## 2 Theoretical Background
The quantities \(d_{s}\) and \(R_{s}\) are the thickness and radius of the shell respectively [18]. The quantities \(A_{2}\) and \(\alpha\) are calculated by Equations (5) and (6) where \(\mu_{s}\) and \(\sigma_{s}\) are the permeability and conductivity of the shell. \[m_{s}=-2\pi R_{s}^{3}A_{2}H_{0} \tag{4}\]  
\[A_{2}=-\frac{2\mu_{s}(sinh(\alpha)-\alpha cosh(\alpha))+\mu_{0}\big{(}sinh( \alpha)-\alpha cosh(\alpha)+\alpha^{2}sinh(\alpha)\big{)}}{\mu_{s}(sinh(\alpha )-\alpha cosh(\alpha))-\mu_{0}(sinh(\alpha)-\alpha cosh(\alpha)+\alpha^{2} sinh(\alpha))} \tag{5}\]  
\[\alpha=R_{s}\sqrt{j\omega\mu_{s}\sigma_{s}} \tag{6}\]  
The equations presented in this section have a two-fold purpose. First, it is apparent that the net effect is that the induced magnetic moments (\(m_{p}\) and \(m_{s}\)) act like superimposed point dipoles that generate a new magnetic field. This provides an abstraction to help interpret numerical simulations from a quantitative point of view. Secondly, the equations demonstrate that the induced magnetic dipoles are nonlinearly related to the material properties and radius of the object and shield in addition to the thickness of the shield. Although this nonlinearity inherently adds complexity, it is also a promising feature that permits inferring material properties via nondestructive methods. \(H\) and \(H_{s}\) correspond to the total magnetic field in the presence of the hidden object and to the scattered magnetic field in the absence of the hidden object. The z-directional magnetic fields are employed because the normal components of the fields are dominant for nondestructive evaluation [8; 19]. For nondestructive evaluation, \(H_{s}\) is calculated by Equation (7) where \(SE_{H}\) is magnetic shielding effectiveness. \[H_{s}=\frac{H_{0}}{SE_{H}}, \tag{7}\]  
\[SE_{H}=cosh(kd_{s})+\frac{1}{3}\bigg{(}\frac{kR_{s}}{\mu_{r}}+\frac{2\mu_{r}}{ kR_{s}}\bigg{)}sinh(kd_{s}), \tag{8}\]  
\[SE_{H,B\&T}=1+\frac{1}{3}jw\mu_{0}\sigma_{s}d_{s}R_{s},\ \ \bigg{(}if\ \frac{kR_{s}}{\mu_{r}}\gg 1 \ and\ kd_{s}\ll 1\bigg{)}, \tag{9}\]The \(SE_{H}\) is expressed by Equation (8) where \(k\) is a complex wave number [7; 20]. The \(SE_{H}\) in Equation (8) can be simplified to \(SE_{H,B\&T}\) by Equation (9) when the radius of the shell is big and the thickness of the shell is narrow. Figure 2 shows the induced magnetic dipole moment (in-phase component) as calculated by Equations (1)-(3). In Equation (1), \(H_{0}\). is 1 A/m. The amplitude and polarity of an induced magnetic dipole (\(m_{p}\)) are calculated as a function of relative permeability and radius of a sphere object. The incident magnetic field is assumed to be a uniform magnetic field, and the frequencies are 50 Hz, 200 Hz, and 1 kHz in the range of ELF/VLF signal. In the 50 Hz case, the induced dipole is positive in all radius cases at \(\mu_{r}=1\). When the relative permeability increases, the polarity of the induced dipoles is changed from positive to negative, and the object can be considered as a ferromagnetic material. When the radius is incrementally increased from 0.1 to 0.4 m, the amplitude of the induced dipole proportionally increases, and the slope of decreased \(m_{p}\) is steeper. For example, the difference between \(\mu_{r}=1\) and \(\mu_{r}=1000\) at \(R=0.4\) m is 4.5 times higher than at \(R=0.1\) m. When the incident field frequency increases, the amplitude of the induced dipole (\(m_{p}\)) increases but the slope of the changes as a function of relative permeability slowly decreases. At 200 Hz and 1 kHz with \(\mu_{r}=10\) and \(\mu_{r}=100\), the sphere object with 0.4 m radius has positive induced \(m_{p}\) and is considered as a conductive material. In all cases, there is a critical permeability at which the dipole moment switches signs. This critical value also typically increases with increasing frequencies. Physically, at low values of permeability, the conductivity tends to dominate, and the induced fields are primarily due to Eddy currents.