## 3 Magnetic Field Measurements of Ceramic Fragments by 3 Fluxgate Sensors in 3 Orthogonal Axes  
The horizontal \(x\)-sensor is oriented vertically to the horizontal component and to the vertical component of the earth magnetic field (Figure 8), measured by the y and \(z\)-sensors, respectively. The sensor reading (1 mV/7 nT) is zeroed with a maximum precision of 0.01 mV or 0.1 nT by adjusting their axial distance from the Al-Ni-Co type cylindrical magnets (400 KA/m), by turning their mounting screws. The excitation area of each sensor is located on the axis that passes through the center of its square cross section of side 2d = 6.5 mm and at a distance of a = 6.5 mm from its curved edge. The best, simplest and cost-effective solution is to attach a magnetic field and possible laboratory interference neutralization system to the three-dimensional sensor apparatus. Active shielding could be used, but is not the best choice, from a practical point of view. Within the area of zero magnetic field, the fragments of various sizes should be placed, beside the sensors, and the lifting, leveling and rotation apparatus should be attached. This would require a particularly bulky and complex construction compared to the proposed layout. Each magnet generates, besides its axial field at the sensor direction, also side fields, which excitate the perpendicular sensors. Figure 8: Illustration of the orthogonal coordinate system in three dimensions of the sensor/magnet apparatus for the compensation of the earth magnetic field and the zeroing of their readings. The measured axial fields \(\mathrm{B_{z}}^{x}\)\(\mathrm{B_{y}}^{y}\),\(\mathrm{B_{x}}^{x}\) (Figure 9) of the cylindrical magnets, with magnetization M, radius b, and height H\({}_{\mu}\), with dipole moment m = M.\(\pi\).b\({}^{2}\).H\({}_{\mu}\) and at axial distances \(\xi_{\mu}{}^{\alpha}\) from the sensors, are approximated (Table 1) by the formula given in [40] (pp. 394-399):  
\[\mathrm{B_{\mu}^{\alpha}=\mu}\ =\ \frac{\mu_{\mathrm{o}}\mathrm{.M.b^{2}}\mathrm{. H_{\mu}}}{2\xi_{\mu}^{3}},(\mu\ =\ x,y,z) \tag{1}\]  
The magnet side fields, \(\mathrm{B_{\mu}}^{\alpha\
eq\mu}\), at axial distances \(\mathrm{g_{\mu}}^{\alpha\
eq\mu}\) and side distances from the sensors, \(\mathrm{h_{\mu}}^{\alpha\
eq\mu}\), are approximated by the formula:  
\[\mathrm{B_{\mu}^{\alpha\
eq\mu}}\ =\ \frac{3\mu_{\mathrm{o}}\mathrm{.M.b^{2}} \mathrm{.H_{\mu}}}{4}\cdot\frac{\mathrm{g_{\mu}^{\alpha}.h_{\mu}^{\alpha}}}{ [(\mathrm{g_{\mu}^{\alpha}})^{2}+(\mathrm{h_{\mu}^{\alpha}})^{2}]^{\frac{5}{2 }}},(\mu,\alpha\ =\ x,y,z) \tag{2}\]  
The summed components of the field \(\mathrm{B_{\mu}}^{\delta}\) of the \(\mu\)=x,y,z magnets and the earth field components at the axial distances \(\delta\)=x,y,z where the sensor readings become zero (Figure 10a), are approximated [40], at the measurement area, by the formulas:  
\[\mathrm{B_{z}^{z}} =-\frac{\mu_{\mathrm{o}}\mathrm{.[PERSON]^{2}}\mathrm{.H_{z}}}{2x^{3}} \tag{3a}\] \[\mathrm{B_{\mu}^{\delta}=\mu}_{\mathrm{x,y}} =-\frac{\mu_{\mathrm{o}}\mathrm{.[PERSON]^{2}}\mathrm{.[PERSON]_{\mu}}}{4} \cdot\frac{2\delta^{2}-d^{2}}{(\delta^{2}+d^{2})^{\frac{5}{2}}}\] (3b) \[\mathrm{B_{\mu}^{\delta}=z}_{\mathrm{u}} =-\frac{3\mu_{\mathrm{o}}\mathrm{.[PERSON]^{2}}\mathrm{.[PERSON]_{\mu}}}{4} \cdot\frac{\mu.d}{(\mu^{2}+d^{2})^{\frac{5}{2}}} \tag{3c}\]  
All the following measurements are taken at the same fixed position with respect to the geomagnetic field by the sensor/magnet apparatus forming a three-dimensional orthogonal system. The induced-magnetization contribution of the ceramic material along \(\mathbf{M}^{*}\)(Figure 10b) to the sensor readings is checked in the following experiment by comparing the measurements (Appendix B.2) taken inside a null magnetic field.