Maxwell’s Equations, Light and the Electromagnetic Spectrum

K Lee Lerner. Scientific Thought: In Context. Editor: K Lee Lerner & Brenda Wilmoth Lerner. Volume 2, Gale, 2009.

Introduction

In the nineteenth century, knowledge of electromagnetism—all those phenomena related to electrical charges, electric currents, and magnetism—moved rapidly from experimental novelty to practical use. At the start of the century, only gas and oil lamps might be found in homes and businesses, but by the end of the century electric light bulbs were common. By 1865, a telegraph cable connected the United States and England. By the end of the nineteenth century, high-energy electromagnetic radiation in the form of x rays was being used to diagnose injury, and radio waves had been discovered, enabling a series of communications revolutions in the early twentieth entury (the latest of which is the spread of cellular phones). The mathematical clarification and unification by Scottish physicist James Clerk Maxwell (1831-1879) of the scientific understanding of electromagnetism prepared the way for the more comprehensive theories of relativity and quantum mechanics in the early twentieth century. Maxwell’s mathematical expressions describing the relationship between electrical and magnetic phenomena are known today as Maxwell’s equations.

Historical Background and Scientific Foundations

Developments Leading Up to Maxwell’s Equations

In the late eighteenth and early nineteenth centuries, philosophical and religious ideas led many scientists to seek a common, fundamental source for the seemingly separate forces of nature (electricity, magnetism, gravitation, etc.). Also, scientists were still concerned with the profound philosophical and scientific questions posed by Isaac Newton’s (1643-1727) Optics (1704) regarding the nature of light. A substance termed ether—imagined as something like an intangible, invisible gas permeating all space—which would support the passage of light waves, much as air supports the passage of sound waves, was thought necessary to explain the wave-like behaviors of light.

Clarification of the relationship between electricity and magnetism was hampered in the latter eighteenth century and early nineteenth century by a rift between the descriptions of nature used by mathematicians and by experimentalists. Advances in electromagnetic theory during the nineteenth century produced unification of these approaches. The culmination of this merger came with the development by Maxwell of a set of four compact equations that accurately described electromagnetic phenomena better than any previous model. Indeed, so accurate are Maxwell’s equations that they are still used routinely throughout science and engineering, although in some phenomena studied by physicists they must be supplemented or replaced by the equations of relativity and quantum mechanics.

In developing his famous equations, Maxwell built upon the mathematical insights of the German mathematician Carl Fredrich Gauss (1777-1855), the rea-sonings and laboratory work of French scientist André Marie Ampère (1775-1836), the observations of Danish scientist Hans Christian Oersted (1777-1851), and a wealth of experimental evidence provided by English physicist and chemist Micheal Faraday (1791-1867). Like most scientific leaps, including Albert Einstein’s (1879-1955) a generation later, Maxwell’s was made possible by a large amount of labor by earlier colleagues and contemporaries.

Simple observations by American statesman and scientist Benjamin Franklin (1706-1790) and other scientists in the mid-eighteenth century continued to intrigue scientists and inventors. As the nineteenth century progressed, electricity came to play an important role as increasingly technology-dependent industrial countries in Europe and North America developed machines with which to meet the needs of their rapidly expanding urban societies.

In 1820, Oersted demonstrated that magnetism has a relationship to electricity by placing a wire near a magnetic compass. When an electric current was passed through the wire, the compass needle showed a deflection (that is, moved from its rest position): This proved the presence of a magnetic field in addition to that of Earth. A year later, inspired by Oersted’s demonstrations, Faraday proved his genius in the practical world of laboratory work by building a device he termed a “rotator,” now credited as the first electric motor. Faraday’s apparatus consisted of a wire carrying an electrical current and rotating about a magnet. Later, Faraday clearly demonstrated the converse phenomenon, the induction of current by magnets being rotated about the wire. The first practical electric motors were all designed according to principles discovered by Faraday, although it was another half century before their widespread application in industry, electric cars, elevators, and appliances. Faraday’smethod of producing electric current with moving magnets and fixed coils—exploiting the phenomenon of electromagnetic induction—is still used by virtually all modern power generators.

Faraday’s 1831 publication of his work on electromagnetic induction was the first in a series of papers eventually collected as Experimental Researches in Electricity (1844, 1847). This work became a standard reference on electricity for nineteenth-century scientists and is credited with inspiring and guiding inventors such as Thomas Edison (1847-1931), the American who invented the light bulb, motion picture, and phonograph record.

During the last decades of the nineteenth century, electric motors powered an increasing number of time-and labor-saving devices, ranging from industrial hoists to personal sewing machines. Electric motors proved safer to manage and more productive than steam or internal-combustion engines. The need for production and distribution of electrical power spawned the construction of dynamos, central power stations, and elaborate electrical transmission systems.

Ampère, a professor of mechanics at the Ecole Poly-technique in Paris, France, was also influenced by Oer-sted’s observations. Ampère’s effect on the theoretical development of electromagnetic theory was as transfor-mative as the influence of Newton on the scientific understanding of gravity. He deepened and clarified the relationship between electrical and magnetic phenomena through a series of brilliantly devised experiments that demonstrated the fundamental principles of electrodynamics (the effects generated by moving electrical charges and magnetic fields). Although Ampère made a number of experimental discoveries, it was his mathematical brilliance that was most important in laying the foundation for later electromagnetic theory. Ampère translated the electromagnetic phenomena observed by Faraday and other experimentalists into the language of mathematics, making possible precise treatment of and further insights into those phenomena.

The culminating fusion of nineteenth-century experimentation in and mathematical abstraction of electromagnetism came with the development of Maxwell’s equations of the electromagnetic field. These equations were more than mere mathematical interpretations of experimental results; Maxwell’s precise expressions predicted new phenomena (i.e., radio waves) and created the necessary background for quantum and relativity theory. Several major early-twentieth-century physicists, such as Max Planck (1858-1947), Einstein, and Neils Bohr (1885-1962), later credited Maxwell with laying the foundations of modern physics.

Maxwell first published his electromagnetic field equations in 1864. In 1873, his book Electricity and Magnetism fully articulated the laws of electromagnetism. Maxwell’s statements about the propagation through space of electric and magnetic fields offered the first working theory of electromagnetic waves. They allowed the unification of known electrical and magnetic phenomena into the electromagnetic spectrum. Visible light, radio waves, x rays, and other phenomena are now known to be electromagnetic waves such as were described by Maxwell, differing only in their rate of vibration (frequency).

The Equations

Although proof of the existence of the electron (the subatomic particle that carries negative electrical charge) did not come until the end of the nineteenth century, and knowledge of the proton (the particle that carries positive electrical charge) was not known until 1918, Maxwell’s equations established that an electric charge is a source of an electric field, and that electric lines of force begin and end on electric charges (though they may also exist in the absence of electric charges, if a changing magnetic field is present).

Because they are mathematically elegant (concise) a full understanding of Maxwell’s equations require an advanced level of mathematical sophistication (above the level of beginning calculus). In addition, several forms of Maxwell’s equations are in common use; although they look different, they all say essentially the same thing.

In summary, Maxwell’s first equation, also known as Gauss’s law, relates the electric field (E) that passes through a particular surface area (A) (e.g., a sphere) to a charge (Q) within that surface. Maxwell’s second equation asserts that there are no “magnetic charges” and that, therefore, magnetic lines of force must always form closed loops. Maxwell’s third equation, also known as Ampère’s law, states that a magnetic field (B) can be induced by a circular loop by charges moving through a wire (i.e., an electric current designated as I) or by a changing electric field (electric flux). Maxwell’s fourth equation, also known as Faraday’s law, states that voltage is generated in a conductor as it passes through a magnetic field or as it cuts through magnetic lines of force (or, as also commonly stated, “cuts magnetic field lines”).

Modern Cultural Connections

Prior to Maxwell’s equations, scientists taught that all waves require a medium of propagation—some kind of flexible substance, whether solid, liquid, or gaseous—which could be set in motion in a way that transferred energy from one point to another. Maxwell’s equations suggested that electromagnetic waves might not require such a medium: A changing electrical field and a changing magnetic field, paired together, might produce each other and move forward in space at the same time, constituting a wave without a medium—an electromagnetic wave. That the ether—a proposed transmissive medium of electromagnetic waves—does not exist was proved in 1887 by an ingenious experiment designed by Albert Michelson (1852-1931) and Edward Morley (1838-1923).

With his electromagnetic field equations, Maxwell was able to calculate the speed of propagation of an electromagnetic wave. This speed arises out of the equations themselves, and is the same for all electromagnetic waves. Maxwell’s value for the speed of electromagnetic propagation fit well with experimental measurements of the speed of light, and Maxwell and other scientists realized that visible light must be electromagnetic in nature, part of a range or spectrum of such waves differing only in frequency of vibration. Exploration of the electromagnetic spectrum resulted in both theoretical and practical advances. For example, German physicist Heinrich Rudolph Hertz (1857-1894) demonstrated the existence of radio waves in 1888. Near the end of the nineteenth century, the discovery by German physicist Wilhelm Röntgen (1845-1923) of high-frequency electromagnetic radiation in the form of x rays found its first practical medical uses.

Maxwell’s mathematical unification of electromagnetism thus laid the foundation of a series of technological, social, scientific, medical, and military revolutions based on wireless (radio) communications. They also made possible the development of relativity and quantum theory in the early decades of the twentieth century.

Maxwell’s equations remain a powerful tool for understanding electromagnetic fields and waves. Although under certain conditions (for example, extremely high gravity, small size-scale, and velocities close to the speed of light) they must be replaced or modified by the equations of relativity and quantum mechanics, they are the working description of electromagnetic phenomenon used daily by most engineers and scientists. They are used, for example, in the design of electrical transmission lines and of radio, television, microwave, and other antennae.

Maxwell was the first scientist to produce a unification theory—a set of equations revealing that two or more apparently distinct forces, in this case the electric and magnetic forces, can be treated as manifestations of a single underlying phenomenon. The next unification of this type was not achieved until the 1960s, when scientists developed equations showing that electricity, magnetism, and the weak nuclear force could all be treated in a unified way. Today, the search for a unified theory of all forces continues.