## 2 Theoretical Background  
Evaluating ELF/VLF magnetic fields as a nondestructive modality will in general require a numerical approach. However, analytical approximations can be utilized to provide a high-level understanding of the important features that are present in the problem. Thus, this section provides closed form expressions that serve as a guide for interpreting results from more sophisticated simulations in Section 3. A magnetic shielding effect is dependent on the complex amplitude of the induced dipole moment. The real part of the induced dipole moment is investigated as a function of relative permeability, the radius of the object, and source frequency. All electromagnetic quantities including the induced dipole moments and magnetic fields are assumed to be time harmonic (\(e^{iwt}\)). All data are calculated when the source field has zero phase (\(t=0\)). Figure 1 shows a simplified geometry for a conductive object and a metal enclosure immersed in a z-directed uniform magnetic field (\(H\)). The object is a sphere with radius \(R\) and the thin enclosure is a metal sphere with radius \(R_{\rm s}\) radius and thickness \(d\). It is assumed that the thickness is small compared to a skin depth of an incident field. The quantity \(H_{0}\) represents the amplitude of the illuminating magnetic field. In Figure 1a, the object is illuminated by the \(H\) field, and the field induces a conduction current and magnetic polarization [7]. The physical process can be equivalently modeled as an induced magnetic dipole (\(m_{p}\)) given by Equation (1). \[m_{p}=-2\pi R^{3}A_{1}H_{0}, \tag{1}\]  
Figure 1: The geometry of spherical (**a**) conductive object, (**b**) thin enclosure, and (**c**) hidden spherical object in thin enclosure using z-directed uniform magnetic field with \(H_{0}\) amplitude. An inside conductive object has \(R\) radius and induces magnetic dipole (\(m_{p}\)). The spherical shell has an \(R_{\rm s}\) outer radius and \(d\) thickness. The incident field (\(H\)) induces magnetic dipole (\(m_{s}\)) and is scattered (\(H_{\rm s}\)). The quantity \(A_{1}\) is given by Equation (2) where \(k\) and \(\mu\) are complex wave number and permeability [17]. The quantity \(k\) is calculated by Equation (3) where \(\epsilon_{0},\ \epsilon_{r},\ \mu_{0},\ \mu_{r},\ c\), and \(\omega\) are the permittivity of free space, relative permittivity, permeability of free space, relative permeability, conductivity, and frequency. The quantities \(\epsilon_{0}\) and \(\mu_{0}\) are 8.854 \(\times\) 10\({}^{-12}\) F/m and 1.257 \(\times\) 10\({}^{-6}\) H/m respectively for free space. \[A_{1}=X+jY=\frac{\Big{(}1+\left(kR\right)^{2}+2\mu_{r}\Big{)}sinh(kR)-\left(2 \mu_{r}+1\right)kR*cosh(kR)}{\Big{(}1+\left(kR\right)^{2}-\mu_{r}\Big{)}sinh( kR)+\left(\mu_{r}-1\right)kR*cosh(kR)}, \tag{2}\]  
\[k=\omega\sqrt{\mu_{0}\mu_{r}\Big{(}\epsilon_{0}\epsilon_{r}-j\frac{\sigma}{ \omega}\Big{)}} \tag{3}\]  
In Equation (2), \(X\) and \(Y\) represent an in-phase (real) and quadrature (imaginary) components, respectively [17]. The \(Y\) quantity can be interpreted as the out-of-phase component of the dipole moment and determines how strongly the induced dipole moment's phase will differ from that of the incident field. Equation (1) is valid when the sphere is small enough and the injected magnetic field is uniform across the sphere. Material properties such as conductivity and permeability induce an in-phase response with an amplitude and a polarity. For example, the in-phase response of conductive metals (e.g., copper and aluminum; \(\mu_{r}=1\)) is always positive, while ferrous metals (e.g., purified iron, permalloy; \(\mu_{r}>1\)) exhibit negative in-phase behavior as a function of frequency [17]. In contrast to the solid object, the thin-shell shield (as shown in Figure 1b) can be modeled via an induced magnetic dipole moment (\(m_{s}\)) that is added to the field response of the isolated object. The moment is calculated by Equations (4) and (5).