# Magnetic Field Penetration Depth in Various Materials and Applications
## 1 Introduction  
Today, the magnetic field has been increasingly incorporated in a variety of fields, from food cooking, health, and environment to engineering and industry. This ubiquity has led to a growing interest in understanding the properties of the magnetic field and its interaction with different materials. Knowledge of the magnetic field's ability to penetrate bulk, substances, and biologists with different densities, compositions, and structures is essential to optimize its multiple applications. In this article, the penetration depth of the magnetic field will be referred to as ppB. The scientific literature reports numerous studies relating ppB to the physical and chemical properties of the material in question. For example, there is research on the application of the magnetic field (MF) as an effective and contamination-free treatment method to improve the characteristics in horticultural and agricultural development [1]. In addition, experiences in coil design and the use of unidirectional magnetic fields in manufacturing have been documented, showing that field strength and frequency, as well as material characteristics, alter the efficiency and performance of devices and production processes. In the field of superconductors and electronic materials, results have shown how ppB varies with temperature, field frequency, and crystalline composition. Studies with conventional and high-temperature superconductors, as well as with single crystals and metallic composites, have revealed that magnetic field penetration determines phenomena such as vortex formation and quantum conduction behavior. This, in turn, enables the development of more efficient technologies in wireless transport, diagnostic biomedical therapies, geophysical exploration, and superconducting device design. For example, the emergence of the Meissner Ochsenfeld effect, showing that superconductivity represents a state of thermodynamic equilibrium, gave rise to the London and Ginzburg-Landau theory [2]. Neodymium magnets (50 and 100 mT) improved regeneration and shoot formation in lemon balm [3]. A MEX nozzle with solenoid was designed to generate a unidirectional magnetic field (10 mT), improving coercivity, remanence, and peak energy of in situ fabricated magnets [4]. With a 35 mT magnetic field, the antioxidants PO and GSH increased in fermentation with nivalenol (0.2 ug/mL), reducing VNI by 56.5% after 48 h and altering alcoholic properties [5]. In Nb\({}_{3}\)Sn, quadrupole resonator and muon spin rotation measurements evaluated lower critical and superheat fields, dependent on the London penetration depth (\(\lambda\)) at low energies [6]. Furthermore, V\({}_{3}\)Si is a multi-banded superconductor, showing hexagonal vortices whose size depends on the field, with a coherence length greater than the value calculated from H\({}_{2}\)[7]. The magnetization of water with a magnetic field enhances colloidal particle aggregation and solid-liquid separation, affecting bacterial activity and biological properties [8]. The utilization of lithium-oxygen-sulfur batteries is improved by a magnetic field through wireless energy transfer, which is influenced by magnetic forces and spin effects [9]. The magnetic field was used to measure pressure in a magnetorheological fluid (MRF) stored in a tube, and it was observed that the pressure generated decreases with an increasing magnetic field frequency (up to 5 Hz) [10]. Regarding the minimum field variation in specific environments, their magnitudes are close to the geomagnetic field between 25 uT and 65 uT, i.e., magnetic fields are sometimes close to zero [11]. The magnetic field compensation with frequency and low magnitude, in the triaxial square Helmholtz coil (TSH) system, is compensated with a geomagnetic field of 39.5 uT and a uniform MF close to zero [12]. The principles of electromagnetic induction in muscle tissue are explained as an example of interaction with parts of the human body [13].