# Magnetic Field Penetration Depth in Various Materials and Applications
## 3 Results
### Mathematical Models  
Mathematical models allow complex systems to be simplified in terms of equations and mathematical relationships, facilitating their analysis of the depth of penetration of the magnetic field (ppB). These models help to identify and quantify the relationships between different variables, such as permeability, resistivity, frequency, temperature, and other factors.  
Because they describe the relationship between frequency, conductivity, permeability, and temperature, the models allow for predicting the behavior of materials under different conditions. In addition, they optimize the design of materials for specific applications, such as superconductors or electromagnetic shielding.  
With these models, it is possible to predict the behavior of the magnetic field under different conditions, anticipating scenarios such as climate change, wireless communication, attenuation, and range of the magnetic field.  
\[\oint\overset{\rightarrow}{B}\times d\overset{\rightarrow}{A}=0 \tag{1}\]  
\[\oint\overset{\rightarrow}{B}\times d\overset{\rightarrow}{1}=\mu_{0}\times I \tag{2}\]  
\[\oint\overset{\rightarrow}{E}\times d\overset{\rightarrow}{1}=\frac{d\theta _{B}}{dt} \tag{3}\]  
\[B=\frac{\mu_{0}\times N\times I}{2\pi\times R} \tag{4}\]  
Equations (1)-(4) correspond literally to [PERSON] (1775-1836) and [PERSON] [74], where B is the magnetic field in (Tesla), R is the radius of the solenoid in (meters), I is the intensity or current (in amperes), N is the number of turns of the wire, dl is the current differential, dA is the area differential, dt is the differential with respect to time, d\(\Omega\) is the electric flux differential, and \(\mu_{0}\) is the vacuum permittivity, which is equivalent to \(\mu 0=4\pi\times 10^{-7}\) T\(\cdot\)m/A  
\[\lambda\times L(T)=\sqrt{\frac{m\times C^{2}}{4\pi\times\rho s\times(T)\times e ^{2}}} \tag{5}\]  
This Equation (5) is a variant of the expression for the London penetration depth (\(\lambda\)), which depends on the temperature (T) through the superconducting carrier density \(\rho\)s (T) [75], where m = is the effective mass of the superfluid charge carrier, C = is the speed of light, \(\rho\)s = is the superfluid carrier density, and e = is the charge of the superfluid charge carrier.  
The mathematical models in the research reviewed were used to calculate the magnetic field penetration, i.e., these equations allow us to study how the penetration depth varies under different conditions, which is useful in applications such as electromagnetic induction, the characterization of materials, and field attenuation. This was performed with the following used to calculate the ppB, called delta \(\delta\), for conducting and insulating materials:  
\[\lambda=\sqrt{\frac{2}{\mu\times\sigma\times\omega}}\rightarrow\omega=2\pi \times f \tag{6}\]  
\[\lambda=\sqrt{\frac{m}{\mu_{0}\times n_{s}\times e^{2}}} \tag{7}\]\[\lambda(T)=\frac{\lambda_{0}}{\sqrt{1-\left(\frac{T}{T_{\rm C}}\right)^{4}}} \tag{8}\]  
Equations (6)-(8) are derived from the theory of London 1935 and BCS theory [76], where delta \(\delta=\) depth of penetration (meters) in materials such as conductors and insulators, rho (\(\rho\)) = material resistivity (ohms meter), mu (\(\mu\)) = material permeability (henries per meter), omega \(\omega=\) angular frequency (radians per second, \(\omega=2\times\pi\times\) f), f = frequency (henries), landa \(\lambda(\Gamma)=\) penetration depth (meters) in superconducting materials, T = temperature, Tc = critical temperature, m = specific mass of charge carriers (kg), n\({}_{\rm s}\) = density of the superconductor (m\({}^{-3}\)), and e = electron charge (C).