Mathematics and the Real World: The Remarkable Role of Evolution in the Making of Mathematics (2014)
CHAPTER IV. MATHEMATICS AND THE MODERN VIEW OF THE WORLD
What can an electric current be useful for? • Is the universe paved with gears? • Are mathematics and physics one and the same? • How did the axiom of parallels affect the theory of relativity? • Who discovered the formula E = mc2? • How can you bend light beams? • Are we actually waves? • How do elementary particles group together? • Are humans made of strings? • How many dimensions do we live in?
24. ELECTRICITY AND MAGNETISM
In the middle of the nineteenth century there was a dramatic development in the use of mathematics to describe the world. It came in the wake of results of experiments in electricity and magnetism. The mathematical explanations of these discoveries led both to further surprising revelations and to a revolution in the approach to the mathematical description of nature. In a certain sense, mathematics that describes physics became physics itself. In this section we will briefly review the experimental discoveries that resulted in that revolution.
Static electricity and magnetism were known in the times of the ancient Greeks and ancient Chinese, and possibly even earlier. Thales of Miletus knew that when amber is rubbed with a cloth, the amber attracts light objects. Today we understand that the rubbing generates static electricity that causes the attraction. The word electricity comes from the Greek word for amber. Magnetism was also a known phenomenon, and the word magnet was taken from the town Magnesia in Turkey, part of Asia Minor, which was then under the Greeks. The Greeks knew that an iron bar suspended from a cord settles in a north-south direction. Compasses based on that property were already in use in the eleventh century. In the spirit of the Greek tradition, however, no experiments were performed to study those phenomena. Throughout ancient times it was thought that magnetism and electricity were totally unrelated.
In the sixteenth century, following the scientific revolution of the modern era led by Galileo, Francis Bacon, and their contemporaries, scientists began performing controlled experiments to study and understand different natural phenomena, including magnetism and static electricity. Among the pioneers in this field was the British physicist William Gilbert (1540–1603), who carried out controlled experiments and was the first to discover that magnets have two poles, north and south. Like poles repel each other, while unlike poles attract each other. Gilbert also found that there were two types of static electricity, which also repel or attract each other like magnets. Yet he did not realize the connection between static electricity and magnetism. More than a hundred years passed, and in the light of Newton's success in formulating the laws of gravity and its uses, scientists tried to find a quantitative expression for magnetic forces. The French physicist Charles-Augustin de Coulomb (1726–1806), after whom the unit of electrical charge (coulomb) is named, discovered that the power of attraction between two magnets and the repulsive force of electrical charges act in a similar fashion to the force of gravity; in other words, the force is proportional to the size of the charge and reduces in proportion to the square of the distance. The mathematical expression was of a familiar form, and hence the law was accepted relatively easily. Moreover, an understanding started crystallizing that perhaps something of deeper significance was taking place, and that was the uniformity of the mathematical forms that describe nature. Further progress in understanding the essence of electricity was made by the Italian physicist Luigi Galvani (1737–1798), who showed that static electricity can cause a mechanical action. Among other things, he connected static electricity to frogs’ legs and found that it made the legs jump. This effect was given the name galvanism, and still today students carry out those experiments in school. The Italian count Alessandro Volta (1745–1827), whose name is used for the unit of electric potential (the volt), showed that if one connects material with static electricity to material without static electricity by means of a metal bar, an electrical current is generated. He also showed how chemical processes can create static electricity and used that to build a primitive electric battery, the principle of which is used still today in the battery industry.
Until the beginning of the nineteenth century nothing was known of a physical connection between electricity and magnetism. The first such connection was brought to light in 1819 by the Danish physicist Hans Oersted (1777–1851). His discovery, apparently serendipitous, was that the needle of a compass changes direction when in the vicinity of an electric current. In other words, the current emits a force around it that affects the magnet. In about 1831, Joseph Henry (1797–1878) in the United States and Michael Faraday (1791–1867), one of England's leading physicists, discovered the second side of the connection between electricity and magnetism. They showed (independently of each other) that when a metal wire is passed close to a magnet, an electric current is produced in the metal. As an aside, we should add that Faraday, who devoted much effort to making science accessible to the public, was famous enough to merit a visit to his laboratory by King William IV. The king saw the experiment and asked, “Professor Faraday, of what use can this discovery be?” Faraday answered, “I don't know, but you will certainly be able collect a lot of tax on the results of this research.”
Faraday's experiments were thorough and extensive. He formulated the quantitative relationship among the strength of the electric current and the speed at which the metal wire moved and the distance from the magnet. It was no great surprise to discover that when the wire moved repeatedly in circles near the magnet, the strength of the current generated acted like the function known to and used by the Greeks to describe the movement of the celestial bodies, and which had been used for about a hundred years to describe the movement of a pendulum, that is, the sine function. The nature of the link between the magnet and the wire was, however, not clear. To understand the connection, Faraday used a concept that he developed himself, that of a magnetic field. No one knew what exactly the magnetic field was or how it worked, but the force it exerted could be measured, so it was not difficult to accept its existence.
The question was still unanswered as to the medium through which the magnetic force moved to create the current in the wire and then from the electric current in the wire to the movement of the magnet of the compass. Faraday suggested that ether was the medium through which the force was transferred, that is, the same material that fills every space around us, which the Greeks had used to explain the movement of the heavenly bodies and which was used later, in Newton's days, in relation to gravity. The formulae that Faraday developed were used to describe those quantities that could be measured and about which direct evidence could be obtained via the senses, extending Newton's approach. The concept of the magnetic field was more abstract but could be accepted because its activity took place in that very same ether. The explanation was that the magnet caused some kind of transformation in the ether that exerted a force on the electric particle, similar to the transformation that gravity apparently exerted in the ether.
The descriptions and measurements of the electrical and magnetic effects in Faraday's time yielded much information about the connection between electricity and magnetism, but the knowledge did not go beyond the measurement of the actual forces that could be measured and a quantitative description of the connection between them. The magnetic field that operated via that elusive material called the ether served to provide a mechanical explanation for the action of the forces, but that was not a mathematical explanation in the spirit of the modern era. There were no mathematical equations that the electrical and magnetic effects satisfied, nor was any purpose defined along the lines of, say, the least action principle that explained those effects.