IOP Publishing
Stigmatic Optics
Rafael G González-Acuña and Héctor A Chaparro-Romo
Chapter 1
The Maxwell equations
In this chapter, we give a brief review of the Maxwell equations for electromagnetic theory, after a concise explication, we obtain the step-by-step electromagnetic wave equation. Maxwell equations are the fundamental basis for optical theory, and therefore to the stigmatic optics discipline.
1.1 Introduction
In this chapter, we are going to review Maxwell’s equations. The main goal is to get the wave equation. From, the wave equation we can get the eikonal equation. From the eikonal equation, we can derive the concept of ray and set the bases of geometrical optics.
The general idea of this book is to take Maxwell’s equations as axioms and their implications as theorems. In this language, the wave equation would be a theorem, a direct consequence of Maxwell’s equations. The equation of the eikonal, under one approximation, is an implication of the wave equation and from it, we develop the theory of stigmatism.
The purest and most exquisite branch of geometric optics is stigmatic optics, the branch to which this book owes its name.
1.2 Lorentz force
Let’s start with the definition of the electric field. An electric field can be described as a vector field in which a point electric charge of value q suffers the effects of an electrical force F⃗ given by the following equation:
F⃗=qE⃗,(1.1)
where E⃗ is the electric field. Electric fields can be originated, from both electrical charges and variable magnetic fields.
Electric fields can be positive or negative. They are positive if they are generated by positive charges, and negative if they are generated by negative charges. Charges with different sign are attracted and with similar charge repel each other. Since the electric field is a vector space it can be represented as vectors, thus it is usually represented as vector lines. The lines emerge from positive charges and end in negative charges, as can be seen in figures 1.1–1.3.
Figure 1.1. Charges with different sign are attracted.
Figure 1.2. Positive charges repel each other.
Figure 1.3. Negative charges repel each other.
A magnetic field is a vector field that specifies the magnetic influence 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)