## 2 State-of-the-Art (Method)
### The Use of Electromagnetic Sensors for Environment Monitoring  
#### 2.1.1 Background  
The first clue of the use of an inductive sensor was the patent of the apparatus for a micro-inductive investigation of earth formations with improved electroquasistic shielding in 1988. This patent was classified as \"G01V3/28\", electric or magnetic prospecting or detecting, measuring magnetic field characteristics of the earth, e.g., declination, deviation, specially adapted for well-logging, operating with magnetic or electric fields, produced or modified either by the surrounding earth formation or by the detecting device using induction coils [23]. Following, a study using inductive sensors was performed in 1989, with an experiment concerning a non-contacting electrical conductivity sensor for remote, hostile environments. In this study, [PERSON], [PERSON] et al. calculated the signal level of the sensor when near the homogenous formation. Equation (1) represented the signal level for the inductive sensor proposed in [24]. \[V_{L}=\ 2\pi w^{2}u^{2}In_{I}n_{r}\sigma G \tag{1}\]  
where \(w\) is the frequency, \(\mu\) is the magnetic permeability of the medium, \(I\) is the current through the transmitter, \(n_{t}\) and \(n_{r}\) are the number of turns of the transmitter and the receiver coils, respectively, \(\sigma\) is the conductivity of the formation, and \(G\) is a geometrical factor and depends on the coil dimensions and the distance to the formation. In our case, the formation is the medium in which the coils are suspended, water with fertiliser, and its \(\mu\) and \(\sigma\) will change as the concentration of fertiliser increases, due to the increase of positive and negative charges of molecules of the fertiliser. The study performed in [25] characterised the circular inductive coils, and studied the effects of signal noise, temperature, and pressure on the device. Finally, they concluded that these sensors are accurate and optimum for conductivity monitoring in hostile environments. On the other hand, the prototypes proposed in this paper are composed of sensors that use the phenomenon of mutual inductance. This principle states that, when in a situation where there is a powered coil (PC), powered with an electric current (EC), a magnetic field will appear. This magnetic field depends on several parameters, such as the number of spires (N) of the coil, the diameter of the wire (\(\OW\)), the diameter of the powered coil (\(\O\)PC), and the used signal to power the coil (including the voltage and the frequency). According to Ampere's Law, the magnetic flux density (B) of a solenoid depends on the permeability of the core (\(\mu_{0}\)), the number of spires (N), and the intensity of the current (I), as shows Equation (1). Nonetheless, Equation (2) is for an infinite solenoid in free space. In our case, the solenoid is introduced in the water, which has its relative permeability (\(\mu_{r}\)). Therefore, the length of the solenoid (I) should be included, as shown in Equation (3). We expect an increase in the permeability of the water when OF is added. Since the permeability of a medium is a measure of its resistance to the creation of a magnetic field, if permeability increases, the magnetic field will increase. Moreover, the increase of the magnetic field, which is maximum in the centre of the solenoid where the ferromagnetic core is generally placed, will have an effect on the electrical conductivity of the medium and increase the flow of electrons. In our device, the magnetic field will be higher in the centre of the powered solenoid and the core will be water with different amounts of fertiliser. \[B=\mu_{0}NI \tag{2}\]  
\[B=\mu_{0}\mu_{r}NI/l \tag{3}\]  
If there is another coil in the proximity of the PC, the aforementioned magnetic field will cause that coil to be induced. This phenomenon is known as mutual inductance. The lines from the magnetic field of the PC will go through the induced coil (IC), thus creating a magnetic flux. The theoretical description of the mutual inductance can be seen in [26].