# Maxwell's Equations

Occasionally, in physics, there is a great fusion of ideas. Thanks to this, a variety of events is shown to stem from a single, underlying principle. This was the case for important mechanical and electrical studies in the 17th and 19th centuries. During these two centuries, a fusion of ideas was mathematically structured in terms of differential equations. These equations determine how physical quantities vary in space and time. They allow us to mathematically describe known events, and to predict new ones. One particularly important set of equations is known by the name of the scientist who found it: Maxwell’s equations. These equations describe events caused by electric and magnetic properties.

All predictions that followed from Maxwell’s equations proved to be right when put to the test. It is now established that Maxwell’s equations describe all electric and magnetic events, called electromagnetic events, for short. Do Maxwell’s equations apply only to electromagnetic events? We wanted to know if there could be some non-electromagnetic events that satisfy Maxwell’s equations. Through our recent research, we found that this is the case.

Electromagnetic and non-electromagnetic events can be described in terms of fields. To explain what the word “field” means in physics, let us consider the gravitational force (Fg) that the Sun exerts on a planet. This force can be described by giving its magnitude (or strength) and its direction. Quantities like this, with magnitude and direction, are called vectors. As the planet moves around the Sun, the force Fg continuously changes its magnitude and direction.

We can graphically represent Fg by an arrow, and we can put such an arrow on each point along the planet’s orbit (see Image A). If we now consider all possible positions on which a planet could be in space, we could put an arrow at each point to represent the corresponding force that the Sun would exert on that planet. Each of these arrows represents a vector. All of the vectors together is what we call a field. In this case, we are talking about the gravitational field of the Sun. This is a useful way of thinking.

However, over time, this way of thinking evolved in the minds of physicists into something else. Indeed, physicists considered that the gravitational field was not just a way of thinking. It was something real, which exists independent of us, like the Sun and the planets. Maxwell’s equations played a central role in this process.

The fields that are ruled by Maxwell’s equations are the electric field, E, and the magnetic field, B. Maxwell and others thought of these fields as disturbances of an underlying, hypothetical substance that extends across the entire universe. This substance was never defined in a precise way. They called it the ether

In connection with the ether, it is important to note that physicists include not only facts, but also hypothetical situations in their theories. They proceed as detectives do when trying to find the perpetrator of a crime. Based on some few facts, they must fill the gaps in a logical and consistent way. In doing so, they make a lot of assumptions and guesses. Moreover, physicists assume that all events are ruled by the “Laws of Nature”. This is similar to the laws of a legal system, which rule over a community. In a court of law, a jury must decide which claim has been proved. In science, too, different theories may be proposed to a “jury”, the established scientific community of each area. Both science and legal systems can only be as good as the people who put them in practice. There is no group of people, which is completely free from some biases. In science, like in a court of law, not only hard facts but also some speaking tactics can be used to convince the jury, as we shall see.

Before Maxwell presented his equations, electromagnetism and optics (the study of light) were seen as two different areas. Maxwell’s equations challenged this understanding. Maxwell impressively showed that E and B, the fields which were ruled by his equations, also obey what is known as a wave equation. A wave equation describes any types of waves, the most common of which is a wave within the ocean. When the ocean is completely calm, there is no wave. When the ocean is disturbed, maybe by some strong winds, waves form and spread. The same happens with any stuff that can be disturbed by something external. As already said, Maxwell thought that electric and magnetic fields are disturbances of a stuff called ether. These disturbances can behave as waves, in which case their behavior is ruled by a wave equation.

Wave equations also include the speed with which the wave moves through space. The wave speed that entered Maxwell’s wave equation turned out to be the speed of light. People were then convinced that light was an electromagnetic wave, and so optics became a subdiscipline of electromagnetism. This finding caused a rejection of the corpuscular theory of light. This theory said that light was made of moving particles, rather than waves. In this theory, light did not consist of any ether. However, due to Maxwell’s results, the corpuscular theory was rejected, and light was seen as a wave (instead of particles). The jury had delivered a clear verdict in favor of the wave theory. But this verdict did not last for long.

The concept of the ether had some problems. It was thought to be the stuff that sustains electromagnetic waves. This is similar to the way water sustains ordinary waves in a sea. However, water offers some resistance to the motion of a ship. If there were no engine to propel the ship, it would come to rest. The all-pervading ether should likewise offer some resistance to the motion of all bodies, including the planets. Hence, the ether should be extremely weak so as to not stop the planets during their billion-year existence. The ether should also be very elastic to sustain the rapidly changing disturbances of which light waves consist of. No satisfactory model of such a substance could be conceived.

Around the same time, a famous physicist named Albert Einstein, proposed an alternative theory called the “theory of relativity”. This theory did not require the assumption that there is an ether. He introduced the concept of space-time that was widely accepted by the scientific community. This caused “ether” to vanish from the physicists’ vocabulary. It was merely a wording change, but it made the scientific community happy. However, there were still some things that needed to be sorted out.

At the beginning of the 20th century, some experimental findings could not be explained by assuming that light was a wave. These findings provided renewed support to the corpuscular theory of light. Other findings, though, supported the wave theory. Exactly the same situation occurred with elementary particles such as the electron. Most physicists then got used to saying that there was a property called 'wave-particle duality'. This means that light and elementary particles have wave-like and particle-like properties at the same time. Some other people, though, found that it was meaningless to assign wave-like and particle-like behaviors to the same object. In our research, we showed that one behavior could be assigned to one entity and the other to a second entity.

A key concept in physics is the “action principle”. In classical physics, it determines the path, or trajectory, that a given particle follows. This path is called an “optimal trajectory”. It can be proved that an optimal trajectory cannot exist in isolation. It belongs to a field of optimal trajectories. Surprisingly, we found that the collective behavior of optimal trajectories is ruled by equations identical to Maxwell’s equations. From this result, we concluded that optimal trajectories obey a wave equation. This is because Maxwell’s equations logically imply a wave equation. This is a purely mathematical connection. There is no need to use the concept of wave-particle duality. What behaves as a wave is the field of optimal trajectories, and what behaves as a particle is the single corpuscle that actually moves along an optimal trajectory. The consequences and possible extensions of these results are yet to be explored. It remains to be seen whether the jury, the physics community, finds this task worthwhile.

Physics is an ongoing endeavor to build a picture of the physical world around us. We continue learning about the electric and magnetic properties of our environments, and new questions arise each day. Maxwell’s equations provide an incredible framework for navigating these questions and remain useful as we continue to discover more.

Written By: Dr. Francisco De Zela

Original Paper

• Title: Non-Electromagnetic Fields that Satisfy Maxwell's Equations

• Journal: Physics Letters A

• Date Published: 5 September 2023

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