# Maxwell’s equations

All electromagnetic phenomena are described by a system of four equations.

By the middle of the 19th century, scientists had discovered a number of laws describing electrical and magnetic phenomena and the connections between them. In particular, the following were known:

• Coulomb’s law, which describes the strength of the interaction between electric charges,
• Gauss’s theorem, which excludes the possibility of the existence of isolated magnetic charges (magnetic monopoles) in nature,
• Biot-Savard’s law, which describes magnetic fields excited by moving electric charges (see also Ampere’s Law and Oersted’s Discovery), and
• Faraday’s laws of electromagnetic induction, according to which a change in magnetic flux generates an electric field and induces a current in conductors (see also Lenz’s rule).

These four groups of laws were generalized by James Clerk Maxwell, who managed to combine them into a harmonious system (which received his name), consisting of four equations and comprehensively describing everything measurable characteristics of electromagnetic fields and electric currents, which is named after him. First of all, we owe Maxwell a rigorous mathematical description of all the known laws of electromagnetism (Faraday, for example, generally formulated all the laws he discovered exclusively in verbal form). Secondly, Maxwell introduced many fundamentally new ideas into the system formulated by him, which were absent in the original laws. Third, he gave all electromagnetic phenomena a rigorous theoretical foundation. And finally, fourth, on the basis of the system of equations he compiled, Maxwell made a number of important predictions and discoveries, including the prediction of the existence of a spectrum of electromagnetic radiation.

Let’s start with the second point. According to the Biot-Savard law, an electric current passing through a conductor excites a magnetic field around it. What if the electric current does not flow through a conductor, but through a flat capacitor? In fact, electrons do not jump from one plate to another, but the current still passes through the capacitor, since the electrons of one plate interact with the electrons of the other plate, being in close proximity to each other, and, due to mutual repulsion, transmit vibrations to each other (for example called oscillations) alternating current, thereby ensuring the flow of current through a seemingly obvious break in the electrical circuit.

Maxwell realized that Ampere’s law in this situation does not explain the passage of current. He also realized that although charges do not transfer from plate to plate, the electric field (the force that would arise if we placed an imaginary electric charge between the plates) increases. Based on this, he postulated that in the world of electromagnetic phenomena, a changing electric field can play the same role in generating a magnetic field as an electric current. Maxwell introduced a fundamentally new concept bias currentby adding it as a separate term to the generalized Ampere’s law – Maxwell’s first equation. And since then, the presence of displacement currents has been unconditionally confirmed by experimental data.

Having made such an important addition to the first of the four equations, Maxwell, on the basis of the system of equations he compiled, purely mathematically derived a prediction that was fantastic for those times: electromagnetic waves, formed as a result of the vibrational interaction of electric and magnetic fields, and the speed of their propagation should be proportional to the force between charges or between magnets. Having solved his differential wave equation, Maxwell was surprised to find that the speed of propagation of electromagnetic oscillations coincides with the speed of light, which by that time had already been determined experimentally. This meant that a phenomenon so familiar to everyone as light is electromagnetic waves! Moreover, Maxwell predicted the existence of electromagnetic waves in the entire known spectrum – from radio waves to gamma rays. Thus, a thorough theoretical study of the nature of electricity and magnetism led to a discovery that has brought innumerable benefits to mankind – from microwave ovens to X-ray units in dental clinics.

 1621 Snell’s Law 1864 Spectrum of electromagnetic radiation 1924 Dispersion: atomic theory
James Clerk MAXWELL
James Clerk Maxwell, 1831–79

Scottish physicist, one of the most prominent theoreticians of the 19th century. Born in Edinburgh, comes from an old noble family. Studied at Edinburgh and Cambridge Universities. He published his first scientific article (on the method of drawing the ideal oval) at the age of 14. Maxwell was a professor in the Department of Experimental Physics at Cambridge University when he died untimely from cancer at the age of 48.

The first major theoretical study of Clerk Maxwell, as he is often called, was his work on color theory and color vision. He was the first to show that the entire range of visible colors can be obtained by mixing the three primary colors – red, yellow and blue; explained the nature of color blindness (a visual defect leading to a disturbance in the perception of color gamut) by a congenital or acquired defect in the retinal receptors. He was the first to invent a real-life color camera (using tartan tape as a photosensitive material) and demonstrated his work at a meeting of the Royal Society of London in 1861. As if casually, he carefully calculated the possible structure of Saturn’s rings and proved that they cannot be liquid, as was previously thought, but must consist of solid particles.

Maxwell made an important contribution to the development of many branches of natural science. But, perhaps, his most important achievement is the development of the theory of electromagnetism and its setting on a solid mathematical basis. Maxwell began to deal with this issue in the mid-1850s. Ironically, Maxwell firmly believed in the existence of the luminiferous ether, and deduced all his equations proceeding from the fact that the ether exists, and in him electromagnetic waves are excited, having, Consequently, the final speed of propagation. Maxwell did not live to see the results of the Michelson-Morley experiment refuting the theory of the existence of the ether. (Just as he did not live to see the unconditional recognition of his theory. The wave nature of light and the correctness of Maxwell’s equations were finally confirmed by Hertz’s experiments only in 1888, and until that time most physicists, including Hertz himself, were suspicious of such a bold theory. Translator’s note.) Fortunately for him and for us, this experiment did not cancel Maxwell’s theory, since Maxwell’s equations are fulfilled regardless of the presence or absence of ether.

Finally, Maxwell made an enormous contribution to the formation of statistical mechanics, finding the velocity distribution of gas molecules, which became the cornerstone of the molecular kinetic theory. Finally, Maxwell himself noticed the imperfection of this theory, formulating a paradox, later called Maxwell’s demon.