of electric charges in relative movement and magnetised materials. A charged that is moving parallel to a current of other charges experiences a force perpendicular to its own velocity described by
where
For a particle subjected to an electric field combined with a magnetic field, the total electromagnetic force or Lorentz force on that particle is given by the combination of equations (1.1) and (1.2),
The Maxwell equations entirely describe the nature of the electromagnetic fields E⃗ and B⃗. In the following sections, we are going to describe them briefly.
1.3 Electric flux
An initial concept needed to enter Maxwell’s equations entirely is electric flux. The electric flux, or electrostatic flux, is a scalar quantity that expresses a measure of the electric field that passes through a defined surface, or expressed in another way, is the measure of the number of electric field lines that penetrate a surface.
The portion of electric flux dΦE⃗ through an infinitesimal area da is given by,
dΦE⃗=E⃗·n⃗da.(1.4)
The electric field E⃗ is multiplied by the component of the area perpendicular to the field. n⃗ is the normal unit vector of the infinitesimal area da.
The electric flux through a surface S is therefore expressed by the surface integral,
ΦE⃗=∫SE⃗·n⃗da,(1.5)
where E⃗ is the electric field and n⃗da is the differential surface vector that corresponds to each infinitesimal element of the entire surface S. Please see figure 1.4.
Figure 1.4. Flux of an electric field through a surface. On the right the normal vector n⃗ of the surface is parallel to the electric field E⃗. On the left there is inclination on the surface. Thus, there is an angle between n⃗ and E⃗.
1.4 The Gauss law
We start with Gauss’s law. Although, there are many ways to express this law and notation differs, the integral form of the Gauss law is customarily given the following expression,
∮SE⃗·n⃗da=qinε0,(1.6)
where n⃗ is the normal unit vector of the closed surface S, qin is the charge inside the closed surface S and ε0 is a constant called the permittivity free space. In the international system of units, where force is in newtons (N), distance in meters (m), and charge in coulombs (C),
ε0=8.85×10−12C2N−1m−2.(1.7)
First, let’s pay attention of the left side of equation (1.6). The left side of this equation is the mathematical representation of the electric flux—the number of electric field lines—crossing into a closed surface S. In the right side the total amount of charge contained within that surface is divided by a constant called the permittivity of free space. Therefore, what Gauss’s law tells us is an electric charge produces an electric field, and the flux of that field passing through any closed surface is proportional to the total charge inside the closed surface.
Let us assume that you have a closed surface S, where the shape and size of S are arbitrary. If there is no charge inside S, then, the electric flux is zero. If there is a positive charge inside S, then, the electric flux through the surface is positive. But, if you add an equal amount of negative charge, thus the total amount of charge inside S is zero, then, the electric flux again is zero.
There is another way to express Gauss’s law using the divergence theorem. The form is the following expression,
∇·E⃗=ρε0,(1.8)
where ρ is the density of charge inside and ∇ is the nabla operator,
∇=∂∂xi⃗+∂∂yj⃗+∂∂zk⃗(1.9)
∇·E⃗ is the divergence of the field E⃗. The divergence of the vector field is a scalar computation that indicates the tendency of the field to flow away from a point. Hence, Gauss’s law tells us that the divergence of the field E⃗ is the density of charge divided by the permittivity free space.
Here we limit ourselves to present this form of the Gauss law because the derivation of the divergence theorem is beyond the scope of the book. For a more detailed analysis of the divergence theorem, the reader is invited to read the references presented in the bibliography of this chapter.
1.5 The Gauss law for magnetism
Gauss’s law for magnetism has the same structure as the Gauss law of the previous section with the condition that there are no magnetic charges.
So, we start with the definition of magnetic flux through a surface S,
ΦB⃗=∫SB⃗·n⃗da,(1.10)
the magnetic field, B⃗, multiplied by the component of the area perpendicular to the field, where n⃗ is the unit normal vector of infinitesimal area da.
Therefore, over a closed surface S, Gauss’s law of magnetism is given by
∮SB⃗·n⃗da=0,(1.11)
As we mentioned in the introduction, there are no magnetic charges. What Gauss’s law of magnetism tells us is that the total magnetic flux passing through any closed surface is zero.
Tacking the knowledge acquired in the last section, we can get the vector form of Gauss’s law of magnetism, hence,
∇·B⃗=0.(1.12)
This happens, as expected because there are no magnetic charges. Therefore, the density of magnetic charge is zero.
1.6 Faraday’s law
Faraday’s electromagnetic induction law establishes that the electromotive force induced in a closed circuit is directly proportional to the speed with which the magnetic flux passing through any surface with the circuit as edge changes in time. Thus,
∮cE⃗·dl⃗=−ddt∫SB⃗·n⃗da,(1.13)
where E⃗ is the electric field, dl⃗ is the infinitesimal element of the length of the circuit
