If you rub an inflated balloon against your hair and then hold it close to your head, you will be able to see some of the hairs rise up. This simple experiment makes electricity visible and is related to the magnets we have stuck to the fridge. Although balloons don’t stick to the fridge and passing a magnet near your head doesn’t make your hair stand on end, these phenomena are two sides of the same coin. Indeed, both are forms in which the same physical object is shown: a electromagnetic field.
However, for centuries, scientists believed that they were two different phenomena, until the mathematical physicist James Clerk Maxwell (1831-1879) proposed his electromagnetism theory. In 1873, now 150 years ago, he published the Treatise on electricity and magnetism, where he offers a full description of his theory. His ideas had already been published in shorter articles, but this book was more accessible for him to consult. It was also intended as a handbook for students at the University of Cambridge. I don’t envy them: it’s over 1,000 pages long and famously devilishly hard to read.
The treatise not only unifies electricity and magnetism, but also the two approaches with which the subject had been approached up to now. Maxwell’s work explains the experimental observations —performed by physicists such as Michael Faraday— and also offers interpretations of abstract theorems —developed by mathematicians, such as Joseph Louis Lagrange, George Stokes and George Green—, which illustrate its importance in practice.
To do this, he proposed a set of equations, now called Maxwell’s equations, which describe the relationship between the variation of a magnetic and an electric field over a period of time. Maxwell’s equations, as they appear in the treatise, are very different from their current version. Originally there were 20 equations, until Oliver Heaviside simplified them to four, which is how they are written today.
Maxwell’s equations show that it is impossible to predict the temporal evolution of the magnetic field without knowing the behavior of the electric field, and vice versa. These should be considered as a single object
They are partial differential equations, that is, equations that contain derivatives with respect to different variables. Derivatives describe the change of a physical quantity due to a small variation of a specific variable. The speed of an object, for example, describes the rate of change of its position, given small variations in time. Objects with a large velocity only require a small amount of time to achieve a large change in position. The velocity of an object is thus the time derivative of its position.
Maxwell’s equations relate the time derivative of the magnetic field to the spatial derivative of the electric field—which describes how the electric field is distributed in space. They show that it is impossible to predict the temporal evolution of the magnetic field without knowing the behavior of the electric field and vice versa. These must be considered as a single object: an electromagnetic field. Also, it follows from the equations that an electric field that changes with time induces a magnetic field and a changing magnetic field generates an electric field.
The resolution of the equations provides a mathematical description of the temporal evolution of the electric and magnetic fields. If we know the electric field and the magnetic field at the initial moment, we can completely predict their evolution in time.
Maxwell offers as an example of an electromagnetic field: light. The applications of this were immense, s equations started a scientific revolution and led to a series of inventions in electronic devices that forever changed our society.
In his treatise, Maxwell also offers an example of an electromagnetic field: light. At that time it was known that light possesses certain properties of a wave, such as those observed in interference phenomena. Furthermore, it was known that light moves at a constant speed, which had been calculated quite accurately. It is easy to show that a wave (mathematically speaking, a trigonometric function, such as cosine or sine), moving at exactly the speed of light, is a solution to Maxwell’s equations. Therefore, Maxwell came to the conclusion that that solution, that electromagnetic wave, is light. Furthermore, the wave solutions of Maxwell’s equations are not only visible light frequencies: they are also radio waves or X-rays.
The applications of this are immense. Before Maxwell, if you wanted light in your home, you had to light a candle. If you wanted to communicate with someone, who was not in the same room, you had to write a letter. There were no cars, much less electric cars. Maxwell’s equations ignited a scientific revolution and led to a series of electronic device inventions that forever changed our society.
Albert Einstein—another great unifier whose theory of relativity unified space and time—was once asked if he considered himself shoulders of Newton, that is, if he saw his work as a continuation of that of Isaac Newton, the unifier of gravity -terrestrial- and astronomy -celestial-. His response was: “I am standing on Maxwell’s shoulders.”
Benjamin Bode He is a postdoctoral researcher at the ICMAT.
Coffee and Theorems is a section dedicated to mathematics and the environment in which they are created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest advances in this discipline, share meeting points between the mathematics and other social and cultural expressions and remember those who marked their development and knew how to transform coffee into theorems. The name evokes the definition of the Hungarian mathematician Alfred Rényi: “A mathematician is a machine that transforms coffee into theorems.”
Edition and coordination: Ágata A. Timón G Longoria (ICMAT).
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