# Magnetic Field Penetration Depth in Various Materials and Applications
## 3 Results
### Penetration Depth
In addition, Figure 8 details heavy superconductors such as CeCoIn5 and CeRhIn5 are materials in which the charge-carrying electrons have an effective mass much greater than the mass of the free electron, i.e., the magnetic field penetration depths are much greater than conventional superconductors such as NbTi and NbftTa. The correlation coefficient of \(-\)0.324 between the variables \(\lambda\) (the penetration length in nanometers) and Tc (the critical temperature in K) indicates a weak negative relationship, i.e., as Tc increases, \(\lambda\) tends to decrease slightly, but this trend is not very pronounced. The \(p\)-value of 0.081, compared to a significance level of 5% (\(\alpha\) = 0.05), indicates that the relationship shown is not statistically significant. Table 3 presents the information that there is insufficient evidence to conclude that there is a linear correlation between \(\lambda\) and Tc, i.e., the null hypothesis is not rejected. Figure 8: Depth of penetration of (\(\lambda\)) in type II superconductors. For insulating material data, the Kolmogorov-Smirnov normality test was performed, with a significance value (\(p\)-value) of 0.200 and 5 degrees of freedom (gl = 5), indicating that the significance value is greater than the common threshold of 0.05. Therefore, the data follow a normal distribution for the given sample size and significance level. It is noted that the \(\lambda\) lambda values were calculated using the standard formula and the specific properties of each material. Materials with low electrical conductivity (such as rock and dry soil) show significantly higher penetration depths, whereas materials with high conductivity (such as brain and meat) have lower values, reflecting the rapid attenuation of the magnetic field. There is an inverse correlation; as Tc increases, \(\lambda\) also tends to be higher. This suggests that superconductors with higher Tc have weaker coupling of electrons in the superconducting state, allowing the magnetic field to penetrate deeper. Also, type I superconductors are ideal for applications where perfectly diamagnetic behavior is required, but their usefulness is limited by the low Tc and their abrupt transition between the superconducting state and the normal state under a critical magnetic field. The data in Table 4 show that the penetration depth (\(\lambda\)) of the magnetic field varies significantly as a function of the electrical conductivity (\(\sigma\)) and frequency of the electromagnetic wave. In particular, as the conductivity and frequency increase, the penetration depth decreases significantly, a fundamental aspect for applications in geophysics, biomedicine and communications. In Figure 9, Type I superconductors include pure elements such as niobium (Nb), mercury (Hg), lead (Pb) and aluminium (Al). These materials exhibit superconductivity with an abrupt transition and total expulsion of the magnetic field (Meissner effect). They have a single critical field (Hc) and are used in low temperature applications. \begin{table}
\begin{tabular}{l l l l} \hline
**Materials** & **Conductivity \(\sigma\) (S/m)** & **Frequency (Hz)** & **Depth (\(\lambda\)) m** \\ \hline Rock & \(1\times 10^{-12}\) & \(3.0\times 10^{3}\) & \(5.0329\times 10^{5}\) \\ Brain (human body) & \(0.1\) & \(3.0\times 10^{3}\) & \(1.59\) \\ Coal & \(0.001\) & \(3.0\times 10^{3}\) & \(5.03\) \\ Wet clay & \(0.001\) & \(3.0\times 10^{3}\) & \(15.92\) \\ Dry earth & \(1\times 10^{-14}\) & \(1.0\times 10^{4}\) & \(5.0329\times 10^{6}\) \\ Bone & \(0.0001\) & \(1.0\times 10^{4}\) & \(50.33\) \\ Head (human body) & \(0.1\) & \(3.0\times 10^{5}\) & \(1.59\) \\ Carbon fiber (CFRP) & \(0.01\) & \(6.0\times 10^{6}\) & \(5.03\) \\ Fresh water & \(0.001\) & \(6.4\times 10^{7}\) & \(15.92\) \\ Flesh & \(0.01\) & \(3.0\times 10^{8}\) & \(5.03\) \\ Moist soil & \(0.01\) & \(1.0\times 10^{9}\) & \(5.03\) \\ \hline \end{tabular}
\end{table} 
Table 4: Depth \(\lambda\) (mm) in natural materials.