{th}, for each material exists a linear domain where the linear relation (\ref{linearity}) is maintained.
Thus, a linear domain should appear also on the graphs $(U_x,U_y)$ as theese voltages are proportionnals to $H$ and $B$.
Then, whit a linear regression in this domain and using the second relation in (\ref{mu}), $\mu_r$ can be easily determined for different samples.
To finish with, for each sample its permeability is measured as $\chi_m = \mu_r - 1$.
\section{Results of the experiment and Discussions}
\begin{minipage}{\textwidth}
\begin{minipage}{0.48\textwidth}
\resizebox{\textwidth}{!}{
\input{graphs/ferroduro.tex}
}
\end{minipage}
\hspace{0.02\textwidth}
\begin{minipage}{0.48\textwidth}
\resizebox{\textwidth}{!}{
\input{graphs/acciaio_lega_est.tex}
}
\end{minipage}
\end{minipage}
\subsection{Determination of $\mu_r$}
A cylindrical transformer, consisting in a primary coil with $N_1 = 450$ and a secondary one with $N_2 = 4980$, both with $l = 11 \pm{0.5}$ \si{\centi\metre}, has been used. The first measurment has been done with air in it, to help calibrating the setup and tension scales as explained in section
\ref{calibration}. The results for this measurement are shown on Figure (\ref{}).
%TODO inserire grafici per qualche campione interessante, gli altri metterli in annessi per non occupare troppo spazio
Figures (\ref{}) % TODO ()etc.
shows the graph $(U_x,U_y)$ for the studied sample and, knowing the data, $\mu_r$ and $\chi_m$ are computed as explained in section \ref{mucomputation} and showed on Table (\ref{mutab}).
%TODO inserire tabella mu_r e chi_m
%TODO Commentare tabella
The difference between this values and the accepted ones on Table (\ref{mureal}), could be caused mostly by the integrator, as the incertainty
on Table (\ref{}) considers any error coming from $H$ measurment.
In fact, the integrator is an electronical circuits with resistances capacitors and operational implifiers that integrates signals. All its components can
increase uncertainty. In addition, the drift isn't perfectly compensated, and it contributes in increasing the error on the measurement of the induction field
and of course, on the linear slope of the collected data.
\subsection{Interesting value for a ferromagnetic sample}
The graph in Figure (\ref{interesting_points}) shows a clear case of {\it Magnetic Histeresis}. The values for the highlighted points
are given in the Table (\ref{table_points}).
%TODO inserire grafico nickel con punti interessati in evidenza più tavola con valori
%TODO commentare
\subsection{Hard ferromagnetic material}
The following graph in Figure (\ref{hard_ferrom}) shows the {\it Magnetic Histeresis} for a {\it Hard Ferromagnetic Material}.
%TODO inserire grafico con ferromagnetico duro magari sovrapposto ad uno normale, per notare meglio la differenza
On the graph it can be seen that, for a hard ferromagnetic material, the residual induction $B_r$ and the coercitif field $H_c$ are bigger
than for a simple ferromagnetic material. This means that the magnetization $\vec{M}$ is much bigger for such a material and that it
can be hardly demagetized. It also suggests
that such a material could be easily used as a strong magnet.
\section{Conclusion}
\section{Annexes}
\subsection{About the error}
%TODO inserire calcolo errore e errori per tutti i sample
\subsection{Accepted values fo $\mu_r$}
The following table shows the accepted values of $\mu_r$ for each of the studied samples \cite{ref3}.
\begin{table}[H]
\center
\begin{tabular}{|c|c|c|}
\hline
{\bf Medium} & $\mu_r$ & $\chi_m$ \\
\hline
Aluminium & 1 & $2.2 \cdot 10^{-5}$\\
\hline
Nickel & (1 to 6) $\cdot 10^2$ & (0.9 to 5.9) $\cdot 10^2$\\
\hline
Copper & $9.9 \cdot 10^{-1}$ & $ -0.6 \cdot 10^{-5}$ \\
\hline
Air & 1 & $3.7 \cdot 10^{-7}$\\
\hline
\end{tabular}
\caption{accepted $\mu_r$ for differents mediums}
\label{mureal}
\end{table}
\section{Literature References}
% Bibliographie
\begin{thebibliography}{99}
\bibitem{ref1}
Notices des TP de physique G1 : Cycle d'hystérèse magnétique.
\bibitem{ref2}
HARALD, Brune, Cours de Physique III, Faculté des Sciences de Base, Section de Physique, Ecole Polytechnique Fédérale de Lausanne.
\bibitem{ref3}
WIKIPEDIA.ORG, Permeability(Electromagnetism), \url{https://en.wikipedia.org/wiki/Permeability_(electromagnetism)#Relative_permeability_and_magnetic_susceptibility} visited the 6th of December 2018.
\end{thebibliography}
\end{document}
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