# Quick Calculation of Magnetic Flux Density in Electrical Facilities
## 2 Physical Principles  
The MFD \(\overset{\rightarrow}{B}\) created at point P by electric charge Q, which moves in vacuum with velocity \(\overset{\rightarrow}{v}\), is:  
\[\overset{\rightarrow}{B}=\frac{\mu_{0}}{4\pi}\cdot Q.\overset{\overset{ \rightarrow}{v}\wedge\overset{\rightarrow}{r}}{r^{3}}, \tag{1}\]  
where r is the vector from the charge to the point.  
In the International System of Units (ISU) \(\mu_{0}=4\pi\cdot 10^{-7}\) H/m o N/A\({}^{2}\).  
From this expression it is easy to obtain the MFD at point P created by an intensity that circulates through a filiform conductor:  
\[\overset{\rightarrow}{dB}=\frac{\mu_{0}}{4\pi}\cdot I.\overset{\overset{ \rightarrow}{d}l\wedge\overset{\rightarrow}{r}}{r^{3}}. \tag{2}\]  
Thus, for a rectilinear conductor of infinite length, the MFD created at point P, at distance r from the conductor, is a vector perpendicular to the plane defined by the conductor and point P, whose module is calculated using Biot and Savart's Law:  
\[B=\frac{\mu_{0}}{2\pi}\cdot\frac{I}{r}. \tag{3}\]  
Similarly, an analytical equation can be derived for a straight finite conductor.  
These expressions can be used outside the conductor, not inside them. Nonetheless, this is not important for the study of the magnetic fields produced by the conductors of facilities in their neighborhoods.  
Therefore, the MFD in P created by a circuit with a current I will be:  
\[\overrightarrow{B}=\frac{\mu_{0}}{4\pi}\cdot I\cdot\oint\frac{\overrightarrow{ dl}\wedge\overrightarrow{r}}{r^{3}}. \tag{4}\]  
If there are k conductors covered by intensities \(I_{1}\), \(I_{2}\),..., \(I_{k}\), each one will create an MFD at P, \(\overrightarrow{B_{1}}\), \(\overrightarrow{B_{2}}\),..., \(\overrightarrow{B_{k}}\). These MFDs are vectors, so the total MFD will be:  
\[\overrightarrow{B}=\sum_{1}^{k}\overrightarrow{B_{j}}. \tag{5}\]  
If sinusoidal electric currents are considered, magnetic flux will also be a sinusoidal function. In this case, the effective value of the MFD at P must be computed. It may also be useful to know the components of \(\overrightarrow{B}\) along three orthogonal axes (X, Y, Z), which are called B\({}_{X}\), B\({}_{Y}\), and B\({}_{Z}\). These components may allow the identification of the source of the field that is measured at a certain point.  
If a varying magnetic field crosses a metallic object, electromotive forces are induced in that object. As a consequence of these electromotive forces, electric currents flow through the object (Foucault or eddy currents). These currents heat the object. The induced currents, in turn, produce magnetic fields. Induced currents are opposed to the cause that produces them, so the resulting MFD tends to decrease, thus producing a shielding effect. Unless the metallic object is significantly thick, the magnetic shielding effect is weak. However, this effect is greater with screens of ferromagnetic materials.  
In the case of a thin metal plate, close to the primary source of the magnetic field, disregarding the effect of the surface charge on the plate, the MFD produced by the eddy currents on a close point P will be:  
\[\overrightarrow{B}=\frac{\mu_{0}}{4\pi}\cdot\frac{\overrightarrow{J}_{e}(s) \wedge\overrightarrow{r}}{r^{3}}dv. \tag{6}\]  
where \(\overrightarrow{J}_{e}\) is the vector of eddy current density in the element \(dv\), at the coordinates s(x,y,z) and \(\overrightarrow{r}\), the vector from that element to point P. Due to the small thickness of the plate, it is admissible, in general, to consider it as a plane and convert equation (6) into a surface integral. The influence of this phenomenon depends on many factors (metal resistivity, the direction of the MFD inside the plate, and so on) and decreases when moving away from the plate. Experimentally, this effect can be taken into account with an MFD attenuation coefficient.