## 3 Calibration and Results  
To estimate the magnetization of the material, we need a mathematical model. The simplest is the single magnetic dipole located at the center of the sample. In this case, we should assume that the sample is homogeneous and neglect the fact that we are measuring too close to it. However, this is a first-order estimate [12; 13]. The magnetic field induced by a dipole depends on the magnetic moment \(\overrightarrow{m}\) and the position in which it is measured \(\overrightarrow{r}\)[15]. When locating the origin of the coordinate system in the magnetic dipole, the field generated by it is given by  
\[\overrightarrow{B}\left(\overrightarrow{r},\overrightarrow{m}\right) = \frac{\mu_{0}}{4\pi}\left[\frac{3\left(\overrightarrow{m} \cdot\overrightarrow{r}\right)}{r^{5}}-\frac{\overrightarrow{m}}{r^{3}}\right] \tag{2}\] \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**B\({}_{\mathbf{z}}\) (T)** & 0.46 & 0.40 & 0.29 & 0.19 & 0.14 & 0.10 & 0.08 & 0.06 & 0.045 & 0.041 & 0.035 \\ \hline
**z(mm)** & 0.0 & 2.0 & 4.0 & 6.0 & 8.0 & 10 & 12 & 14 & 16 & 18 & 20 \\ \hline \end{tabular}
\end{table} 
Table 1: Magnetic field applied by the magnet Bz in Gauss in relation to the distance from the \(z\) axis in mm. Figure 4: (a) Calibration curve of the AD-AD22151 Hall effect sensor obtained from the magnetometer built at PUC-Rio. (b) Noise graph of the AD-AD22151 Hall effect sensor, at the highest sensitivity S1 and the lowest sensitivity S2. (c) Graph that represents the dependence, theoretical and experimental, of the magnetic field of the magnet with the displacement in space in the \(z\) axis. where \(\mu_{0}\) is the free space permeability. Our configuration applies the magnetic field in the \(z\) direction, the detects the \(x\)-axis and \(y\)-axis components of the response (\(x_{1},y_{1},z_{1}\)) and \(m_{z}\) represents the magnetic moment only from the \(z\) direction, so we are measuring,  
\[B_{y}(x_{1},y_{1},z_{1})\ =\ \frac{\mu_{0}}{4\pi}m_{z}\left[\frac{(y_{1}z_{1})}{ \left(x_{1}^{2}+y_{1}^{2}+z_{1}^{2}\right)^{5/2}}\right] \tag{3}\]  
Magnetization is the magnetic dipole moment normalized by the mass or volume of the sample. In our case, \(M_{z}=\frac{m_{z}}{2\pi\theta^{3}}\) defined where \(a\) is the sample radius. Therefore, with the help of a handheld microscope (Peak, Inc.) and using the sensor data sheet, we could estimate the relative distances between the center of the sample holder and the active area that lies within the sensor like y\({}_{1}\) = 0.63 \(\pm\) 0.06 mm and z\({}_{1}\) = 2.30 \(\pm\) 0.06 mm. We assume that the center of the sphere (3.00 mm diameter) is located in the center of the sample holder, which is the origin of the coordinate system used. Unfortunately, due to the system geometry, we were unable to use the microscope to estimate the position on the \(x\)-axis, \(x_{1}\). Thus, finally, using \(m_{z}\) = 55.25 Am\({}^{2}\)/kg as the mass magnetization of Ni at 0.4 T [1; 8; 12; 13], we found \(x_{1}\) = 1.94 mm. Figure 5 shows the measurements of the sphere in the magnetometer we developed. From this curve, we can estimate some magnetization values for the nickel sphere, close to zero field, as can be seen in Figure 5. These values were compared with the measurements obtained by a commercial magnetometer (MPMS SQUID, Quantum Design Inc., San Diego, CA, USA) in the same sphere. Comparing the two devices (SQUID and PUC-Rio), the average relative error from the farthest point, in relation to the same magnetic field applied, was approximately 38%, whereas in the saturation region, the largest error was only 17% (See Table 2). Figure 5: Continuous curve in the red color of a 99% pure nickel sphere, obtained by a magnetometer built at PUC-Rio. Continuous curve in the blue color of a 99% pure nickel sphere, obtained by a commercial magnetometer (MPMS SQUID, Quantum Design Inc.). In the inset of the figure, the sample holder with a 99% pure nickel sphere is highlighted.