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’sThe Long Road to Equations How four enthusiasts helped bring the theory of to

Should you wish to pay By James C. Rautio elegant equations. Today, these are ill us tration by L orenzo Petrantoni homage to the great James known collectively as Maxwell’s Clerk Maxwell, you wouldn’t lack for locales in which to equations, and they can be found in just about every do it. There’s a memorial marker in London’s ­West­minster introductory and physics textbook. Abbey, not far from Isaac ’s grave. A magnificent It could be argued that these equations got their start statue was recently installed in Edinburgh, near his birth- 150 years ago this month, when Maxwell presented his place. Or you can pay your respects at his final resting place theory uniting and before the near Castle Douglas, in southwestern Scotland,­ a short Royal Society of London, publishing a full report the distance from his beloved ancestral estate. They’re fitting next year, in 1865. It was this work that set the stage for monuments to the person who developed the first uni- all the great accomplishments in physics, telecommuni- fied theory of physics, who showed that electricity and cations, and that were to follow. magnetism are intimately connected. But there was a long gap between the presentation But what these landmarks don’t reflect is the fact that, at and the utilization. The mathematical and conceptual the time of Maxwell’s death in 1879, his electro­magnetic underpinnings of Maxwell’s theory were so compli- theory—which underpins so much of our modern tech- cated and counterintuitive that his theory was largely nological world—was not yet on solid ground. neglected after it was first introduced. An extraordinary amount of information about the It took nearly 25 years for a small group of physi- world—the basic rules by which light behaves, current flows, cists, themselves obsessed with the mysteries of and magnetism functions—can be boiled down to four electricity and magnetism, to put Maxwell’s ­theory

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12.MaxwellEquations.NA [P].indd 37 11/14/14 3:09 PM on solid footing. They were the ones who carrying wire. Soon after, André‑Marie His work was interrupted by a flurry gathered the experimental evidence Ampère showed that two parallel of distractions. He took a job in 1856 at needed to confirm that light is made up current-­carrying wires could be made Marischal College, in Aberdeen, Scotland; of electromagnetic waves. And they were to exhibit a mutual attraction or repul- devoted several years to a mathematical the ones who gave his equations their sion depending on the relative direction study of the stability of the rings of Saturn; present form. Without the ­Herculean of the currents. And by the early 1830s, was laid off in a college merger in 1860; efforts of this group of ­“Maxwellians,” had shown that just as and contracted smallpox and nearly died so named by historian Bruce J. Hunt, of electricity could influence the behavior before finally taking a new job, as a pro- the University of Texas at Austin, it might of a , a magnet could affect elec- fessor at King’s College London. have taken decades more before our mod- tricity, when he showed that drawing Somehow, in all of this, Maxwell found ern conception of electricity and magne- a magnet through a loop of wire could the time to flesh out Faraday’s field the- tism was widely adopted. And that would generate current. ory. Although not yet a complete theory have delayed all the incredible These observations were piecemeal of electromagnetism, a paper he pub- and technology that was to follow. evidence of behavior that no one really lished in several parts in 1861 and 1862 understood in a systematic or compre- proved to be an important stepping-stone. hensive way. What was Building on previous ideas, Maxwell oday, we learn early really? How did a current-carrying wire envisioned a kind of molecular medium on that visible light is just reach out and twist a magnet? And how in which magnetic fields are arrays of spin- one chunk of the wide elec- did a moving magnet create current? ning vortices. Each of these vortices is sur- tromagnetic spectrum,­ A major seed was planted by Faraday, rounded by small particles of some form whose is made who envisioned a mysterious, invisible that help carry spin from one vortex to Tup of oscillating electric and magnetic “electrotonic state” surrounding the mag- another. Although he later laid it aside, fields. And we learn that electricity and net—what we would today call a field. He Maxwell found that this mechanical vision magnetism are inextricably linked; posited that changes in this electrotonic helped describe a range of electromag- a changing creates an state are what cause electromagnetic netic phenomena. Perhaps most crucially, , and current and changing phenomena. And Faraday­ hypothesized it laid the groundwork for a new physical electric fields give rise to magnetic fields. that light itself was an electromagnetic concept: the . We have Maxwell to thank for these wave. But shaping these ideas into a com- Displacement current isn’t really cur- basic insights. But they didn’t occur to plete theory was beyond his mathemati- rent. It’s a way of describing how the him suddenly and out of nowhere. The evi- cal abilities. That was the state of affairs change in electric field passing through a dence he needed arrived in bits and pieces, when Maxwell came on the scene. particular area can give rise to a magnetic over the course of more than 50 years. In the 1850s, after graduating from field, just as a current does. In Maxwell’s You could start the clock in 1800, when the University of Cambridge, in England, model, the displacement current arises physicist reported the Maxwell set about trying to make mathe- when a change in electric field causes a invention of a battery, which allowed matical sense of Faraday’s observations momentary change in the position of the experimenters to begin working with and theories. In his initial attempt, an particles in the vortex medium. The move- continuous . Some 20 years 1855 paper called “On Faraday’s Lines ment of these particles generates a current. later, Hans Christian Ørsted obtained of Force,” Maxwell devised a model by One of the most dramatic manifes- the first evidence of a link between elec- analogy, showing that equations that tations of displacement current is in tricity and magnetism, by demonstrat- describe incompressible fluid flow could the capacitor, where in some circuits ing that the needle of a compass would also be used to solve problems with the energy stored between two plates

move when brought close to a current- unchanging electric or magnetic fields. in a capacitor oscillates between high I ma g es A rc h ive/Getty History Universal (2); Wikipedia le f t: From Maxwellian Milestones

1785 Charles-Augustin de 1820 André-Marie Ampère reports that the force between two shows that two parallel charges varies with the inverse square current-carrying wires can of the distance. attract or repel each other, depending on the relative 1800 direction of their currents. Alessandro Volta 1820 Hans Christian Ørsted invents the first shows that the needle of a 1855 1861 & 1862 Maxwell battery, which allowed compass can move when brought 1831 Michael Faraday, Maxwell’s first publishes a four-part paper, “On Physical experimenters to begin close to a current-carrying wire, who conceived of electrical paper on Faraday’s Lines of Force.” It introduces the core idea working with continuous the first evidence of a link between and magnetic fields, discovers 1831 observations and that a change in through a direct current. electricity and magnetism. electromagnetic induction. is born in Edinburgh. theories debuts. surface can create a magnetic field.

1800 1810 1820 1830 1840

12.MaxwellEquations.NA [P].indd 38 11/14/14 3:09 PM spectively. And there describes how both move in concert to make an Four golden rules are two other fields, current and changing the displacement electric fields can electromagnetic wave. Today, the relation- fieldD and the mag- give rise to magnetic This work is the foundation of ship between elec- netic fieldH . These fields. The symbols tricity and magnetism, fields are ­related to at the beginning of our modern understanding of along with the wave E and B by constants each equation are electromagnetism, and it pro- nature of light and that reflect the differential operators. electromagnetic nature of the medium These compactly vides and engineers radiation in general, that the fields pass encode­ calculus­ with all the tools they need to is encoded in the four through (the values that involves vectors, “Maxwell’s equations” of these constants quantities that have calculate the relationships shown at left. The in can be a directionality and among charges, electric fields, equations can be combined to give the thus x, y, and z com- written in different speed of light). The ponents. Maxwell’s currents, and magnetic fields. ways. Here, J is the displacement fieldD original formulation of But what should have been . E and was one of Maxwell’s his electromagnetic B are the electric and key contributions, theory contained a coup was actually met with magnetic fields, re- and the last equation 20 equations. extreme skepticism, even from Maxwell’s closest colleagues. One of the most vocal skeptics was Sir ­William Thomson (later and low values. In this system, it’s fairly Much as a spring constant determines Lord ). A leader of the British sci- easy to visualize how Maxwell’s mechan- how quickly a spring rebounds after it’s entific community at the time, Thomson ical model would work. If the capacitor stretched or compressed, these constants simply didn’t believe that such a thing as contains an insulating, mate- can be combined to determine how fast an displacement current could exist. rial, you can think of the displacement electromagnetic wave travels in free space. His objection was a natural one. It was current as arising from the movement After others had determined their val- one thing to think of a displacement cur- of that are bound to the nuclei ues using experiments on capacitors and rent in a dielectric filled with atoms. It of atoms. These swing back and forth inductors, Maxwell was able to estimate was quite another to imagine it forming from one side to another, as if attached the speed of an electromagnetic wave in in the nothingness of a vacuum. With- to stretched rubber bands. But Maxwell’s vacuum. When he compared the value to out a mechanical model to describe this displacement current is more fundamen- existing estimates of the speed of light, he environment and without actual moving tal than that. It can arise in any medium, concluded from their near equality that electric charges, it wasn’t clear what dis- including the vacuum of space, where light must be an electromagnetic wave. placement current was or how it might there are no electrons available to create arise. This lack of a physical mechanism a current. And just like a real current, it was distasteful to many physicists in the gives rise to a magnetic field. Maxwell completed the last . Today, of course, we’re

/Getty I ma g es /Getty With the addition of this concept, key pieces of his electromagnetic the- willing to accept physical theories, such Maxwell had the basic elements he ory in 1864, when he was 33 (although as quantum mechanics, that defy our needed to link measurable circuit prop- he made some simplifications in later everyday physical intuition, so long as erties to two, now out-of-use, constants work). In his 1864 talk and the paper they are mathematically rigorous and that express how readily electric and that followed, he left the mechanical have great predictive power. magnetic fields form in response to a model behind but kept the concept of Maxwell’s contemporaries perceived or a current. (Nowadays, we displacement current. Focusing on the other big shortcomings in his theory. formulate these fundamental constants mathematics, he described how elec- For example, Maxwell postulated that differently, as the and per- tricity and magnetism are linked and oscillating electric and magnetic fields

t: SSPL R i gh t: I ma g es; P opper f oto/Getty L e f t: meability of free space.) how, once properly generated, they together form waves, but he didn’t Maxwellian Milestones

1864 Maxwell 1879 Heinrich ’s 1885 1940 presents new work before the Royal interest in Maxwell’s work OLIVER gives the term “Maxwell’s​ is piqued by a Prussian Society of London, published the next year. HEAVISIDE equations” a boost with his Academy of competition to It suggests that electric and magnetic fields publishes a monograph “Considerations obtain experimental evidence for or against can move through space in waves and that condensed Concerning the Fundaments of electromagnetic waves. light itself is such a wave. 1879 version of Theoretical Physics.” Maxwell’s 1861 & 1862 Maxwell 1873 Maxwell publishes his MAXWELL equations, 1888 Hertz, several years after publishes a four-part paper, “On Physical magnum opus, A on dies of reducing the moving to a well-equipped laboratory Lines of Force.” It introduces the core idea Electricity and Magnetism, which stomach equation count in Karslruhe, reports confirmation of that a change in electric flux through a contains further mathematical and cancer at from 20 to four. the existence of the electromagnetic surface can create a magnetic field. interpretive work. age 48. waves predicted by Maxwell.

1860 1870 1880 1890 1900

12.MaxwellEquations.NA [P].indd 39 11/14/14 3:09 PM Maxwell’s theory It was Heaviside, working largely in today can be summed seclusion, who put Maxwell’s equations in up by four equations. their present form. In the summer of 1884, But his formulation took Heaviside was investigating how energy the form of 20 simulta- moved from place to place in an electri- neous equations, with cal circuit. Is that energy, he wondered, 20 variables. The dimen- carried by the current in a wire or in the sional components of his surrounding it? equations (the x, y, and Heaviside ended up reproducing a z directions) had to be result that had already been published spelled out separately. by another British physicist, John And he employed some Poynting. But he kept pushing further, counterintuitive vari- and in the process of working through ables. Today, we are the complicated vector calculus, he accustomed to think- happened upon a way to reformulate ing of and working with Maxwell’s score of equations into the electric and magnetic four we use today. magic: Heinrich Hertz used the coil fields. But Maxwell worked primarily The key was eliminating Maxwell’s [left] and the antennas [right] to produce and with another kind of field, a quantity strange magnetic vector potential. detect electromagnetic radiation outside the visible range. he called electromagnetic momen- “I never made any progress until I threw tum, from which he would then cal- all the potentials overboard,” Heaviside describe how they move through space. culate the electric and magnetic fields later said. The new formulation instead All waves known at this time required a that Faraday­ first envisioned. ­Maxwell placed the electric and magnetic fields medium in which to travel. Sound waves may have selected that name for the front and center. travel in air and water. So if electro­ field—today known as magnetic vector One of the consequences of the work magnetic waves existed, physicists of the potential—because its derivative with was that it exposed the beautiful sym- time reasoned, there must be a medium to respect to time yields an electric force. metry in Maxwell’s equations. One of the carry them, even if that medium couldn’t But the potential does us no favors when four equations describes how a chang- be seen, tasted, or touched. it comes to calculating a lot of simple ing magnetic field creates an electric Maxwell, too, believed in such a electromagnetic behavior at boundar- field (Faraday’s discovery), and another medium, or ether. He expected that it ies, such as how electromagnetic waves describes how a changing electric field filled all of space and that electromag- reflect off a conductive surface. creates a magnetic field (the famous dis- netic behavior was the result of stresses, The net result of all of this complexity is placement current, added by Maxwell). strains, and movements in this ether. that when Maxwell’s theory made its debut, This formulation also exposed a mys- But in 1865, and in his later two-volume almost nobody was paying attention. tery. Electric charges, such as electrons Treatise on Electricity and Magnetism, and ions, have lines of electric field Maxwell presented his equations with- around them that radiate from the out any mechanical model to justify how But a few people were. And one charge. But there is no source of mag- or why these mystical electromagnetic of them was . Once netic field lines: In our known universe, waves could possibly propagate. For described by a friend as a “first rate oddity,” magnetic field lines are always continu- many of his contemporaries, this lack Heaviside, who was raised in extreme pov- ous loops, with no start or end. of a model made Maxwell’s theory seem erty and was partially deaf, never attended This asymmetry troubled Heaviside, grievously incomplete. university. Instead, he so he added a term rep- Perhaps most crucially, Maxwell’s own taught himself advanced post your resenting a magnetic description of his theory was stunningly science and mathematics. comments “charge,” assuming that complicated. College students may greet Heaviside was in his at http://spectrum. it had just not yet been ieee.org/maxwell1214 the four Maxwell’s equations with terror, early 20s and working as discovered. And indeed but Maxwell’s formulation was far mess- a telegrapher in Newcastle, in northeast it still hasn’t. Physicists have since con- ier. To write the equations economically, England, when he obtained Maxwell’s ducted extensive searches for such we need mathematics that wasn’t fully 1873 Treatise. “I saw that it was great, magnetic charges, also called mag- y A rc h ives T ec h nolo g y mature when Maxwell was conducting greater and greatest,” he later wrote. netic monopoles. But they have never his work. Specifically, we need vector “I was determined to master the book been found. calculus, a way of compactly codifying and set to work.” The next year, he left Still, magnetic current is a useful the differential equations of vectors in his job and moved in with his parents artifice for solving electromagnetic

three dimensions. to learn Maxwell. problems with | continued on page 54 o f I nstit u te K arlsr uh e

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continued from page 40 | some kinds higher- light, but the effort fiz- of geometries, such as the behavior of zled when Lodge and ­FitzGerald realized radiation moving through a slit in a con- their schemes would create radiation of ductive sheet. too low a frequency to be detected by eye. If Heaviside modified Maxwell’s equa- Nearly a decade later, Lodge was tions to this degree, why don’t we call performing experiments on lightning them Heaviside’s equations? Heaviside protection when he noticed that dis- answered this question himself in 1893 charging capacitors along wires pro- in the preface to the first volume of duced arcs. Curious, he changed the his three-volume publication, Electro­ wire lengths and found that he could magnetic Theory. He wrote that if we realize spectacular sparks. He correctly Follow Max on his have good reason “to believe that he deduced that this was the action of an animated adventures [Maxwell] would have admitted the electromagnetic wave in resonance. He necessity of change when pointed out to found that with enough power, he actu- him, then I think the resulting modified ally could see the air becoming ionized theory may well be called Maxwell’s.” around the wires, a dramatic illustra- tion of a . Now confident that he was generating athematical ele- and detecting electromagnetic waves, gance was one thing. But Lodge planned to report his astounding finding experimental evi- results at a meeting of the British Asso- dence for Maxwell’s the- ciation, right after he returned from a Mory was something else. When Maxwell vacation in the Alps. But while reading passed away in 1879, at age 48, his the- a journal on the train out of ­Liverpool, Catalog on adhesives ory was still considered incomplete. he discovered he’d been scooped. In the for the aerospace industry There was no empirical evidence that July 1888 issue of , he light is composed of electromagnetic found an article entitled “Über elektro­ waves, aside from the fact that the speed dynamische Wellen im Luftraum und of visible light and that of electromag- deren Reflexion” (“On electrodynamic

stry ce Indu Aerospa netic radiation seemed to match up. In waves in air and their reflection”) writ- addition, Maxwell did not specifically ten by a little-known German researcher, address many of the qualities electro- Heinrich Hertz. magnetic radiation should have if it Hertz’s experimental work on the sub- makes up light, namely behaviors like ject began at the Technische ­Hochschule reflection and . (now the Institute of Technol- Physicists George Francis FitzGerald ogy) in Karlsruhe, Germany, in 1886. He and worked to strengthen noticed that something curious hap- the link to light. Proponents of ­Maxwell’s pened when he discharged a capaci- 1873 Treatise, the pair met the year tor through a loop of wire. An identical before Maxwell’s death at a meeting of loop a short distance away developed Products meet NASA the British Association for the Advance- arcs across its unconnected terminals. low outgassing standards ment of Science in Dublin, and they Hertz recognized that the sparks in the began collaborating, largely through unconnected loop were caused by the the exchange of letters. Their corre- reception of electromagnetic waves that spondence with each other and with had been generated by the loop with the ­Heaviside helped to advance the theoret- discharging capacitor. ical understanding of Maxwell’s theory. Inspired, Hertz used sparks in such As historian Hunt outlines in his book, loops to detect unseen radio-frequency The Maxwellians, Lodge and FitzGerald­ waves. He went on to conduct experi- also hoped to find experimental evi- ments to verify that electromagnetic dence to support the idea that light is waves exhibit lightlike behaviors of an electromagnetic wave. But here they reflection, refraction, diffraction, and WATCH THIS ad come to life didn’t have much success. In the late . He performed a host of 1870s, Lodge developed some circuitry experiments both in free space and 154 Hobart Street, Hackensack, NJ 07601, USA that he hoped would be capable of con- along wires. He molded a meter-long +1.201.343.8983 ∙ main masterbond.com verting lower-frequency electricity into made of asphalt that was transpar- www.masterbond.com

12.MaxwellEquations.NA [P].indd 54 11/14/14 3:09 PM ent to radio waves and used it to observe Earn PDHs by relatively large-scale examples of reflec- attending a Webinar! tion and refraction. He launched radio waves toward a grid of parallel wires and showed that they would reflect or pass through the grid depending on the grid’s orientation. This demon- The brightest minds discussing the biggest topics. strated that electromagnetic waves were transverse:­ They oscillate, just as light does, in a direction perpendicu- New Webinars in December lar to the direction of their propaga- 4 Dec Design and Placement tion. Hertz also reflected radio waves for the Automotive Market off a large sheet of , measuring the This webinar will demonstrate how the various simulation tools in distance between canceled-out nulls in CST STUDIO® (CST MWS) can be used in tandem to help the resulting standing waves in order to the antenna engineer and the system integrators at each stage of the determine their wavelengths. design process, and how meaningful “what-if?” analyses can be devised With this data—along with the fre- and executed to improve the overall quality of the nal product. http://spectrum.ieee.org/webinar/antenna-design-and-placement- quency of the radiation, which he cal- for-the-automotive-market culated by measuring the 11 Dec Introduction to Antenna Simulation with COMSOL and of his circuitlike trans- This webinar will highlight the capabilities of COMSOL Multiphysics mitting antenna—Hertz was able to cal- for RF applications including impedance, far- eld, and near- eld culate the speed of his invisible waves, calculations. A live demonstration will show you how to model a dipole which was quite close to that known antenna and simulate its radiation pattern using the COMSOL simulation software http://spectrum.ieee.org/webinar/-introduction-to- for visible light. antenna-simulation-with-comsol Maxwell had postulated that light was 16 Dec Thinking Across the Plug: an electromagnetic wave. Hertz showed Integrating Electric Vehicles and the Smart Grid that there was likely an entire universe Speakers will discuss what we’re discovering about the interaction of invisible electromagnetic waves that between next-generation electric vehicles and the electric distribution behave just as visible light does and that system that powers them. http://spectrum.ieee.org/webinars move through space at the same speed. This revelation was enough, by infer- On-Demand Webinars ence, for many to accept that light itself is an electromagnetic wave. Xerox Talks Best Practices for Open Source Governance Lodge’s disappointment at being Join Robert Levine, Licensing Executive for Xerox, and Black Duck as they discuss open scooped was more than compensated by source best practices, common challenges, and emerging trends. http://spectrum.ieee. the beauty and completeness of Hertz’s org/webinar/xerox-talks-best-practices-for-open-source-governance work. Lodge and FitzGerald worked to Integrating Open Source Governance and Build Automation popularize Hertz’s findings, present- for Competitive Solutions ing them before the British Associa- Join for a discussion on how integrating governance and build automation empowers software development organizations. http://spectrum.ieee.org/webinar/reinforcing- tion. Almost immediately, Hertz’s work open-source-software-governance-with-build-automation went on to inform the development of Eliminate Pitfalls of DDR Memory Testing . The earliest incar- Register for this free webinar on using an oscilloscope for validation and debug of nations of the technology employed DDR2/3/4 physical layer measurements. http://spectrum.ieee.org/webinar/eliminate- much like the broadband pitfalls-of-ddr-memory-testing spark-gap devices Hertz used. Bring Medical Devices to Market Faster with Reusable Components Eventually scientists accepted that This webinar will show attendees how to think strategically about design components, identify which elements should be standardized and re-used, and more. http://spectrum. waves could travel through nothing at ieee.org/webinar/bring-medical-devices-to-market-faster-with-reusable-components all. And the concept of a field, at first dis- tasteful because it lacked any mechan- ical parts to make it work, became Sponsors central to much of modern physics. There was much more to come. But even before the close of the 19th cen- Sign up today! tury, thanks to the dogged efforts of a www.spectrum.ieee.org/webinar few dedicated enthusiasts, Maxwell’s legacy was secure. n

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