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Maxwell's Equations

Ibrahima SAKHO

Université de Thiès, Senegal

1.1. Maxwell's equations in a vacuum

Our aim is to introduce the fundamental equations of electromagnetism followed by the four Maxwell equations in vacuum. These equations express the local relationships between the electric and magnetic fields and their sources constituted by the free charge and current densities. From an educational point of view, the formulation of Maxwell's equations in vacuum necessarily involves the study of the three fundamental parts of electromagnetism, which are electrostatic, magnetostatic and induction. The study of these three parts will establish Gauss's theorem, Ampère's theorem, Faraday's law and the local law of the magnetic field translating the conservation of magnetic flux. From this study we will deduce Maxwell's equations for their applications to the study of the propagation of electromagnetic waves in vacuum and antenna radiation.

1.1.1. Electrostatics1

1.1.1.1. Coulomb's law: electrostatic field

A charged particle, assumed to be a point particle, carries an electric charge denoted q. Two point charges q1 and q2, spaced r apart and placed in a vacuum, are in electrostatic interaction. According to the principle of interactions, the electrostatic force exerted by charge q1 on charge q2 is equal and opposite to the electrostatic force exerted by charge q2 on charge q1. According to Coulomb's law:

In equation [1.1], k is the electric constant and e0 denotes the dielectric permittivity of the vacuum: e0 ~ 8.84 ×10 -12 F · m -1.

Figure 1.1. Coulomb forces between two point and fixed charges q1 and q2

The electrostatic forces and are repulsive if the charges q1 and q2 have the same sign and attractive if the two charges have opposite signs (Figure 1.1).

Now consider a test charge denoted q0 fixed at a point O in domain free of charges and currents. Let us place at a distance r from charge q0 a charge q1 fixed at point M (OM = r). The charge q1 is subject to the electrostatic force from q0. Let us remove the charge q1 and place another fixed charge q2 at the distance r from charge q0. It is also subject to the electrostatic force from q0. Let's repeat the experiment several times, taking care to leave in domain only charge q0 and a single charge qi (i =1, 2, 3, .). Each of the point charges qi is subject to an electrostatic force in domain such that:

Let's express the ratios . Using [1.2], we obtain:

The result [1.3] shows that the ratios are equal and depend only on the charge q0 and the distance r between charge q0 and charge qi. In domain there is therefore a vector field created by the charge q0 known as the electrostatic field, denoted .

By definition, the electrostatic field is equal to the ratios , so according to [1.3] (with q0 = q):

Figure 1.2. Electrostatic field created by a fixed point charge q

The electrostatic field lines are oriented curves and tangent at each point to the electrostatic field. For a point charge at point O, the field lines are divergent if q > 0 and convergent if q < 0. On Figure 1.2, a single field line (in green) has been shown to coincide with the direction of the electrostatic field. The expression [1.4] shows that there is an electric monopole, the source of the electrostatic field . In addition, according to [1.4], the electrostatic field created by the point charge q at point M in space is a radial field in 1/r2. The electrostatic field is divergent if q > 0 (Figure 1.2a) and convergent if q < 0 (Figure 1.2b). Furthermore, according to [1.3] and [1.4], . Thus, a point charge q in an area of space where an electrostatic field prevails is subjected to the Coulomb force:

1.1.1.2. Electrostatic field circulation: electrostatic potential

Let us determine the circulation C of the electrostatic field according to direction OM between point M at a distance of r from point O and point M' at a distance of distance r' from the same point O (r' > r). By definition:

Using [1.4], we obtain the following, according to [1.6]:

By definition, the circulation C of the electrostatic field between points M and M' is equal to the potential difference between these two points, i.e:

By definition, the electrostatic potential created by a point charge q fixed at point M in space characterizes the electrical state of this point. It is given by the expression:

In equation [1.8], K is a constant defining the origin of the electrostatic potential (generally, we choose K = 0 to infinity).

Is there an equation between an electrostatic field and electrostatic potential? To answer this question, let us express the gradient of the electrostatic potential VM. In spherical coordinates, the radial component of the gradient of a scalar function f is written:

Using [1.8] and [1.9], we obtain the following, considering [1.4]:

The result above allows us to write the important local equation linking the electrostatic field and the electrostatic potential:

Knowing the expression of the electrostatic field, we deduce the potential V from the local law [1.10] and vice versa.

Box 1.1. Coulomb (1736-1806)

Charles-Augustin Coulomb was a French officer, engineer and physicist. He is best known for his many experiments in electrostatics using a torsional balance to determine the force exerted between two electrical charges. He became famous following the publication of Coulomb's law in 1785, which forms the basis of electrostatics.

Using equation [1.6], we express the circulation of the electrostatic field on a closed contour (C) (Figure 1.3). We obtain:

Figure 1.3. Circulation of the electrostatic field about a closed contour (C)

The circulation of the electrostatic field about a closed contour is therefore zero.

Let (S) be a surface based on contour (C). Using Stokes' theorem, we obtain the following, according to [1.11]:

With this equation being verified regardless of the contour or surface (S), we establish one of the local laws satisfied by the electrostatic field:

Box 1.2. Stokes (1819-1803)

George Gabriel Stokes was a British physicist and mathematician. He is especially famous for having established Stokes' theorem in 1854, which is one of the fundamental theorems of integral transformation linking the rotational of a vector field to the circulation of the same field along a closed contour. Stokes' theorem is widely used in electrostatics and magnetostatics.

1.1.1.3. Electrostatic field and potential of a continuous charge distribution

For a continuous charge distribution, we define an infinitesimal charge dq associated with a linear ?, surface s or volume ? charge:

• - linear distribution over a wire with a length dl: dq = ?dl; ? in C · m-1;
• - surface distribution over a surface dS: dq = s dS; s in C · m-2;
• - volume distribution in a volume dV: dq = ? dV; ? in C · m- 3.

Let us consider various continuous charge distributions (Figure 1.4).

Figure 1.4. Continuous charge distributions

For these charge distributions, the electrostatic field and the electrostatic potential V are given by the following expressions:

• - linear distribution l (Figure 1.4a):

• - surface distribution S (Figure 1.4b):

• - volume distribution V (Figure...