1
Magnetic Field

1.1. Overview of history

Figure 1.1a. Ørsted and his historical experiment

In April 1820, during a class on electricity for his students, Ørsted discovered the relationship between electricity and magnetism.

He showed, through an experiment, that a wire carrying current was capable of moving the magnetized needle of a compass (another version: one of his students would have made this observation during a practical session). Ørsted did not suggest any satisfactory explanation of the phenomenon, nor did he attempt to represent the phenomenon in a mathematical framework. On July 21, 1820, however, he published his experimental results in a four-page article entitled: Experimenta circa effectum conflictus electrici in acum magneticam. His writings were translated and disseminated to all European scientific communities, and his results were strongly criticized.

For a color version of all figures in this book, see www.iste.co.uk/gontrand/electromagnetism.zip.

Figure 1.1b. André-Marie Ampère took note of these results in September 1820 and quickly (within a few weeks) developed (Polymieu, Rhône, 30 km from Lyon) the theory that would allow the emergence of electromagnetism

Figure 1.1c. Charles-Augustin Coulomb: 1736-1806

Figure 1.1d. Pierre Simon de Laplace: 1749-1827

Figure 1.1e. These two men initiated measurements on magnetic fields (1820)

In the 4th century BC, the properties of magnetite, which attracts iron filings, were cited and reported by Thales of Miletus (Greece), as well as in China from the 3rd century BCE where these minerals were called "tzhu shih" - ??? - or "loving stones" (aimants). Pliny the Elder (1st century, Rome) knew that this property could be transferred, by influence, to a piece of iron.

For several centuries, nothing was added to the knowledge of these phenomena. It was the spread in the West, in the 12th century, via the invention of the compass, due to the Chinese about two centuries before, which aroused new interest for the latter. The rediscovery in the West of the magnetic declination (deviation of the needle of a compass from the geographical North) is attributed to navigators contemporary to Christopher Columbus (1492). Fundamental notions for describing the properties of magnets were introduced by Pierre de Maricourt (1269), who discovered the concept of poles and therefore of the dipole nature of magnets. Decisive progress was made by William Gilbert in his book De Magnete (1600). Gilbert draws a clear distinction between the attractive properties of magnets and the forces of electrostatic interaction. He describes the properties of magnets, the phenomena of magnetization, as well as Earth's magnetic field.

The quantitative description of magnetic forces did not begin, however, until the late 18th century with the work of Charles-Augustin Coulomb (1777), who recognized that there are no free magnetic poles, that the magnetization of a body is defined by its magnetic moment and that it probably results from the individual moments of all its "molecules".

Poisson, in his Mémoire sur la théorie du magnetisme (1824), laid the foundations for the theory of magnetism. His work led him to define the state of a magnetic body by its magnetization, a vector size that can vary at any point and measure the magnetic moment per unit volume. He was also able to calculate, at any external point of a magnetized volume defined by the distribution of its magnetization, what we now call the magnetic field. These first notions of magnetism, derived from studies of magnetic matter, developed independently, after Gilbert's observations, from studies of electrical phenomena.

The link between magnetism and electricity was established in 1820 by the fundamental experiment of Ørsted (and others), which highlighted the action exerted by a wire through which a current passes on the magnetic needle of a compass directed parallel to this wire. In the same year, by measuring the period of oscillation of a magnetized needle as a function of its distance to a wire in which a current flows, Biot and Savart studied the interaction forces between magnets and currents. It was from these experiments that Laplace deduced what is now called the Biot-Savart "law". From the same date, André-Marie Ampère published a series of books, the culmination of which was his 1827 dissertation, Théorie mathématique des phénomènes électrodynamiques, uniquement déduite de l'expérience, in which he described and analyzed the interactions between circuits that electric currents pass through and hypothesized magnetism due to molecular currents in matter. The theoretical expression of the laws of magnetostatics was therefore established from the beginning of the 19th century. It is these laws that we present here with the help of a somewhat modernized formalism.

During the 19th century, knowledge of magnetic media gradually became clearer; but it was not until 1895 that Pierre Curie (May 15, 1859 to April 19, 1906; Paris) distinguished between diamagnetism and paramagnetism and highlighted the transition from ferromagnetism to paramagnetism by temperature rise. In 1906, Pierre Weiss (March 25, 1865, in Mulhouse to October 24, 1940, in Lyon) developed the theory of ferromagnetism.

Subsequently, quantum theory paved the way for a decisive advance in the description of the magnetic properties of matter. In 1928, Werner Heisenberg (December 5, 1901 to February 1, 1976) demonstrated that the origin of ferromagnetic interactions could be found in interatomic electronic exchanges. In 1932, John Hasbrouck Van Vleck (March 13, 1899 to October 27, 1980, US) established the quantum theory of paramagnetism and diamagnetism. In 1930, Félix Bloch (October 23, 1905 to September 10, 1983, Zurich) discovered the fields of ferromagnetic materials and described the outlines of these domains. Finally, Louis Néel (November 22, 1904, Lyon, to November 17, 2000, Brive-la-Gaillarde) established and explained the concepts of anti-ferromagnetism (1932) and ferrimagnetism (1947).

Figure 1.2. Magnetism history synopsis

Figure 1.3. Magmetism history synopsis (continued)

Figure 1.4. Field sources trio and the equivalence of magnets and currents for both the actions created and the actions undergone

1.2. Magnetic fields and magnetic forces

Figure 1.5. Field sources

Figure 1.6. Field sources (continued)

A charged particle (of charge q) generally interacts with other charges according to a general law of the form:

(Vectors appear in bold, or are marked by an arrow throughout the text.) with:

• - f: the force exerted on the particle: electric force + Lorentz force;
• - q: its charge;
• - v: its velocity;
• - E: models the interaction with fixed charges: this is an "electric field" calculated from the location of these fixed charges;
• - B: models the interaction with moving charges (currents).

This is a magnetic field. B was first called a magnetic field, in propedeutics. Now, it is called magnetic induction, due to the AFNOR (Association française de normalisation) standard. First, we will look at the magnetic field created in the vacuum by moving charges, i.e. currents.

1.2.1. First experiments

The first precise determination of the force of interaction between circuits, through which current flows, dates back to the 1800s (Biot, Savart, etc. experiments of 1820-1821); then Ampère developed the theory in 1820, which does not need to be amended despite advances in contemporary physics. He carried out experiments in a few weeks and developed the theory in about the same time.

The total field is calculated by the addition (a priori vectorial) of elementary contributions.

The formalism of magnetostatics can generally be developed from the Biot-Savart law or the Ampère theorem (see below).

1.2.2. Topography: invariances and symmetries

Ab initio, invariances and symmetries/anti-symmetries of the system and associated charges or currents are studied.

This is unacceptable in propedeutics, where a Platonic world reigns, where we encounter spheres, planes and cylinders. This is the world of Harmony, of Symmetries (/Anti-symmetries).

In this case, it makes it possible to simplify calculations, to solve equations analytically, manually. Depending on the field, in our contingent life, solving these equations often uses numerical analysis, and therefore its preferred tool: the computer. We no longer have to deal with the continuum, as equations are discretized/digitized. However, to move on to the continuum, the powerful concept of limit is used. As for digital technology, the opposite path is taken: from the continuum to the discreet. In practice, it is often necessary to solve integer- differential equations; the derivative, of the first...