## 3 Magnetic Field Measurements of Ceramic Fragments by 3 Fluxgate Sensors in 3 Orthogonal Axes
Experimental Results: Modelling the Ceramic Region generating the Magnetic Field detected by the Sensors for the Calculation of Magnetization in Base Vase Specimens  
The remanent magnetization dependence and computation on the magnetic sensor measurements, is studied in Appendix B through a series of experiments (Appendices B.1-B.5) in irregular base fragments (Appendix B.1) and mainly in cylindrical specimens (Appendices B.2-B.5) from the base of vases 4-6 (Figure 4a). The angles, \(\theta_{\text{B}}\), \(\gamma>\theta_{\text{B}}\), between the field, **B**, and the magnetization, **M**, increase as the distance to the measurement position from the edge of the fragment decreases. From the experimental results of the measurements on specimens and fragments from the base vases 4-6, it is found that:  
* The sensors detect the axial field **B** at the edge of a cylindrical region of ceramic material (Appendices B.1, B.2 and B.5) in the direction of the remanent magnetization, M.
* The measured magnetic field is only due to the remanent magnetization of the ceramic material (Appendix B.2), and it is not altered by its induced magnetization. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Magnets** & \multicolumn{4}{c}{**Fields (nT) at the Sensors**} & \multicolumn{4}{c}{**Fields (uT) Inside the Measurement Area**} \\ \hline \multicolumn{3}{c}{**Distances (mm)**} & **z** & **y** & **x** & **z** & **y** & **x** \\ \hline \multirow{3}{*}{\(Z\)} & \multirow{3}{*}{\(Z\)} & \(-61,830.98\) & & & \(-22.2\) & & \\ \cline{3-8}  & & & & 60,880.61 & & & \\ \cline{3-8}  & & & & 60,880.61 & & & \\ \hline \multirow{3}{*}{\(Y\)} & \multirow{3}{*}{\(Y\)} & \(17,197.76\) & & & & & \\ \cline{3-8}  & & & \(-112,658.65\) & & \(-4.3\) & \(-19.9\) & \\ \cline{3-8}  & & & & 35,669.95 & & & \\ \cline{3-8}  & & & 12,633.18 & & & & \\ \cline{3-8}  & & & 24,378.01 & & \(-3.2\) & & \(-13.2\) \\ \cline{3-8}  & & & & \(-96,550.59\) & & & \\ \hline \multirow{3}{*}{Earth’s field} & \(32,000\) & & & & & & \\ \cline{3-8}  & & & 27,400 & & 32.0 & 27.4 & \\ \cline{3-8}  & & & 0.0 & 0.0 & & & \\ \cline{3-8}  & & & & 2.3 & 7.5 & \(-13.2\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Computation of the fields at the sensors at the distances x,y,z that zero their readings. The magnetic field is computed at those distances, inside the measurement area. The magnet fields at the sensors and in the measurement area in the Figures 9 and 10 above and in the table are marked with the same color. Figure 10: Illustration, (**a**) of the individual fields, \(\text{B}_{\text{u}}\)\({}^{8}\) of the \(\mu=x\),y,z magnets at axial distance \(\delta=x\),y,z, and of the earth and (**b**) of the total field inside the measurement area, along the direction of which the induced magnetization M\({}^{\star}\) of the ceramic material is oriented. * The height, \(\ell\), of the magnetically detecting area of the ceramic material by the sensors (Appendices B.2-B.5 and B.7), depends of the angle, \(\gamma\), between the magnetization \(\mathbf{M}\) and the vertical z-axis on specimens' surface, each specimen thickness, L, and the distance, D, between the sensors and the fragment edge along \(\mathrm{B_{xy}}\), when the vertical \(z\)-sensor reading is doubled: When \(\mathrm{D>d=L.tan\theta_{B}}\) the height, \(\ell=\mathrm{L/cos\theta_{B}}\), depends on the thickness, L, of the fragment. When \(\mathrm{D<d}\) the height, \(\ell=\mathrm{D/sin\theta_{B}}\) is dependent on the distance D.
* The radii (Appendix B.5), \(\mathrm{r^{i}=2\alpha^{i}.cosy\ (i=\omega/\beta)}\), of the ceramic cylindrical areas depends on the directionality (\(\gamma\)) of magnetization and of the parameters \(\alpha^{i}\),\(\alpha^{i}\), which, geometrically, are the sensitivity radius of sensors around the measurement position of the vertical and the parallel sensors, respectively.