## Maxwell's Contributions to Science

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In addition to his revolutionary work in electromagnetism, Maxwell made contributions to many areas of science. These include geometry, color perception, astronomy, statistical mechanics, thermodynamics, elasticity, control theory, and dimensional analysis. In this section we will look at some of these.

### Geometry

Maxwell's first contributions to science were in mathematics, in particular geometry. His first paper was entitled *On the description of Oval Curves, and those having a plurality of foci* and was published in the Proceedings of the Royal Edinburgh Society. Maxwell was fourteen at the time. In this paper he discussed oval curves that could be drawn using pins, a piece of string and a pencil. It was well known that an ellipse consists of those points such that the sum of the distances from each point to two fixed points (foci) is a constant. An ellipse can be drawn as shown in Figure 9.

Here a piece of string is pinned at its two ends, the pencil is pushed against the string and moved keeping the string taught. Maxwell generalized this construction method to other types of ovals. For example, if you un pin one of the ends, tie this end of the string to the pencil, loop the string around the unused pin, press the pencil against the loop of string, and draw, you get an oval in which the sum of twice the distance to one of the pins plus the distance to the other pin is a constant. Using multiple loops he obtained many additional ovals. He also considered ovals constructed using three, four and five pins. Maxwell was deemed too young to present the paper to the society, so the paper was read by professor Forbes of the University of Edinburgh. It was discovered after this paper was published that many of the ovals Maxwell generated had been discovered previously by the great French mathematician Descartes. However, Maxwell's paper was more general and his construction methods simpler.

Maxwell's second paper was presented to the Royal Edinburgh Society when he was eighteen. It was entitled *Rolling Curves*. A simple example of a rolling curve is the cycloid generated by a fixed point on a circle as it rolls along on a line (see Figure 10). Maxwell considered general properties of curves generated by a point on one curve as it rolls on a second curve.

### Color Theory

Today James Clerk Maxwell is best known for his electromagnetic equations, but during his lifetime he was better known for his work on color perception. Newton was the first to develop a color theory. He separated colors into a rainbow-like spectrum using a prism, and showed that white was the combination of all the spectral colors. He also postulated that any color could be obtained by mixing the spectral colors in the correct proportions. However, artists commonly obtained the colors they desired by mixing just three colors of paint (usually red, yellow, and blue).

In the early nineteenth century, an English doctor and physicist Thomas Young postulated that the human eye contains three receptors each sensitive to a particular color. He also suggested the idea of representing colors in a triangle with the primary colors at the vertices, however, he didn't follow up on this idea. Maxwell began his work on color in Professor Forbes laboratory while he was a student at Edinburgh University (1848–1850). They were trying to show that any color could be obtained as a mixture of three primary colors. Various colors were produced by spinning a circular disk having colored sectors. If the disk is spun rapidly enough, the eye can not separate the colors and the disk appears to be of a uniform color. They tried to obtain white by using sectors colored red, yellow, and blue (the colors used by painters), but no combination of these three colors produced white. They also used yellow and blue sectors in an attempt to obtain green. To their surprise, the resulting color was a dull pink.

James soon found out the reason for these discrepancies. There is a fundamental difference between mixing colored lights and mixing paint pigments. Paint pigments tend to absorb certain colors, so that the color you see consists of the colors not absorbed. Thus, mixing paint pigments is a subtractive process. However, mixing colored lights is an additive process. When James went to red, green, and blue sectors he was able to produce white and a variety of other colors. He also showed that any three colors could be used as primaries as long as white could be obtained as some combination of these colors.

Maxwell later improved on the spinning disk by adding a smaller sectored disk in the center. He could then easily compare the color of the outer portion of the disk with the inner portion. This device is usually called Maxwell's top and is pictured in Figure 11. Maxwell could vary intensity as well as color by adding black sectors to the inner or outer disk. For example, various intensities of white (shades of gray) could be obtained in the center by using black and white sectors as shown in Figure 11.

Maxwell realized that color was not purely a physical property of an observed object, but that it also depended on how the light was processed by the eye. He agreed with Young that the eye must contain three type of color sensors. However, he believed that while each of these sensors was most sensitive to one color, they were also sensitive to a lesser degree to nearby colors. Maxwell also was aware that certain colors could only be matched by having a negative amount of at least one primary. He achieved this with his top by combining the color to be matched with one of the primaries on either the inner or outer disk and then matching a combination of the other two primaries on the other portion of the disk.

As useful as the color top was, Maxwell realized that it was limited by the relatively few colors of paper that were available. He designed several color boxes that produced spectral colors using a combination of prisms. The prisms spread out the spectral colors spatially and the desired colors and their amounts were controlled by precisely calibrated adjustable slits. The colors were combined by means of a series of mirrors and lenses. He was able to obtain much more accurate measurements using these boxes. He was also able to associate the colors mixed with their wavelengths. Taking measurements on a large number of subjects, he was able to determine the cause of common types of color-blindness. Those who were color blind were deficient in one of the three color receptors, usually the red receptor. Maxwell even invented one of the first optical instruments to view the retina. Maxwell is also credited with producing the first color photograph. He took photographs using red, green, and blue filters. He the combined them by projecting the photographs using the same filters. However, Maxwell's biggest contribution to color perception was his development of a mathematical model for describing colors and their combinations.

The most complete description of his model was contained in the article [James Clerk Maxwell, *On the Theory of Compound Colours, and the Relations of the Colours of the Spectrum*, Phil. Trans. R. Soc. Lond., **150**, pp 57–84, 1860]. Maxwell represented the three primary colors by three points in a plane that are the vertices of a triangle. He selected an origin that doesn't lie in the plane of the triangle. Arbitrary colors are represented by points in three-space. Each point has an associated vector from the origin to the point. In Maxwell's model colors are combined in the same way that force vectors are combined in mechanics. Thus, the vector associated with any color can be represented by a linear combination of the vectors associated with the three primaries. The vector associated with a color has both a direction and a magnitude. The direction is determined by the point where an extension of the vector intersects the plane of the color triangle. This point of intersection he called the quality of the color. The ratio of the length of the vector associated with a color to the length of the vector associated with the associated quality point is called the quantity of the color. The quality is a measure of the shade or tint of a color and the quantity is a measure of its intensity. All the points in the plane of the triangle have a quantity of one. A more detailed description of Maxwell's color model is contained in Appendix A. Figure 12 shows the colors in a Maxwell triangle (It doesn't show the colors that lie outside the triangle).

Maxwell received the Rumford medal from the Royal Society of London in 1860 for his work on color vision.

### Saturn's Rings

Saturn, with its large collection of flat rings, had been a mystery to astronomers for centuries. How could such a strange arrangement be stable? Why didn't the rings collapse into Saturn or drift off into space? In 1855 St. John's College of Cambridge posed this topic for their Adams' prize. They asked under what conditions the rings would be stable if they were a solid, a fluid, or were made up of independently orbiting pieces of matter. Responses were due by December 1857. Maxwell worked a whole year on this problem and was the only participant to enter a solution. Using very complicated mathematical calculations, he was able to show that solid rings would be unstable except for one unreasonable situation in which four-fifths of the mass was concentrated on one point of the circumference and the rest was evenly distributed. For the fluid case Maxwell used Fourier analysis to study the various types of waves that could exist. He showed that the fluid rings would eventually break up into separate blobs. By a process of elimination he had shown that the only possible configuration was for the rings to be made up of individually orbiting pieces of matter. He wasn't able to establish stability for this case in general, but he did analyze a special case in which there was a single ring made up of equally spaced particles. He showed that the ring would be stable if its average density was small enough compared with that of Saturn. James was awarded the Adams prize in 1859 and his conclusion about the make-up of the rings was verified by Voyager flybys in the 1980s. Maxwell's essay was so detailed and convincing that the English mathematician and astronomer George Airy wrote

It is one of the most remarkable applications of mathematics to physics that I have ever seen.

### The Kinetic Theory of Gases

If Maxwell had made no other contributions to science, his work on the kinetic theory of gases would have established him as one of the greatest physicists of all time. He was the first to introduce statistical laws into physics. This has led to the use of statistics in fields such as statistical mechanics, thermodynamics, and quantum mechanics. The kinetic theory of gases originated with the work of the eighteenth century mathematician Daniel Bernoulli. He proposed that gases consist of a large number of particles moving in all directions. In addition, he assumed that the pressure exerted by a gas on a surface is due to the impact of these particles and that heat is merely the kinetic energy of these particles. This theory was developed by others, and many of the properties of gases could be explained on this basis. But there was one major difficulty that had yet to be explained. In order for the theory to describe the pressures generated at normal temperatures, the particles must move very rapidly. Why then do gases diffuse so slowly? For example, the spread of odors throughout a room is a slow process. In 1859 Maxwell read a paper by the German physicist Rudolf Clausius who proposed that the slow rate of diffusion was due to a large number of collisions between the particles. This paper stimulated Maxwell's interest in the subject. Maxwell realized that it was impossible to describe all the motions and collisions exactly using standard Newtonian mechanics. He therefore derived a statistical law for the distribution of velocities. He represented each particle velocity in terms of its three components relative to an arbitrary set of orthogonal axes. The speed of the particle would then be the square root of the sum of the squares of the three components. He assumed that the three components of velocity were statistically independent and that each component had the same probability distribution, which he took as Gaussian or normal. The distribution $f(v)$ of particle speeds then has the form

\begin{equation*} f(v)=Av^2 e^{-\alpha v^2}. \end{equation*}This distribution is known as the Maxwell distribution or the Maxwell-Boltzmann distribution of velocities. Maxwell used his formulation of the kinetic theory to derive an unexpected result, namely that the viscosity of a gas is independent of both pressure and density. The same was true of thermal conductivity. These results he later confirmed experimentally. Maxwell's work inspired a young Austrian physicist named Ludwig Boltzmann and he, along with Maxwell, further developed the kinetic theory of gases.

### Maxwell's Demon

Maxwell's demon is the name given by Maxwell's friend William Thomson (Lord Kelvin) to a hypothetical figure that appeared in a letter Maxwell sent to Peter Tait upon receipt of a manuscript of Tait's book ** Thermodynamics** for review. This hypothetical creature was the main player in a thought experiment involving the second law of thermodynamics (one of the consequences of this law is that heat can not be transferred from a cold to a hot body without the expenditure of work). Here is this thought experiment in Maxwell's own words

Now let A & B be two vessels divided by a diaphragm and let them contain elastic molecules in a state of agitation which strike each other and the sides. Let the number of particles be equal in A & B but let those in A have the greatest energy of motion … . I have shown that there will be velocities of all magnitudes in A and the same in B only the sum of the square of the velocities is greater in A than in B. When a molecule is … allowed to go through a hole in the [diaphragm] … no work would be lost or gained only its energy would be transferred from the one vessel to the other.

Now conceive a finite being who knows the paths and velocities of all the molecules by simple inspection but who can do no work, except to open and close a hole in the diaphragm, by means of a slide without mass. Let him first observe the molecules in A and when he sees one coming the square of whose velocity is less than the mean sq. vel. of the molecules in B let him open the hole & let it go into B. Next let him watch for a molecule in B the square of whose velocity is greater than the mean sq. vel. in A and when it comes to the hole let him draw the slide & let it go into A, keeping the slide shut for all other molecules.

Then the number of molecules in A & B are the same as at first but the energy in A is increased and that in B diminished that is the hot system has got hotter and the cold colder & yet no work has been done, only the intelligence of a very observant and neat fingered being has been employed.

This was one of the first examples of a thought experiment in physics. Maxwell's intent was not to disprove the second law, but to show that this law had only statistical certainty and could not be applied to individual molecules. In Maxwell's review of the second edition of Tait's **Thermodynamics** in 1878 he notes that *the second law of thermodynamics is continually being violated, and that to a considerable extent, in any sufficiently small group of molecules belonging to a real body*.

### Electromagnetic Equations

Maxwell's greatest scientific accomplishment was certainly his development of a general theory for electromagnetism and the use of this theory to predict the existence of electromagnetic waves. This achievement places him alongside Newton and Einstein as one of the greatest physicists of all time. There had been a number of important developments involving electricity and magnetism leading up to Maxwell's discovery.

In 1784 Charles-Augustin de Coulomb discovered that the force of attraction or repulsion between two small electrified bodies varies inversely as the square of the distance between them. In 1799 Alesandro Volta developed a zinc and silver in brine battery that provided a source for continuous electric currents. In 1820 Hans-Christian Oersted observed that a current in a wire caused a nearby compass needle to deflect. Less than a week after hearing of Oersted's discovery, André-Marie Ampére presented a paper before the French Academy of Sciences giving a more complete treatment of this phenomena and demonstrated that parallel current carrying wires attract each other if the currents are in the same direction and repel each other if the currents are in opposite directions. The following year Michael Faraday produced a primitive electric motor based on these results.

In 1827 Ampére showed that two circular current carrying coils behave as magnets with the force of attraction or repulsion varying inversely as the square of the distance between them. In 1831 Faraday discovered that a changing magnetic flux through a wire loop will produce an electric current in the wire. Faraday used this result to construct an electrical generator. During Maxwell's lifetime there were a number of scientists attempting to formulate a general theory for electricity and magnetism. Since the forces between electrical charges or magnetic poles obey inverse-square laws similar to Newton's law for the gravitational attraction between masses, most investigators approached the task using this instantaneous action at a distance force model. It is interesting that Newton himself was troubled by the fact that distant objects could interact instantaneously without any evident connection between them. He once said

That gravity should [be such] that one body can act upon another at a distance through a vacuum, without the mediation of anything else … is to me so great an absurdity, that I believe no man [of] competent faculty of thinking can ever fall into it.

However, he was so successful in predicting the motions of planets and other objects using this approach that scientists soon forgot about the philosophical difficulties. Ampére and Weber were probably the two most influential scientists at the time of Maxwell that were committed to the Newtonian action at a distance approach to electricity and magnetism. Michael Faraday, however, had a different idea. He couldn't believe that forces appeared instantaneously at a distance. He thought that something must be happening in between and that forces must be transmitted between magnetic poles or electrical charges. He was very impressed with the patterns formed by iron filings sprinkled on a paper with a magnet underneath (see Figure 13). He called the lines formed by these particles “lines of force”.

Faraday introduced the idea of electric and magnetic fields acting throughout space. He felt that forces were transmitted through the interaction of these fields. Since Faraday had no mathematical training, his ideas were not embraced by the scientific community. Maxwell, however, was intrigued by these ideas and attempted to give them a mathematical basis. His first paper dealing with this topic was called *On Faraday's Lines of Force*. He sent a copy to Faraday. Faraday was delighted to have someone interested in his ideas and sent the following reply

I received your paper, and thank you very much for it. I do not say I venture to thank you for what you have said about `Lines of Force', because I know you have done it for the interests of philosophical truth; but you must suppose it is work grateful to me, and it gives me much encouragement to think on. I was almost frightened when I saw such mathematical force made to bear upon the subject, and then wondered that the subject stood it so well. I send by this post another paper to you; I wonder what you will say to it …

Maxwell had numerous conversations with Faraday while he was in London, and he developed a great respect for this aging scientist. Faraday was flattered that a young scientist of Maxwell's caliber was interested in his ideas.

In 1861 Maxwell presented a two part paper entitled [*On Physical Lines of Force*, Philosophical Magazine and Journal of Science (March 1861)]. Two more parts of this paper were added in 1862. In this paper he developed a conceptual elastic model involving a network of tiny spinning cells separated by spherical idle wheels. With this model he was able to duplicate all the known behaviors of electricity and magnetism. However, Maxwell knew that this model was merely an analogy and didn't represent actual mechanisms. In fact, Maxwell seemed to sense that the actual mechanisms may be beyond our ability to grasp.

Instead of refining this elastic model, he turned to more of a black-box approach in which the emphasis was more on describing the mathematical relationships between the resulting forces (fields) than on the underlying mechanisms. He was eventually able to develop a model based on electric and magnetic fields that we now refer to as Maxwell's Equations. In deriving these equations Maxwell relied heavily on the emerging field of vector analysis. In fact, Maxwell coined the term “curl” for one of the common vector differential operators. Maxwell originally stated his results in eight equations involving electric and magnetic potentials, but Oliver Heaviside later condensed them to the familiar four equations shown below.

\begin{align*} \text{div }{\mathbf{E}} &= 4\pi\rho\\ \text{div }{\mathbf{B}} &= 0\\ \text{curl }{\mathbf{E}} &= -\frac{1}{c}\frac{\partial\mathbf{B}}{\partial t}\\ \text{curl }{\mathbf{B}} &= \frac{4\pi}{c}\mathbf{J}+\frac{1}{c}\frac{\partial\mathbf{E}}{\partial t} \end{align*}Here $\mathbf{E}$ is the electric field vector, $\mathbf{B}$ is the magnetic induction vector, $\rho$ is the free charge density, and $\mathbf{J}$ is the current density. The term ‘div’ stands for “divergence” and represents a differential operator that measures the tendency, on average, of a vector field to point inward or outward from a point. If the divergence is positive, there is an outward tendency indicating the presence of a source at the point. If the divergence is negative, there is an inward tendency indicating a sink at the point. When the divergence is zero, it indicates that there is neither a source nor a sink at the point. The term ‘curl’ represents a differential operator that measures the tendency of a vector field to wrap around a point. The sign indicates whether it wraps in a clockwise or counter-clockwise direction.

The first of Maxwell's equations is called Gauss' law and is a generalization of Coulomb's inverse square law for point charges. The second equation states that there are no magnetic point sources or sinks. Magnetic lines of force never originate or terminate at a point, but always form closed loops. The third equation is a statement of Faraday's law of magnetic induction. The fourth equation is a statement of Ampere's law for magnetic-like forces generated by currents with an added term that is Maxwell's major contribution. The added term is called the displacement current and represents the current generated by a changing electric field analogous to the way that a changing magnetic field generates a current. Maxwell showed that the constant $c$ appearing in these equations was very close to the measured velocity of light.

Maxwell used these equations to derive wave equations for both $\mathbf{E}$ and $\mathbf{B}$ (see Appendix B). Because of these results Maxwell predicted the existence of electromagnetic waves and postulated that light was such a wave. It should be noted that electromagnetic waves involve both the electric field and the magnetic induction field. There are no purely electrical or purely magnetic waves. Moreover, all electromagnetic waves travel at the speed of light.

The extra displacement current term added by Maxwell was crucial to the derivation of these results. At the time displacement currents had not been measured, but Maxwell felt that they must exist. The American physicist Henry Rowland made a direct measurement of this added term in 1875. It was not until 1887, eight years after Maxwell's death, that Heinrich Hertz produced and measured an electromagnetic wave other than light.

Maxwell's equations play a similar role in electromagnetics as Newton's laws play in mechanics. It is interesting that Newton's laws had to be modified in Einstein's theory of relativity, but Maxwell's equations remained unchanged. In fact, Maxwell's equations were one of the main motivations for Einstein's theory of relativity. The concept of fields introduced by Faraday and refined by Maxwell now plays an important part in all branches of physics. In 1873 Maxwell published the two volumes of his famous work *A Treatise on Electricity and Magnetism*. It is difficult to over state the effect that Maxwell's equations have had on the development of modern physics and engineering.

### Other Contributions

Maxwell published a famous paper On governors in the Proceedings of Royal Society, vol. 16 (1867–1868). This paper is quite frequently considered a classical paper of the early days of control theory. Here governors refer to the governor or the centrifugal governor used in steam engines.

Maxwell was also asked by the British Association for the Advancement of Science to lead a small team whose goal would be to sort out the hodgepodge of units being used in electricity and magnetism. Maxwell realized that the confusion over units was not confined to electricity and magnetism. He proposed a systematic way of defining all physical quantities in terms of mass, length and time (**M**, **L**, and **T**), e.g., velocity has dimension **L**/**T**, acceleration has dimension **L**/**T**${}^2$, and force has dimension **ML**/**T**${}^2$. His work in this regard became the basis for what is now called dimensional analysis. Dimensional analysis is often used to verify that all terms in an equation have the same dimensions. The committee presented their report in 1863. The report contained a standard system of units that was later adopted, almost unchanged, as the first international set of units (misleadingly called Gaussian units). The report also looked at ways of measuring electrical and magnetic quantities that only involved mass, length, and time.

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