Faraday, Maxwell, and the Electromagnetic Field

by Nancy Forbes

  CHAPTER NINE: SOCIETY AND DRILL

  * Quoted passages in this chapter for which the source is not evident from the text or given a numbered note marker are from Campbell and Garnett.

  1. The fellow student who reported on Maxwell's midnight jogging was Charles Hope Robertson. Robertson was also the friend whom Maxwell helped by reading out the next day's bookwork when he had eye trouble.

  2. The complete essay “Are There Any Real Analogies in Nature?” can be found in Campbell and Garnett, pages 235–244.

  3. The fellow student who “never met a man like” Maxwell was W. N. Lawson.

  4. William Hopkins was the most successful Cambridge coach of the time. Seventeen of his pupils became senior wranglers, including George Gabriel Stokes, William Thomson, P. G. Tait, Maxwell's rival E. J. Routh, and Arthur Cayley, who created the theory of matrices.

  5. For the Lagrangian and the Hamiltonian, see note 5 to chapter 13. The Routhian combines elements of the Lagrangian and the Hamiltonian, and the Laplacian is, in vector terminology, the divergence of the gradient of a scalar function.

  6. It was W. N. Lawson who reported to Campbell on Maxwell's geniality and kindness.

  CHAPTER TEN: AN IMAGINARY FLUID

  1. Young Maxwell's question about the blue stone is reported in Campbell and Garnett, page 14.

  2. Maxwell's reference to “different kinds of paint” can be found in W. D. Niven's The Scientific Papers of James Clerk Maxwell, vol. 1, page 127.

  3. An instance of Maxwell's enduring influence is the present-day chromaticity diagram, which differs only in detail from his original color triangle. It uses a right-angled triangle in which the proportions of red and green are plotted, the proportion of blue being implied because the proportions of red, green, and blue always add up to 1.

  4. Maxwell's letters to Thomson at this time are in Harman, vol. 1, pages 254–63 and 319–20.

  5. The full text of Maxwell's comments on Faraday's experimental methods (which he contrasts with those of Ampère) can be found in article 528 of his Treatise on Electricity and Magnetism, on page 176 of vol. 2.

  6. The quoted words from Maxwell's paper “On Faraday's Lines of Force,” can be found in Simpson, page 57.

  7. The quoted words from Maxwell's paper “On Faraday's Lines of Force,” can be found in Simpson, page 60.

  8. Airy's disparaging comment about lines of force is quoted by J. J. Thomson in his essay “James Clerk Maxwell” in James Clerk Maxwell: A Commemoration Volume, page 28.

  9. The flux concentration, or density, in a small region (or, in the limit, at a point) is the amount of flux per unit area of a plane surface perpendicular to the direction of the flux.

  10. Thomas K. Simpson gives a guided study of part 1 of Maxwell's paper “On Faraday's Lines of Force” in his book Maxwell on the Electromagnetic Field.

  11. Maxwell wrote of the power of the subconscious mind in a May 29, 1857, letter to his friend Litchfield. It is given in Campbell and Garnett, page 136.

  12. These lines are from Maxwell's poem “Recollections of Dreamland,” written a few months after his father died. The complete poem can be found in Campbell and Garnett, pages 298–99.

  13. Maxwell's preference for “the rubs of the world” is reported in Campbell and Garnett, page 126.

  14. The rival candidate for whom Maxwell provided a reference was William Swan, later professor of natural philosophy at St. Andrews.

  15. These lines are also from “Recollections of Dreamland,” mentioned above in note 12.

  16. Maxwell made this reference to the “Natural Philosophers of the North” in an October 14, 1856, letter to Cecil Monro, quoted in Campbell and Garnett, page 132.

  CHAPTER ELEVEN: NO JOKES ARE UNDERSTOOD HERE

  *Quoted passages in this chapter for which the source is not evident from the text or given a numbered note marker are from Campbell and Garnett.

  1. Maxwell's comment “No jokes are understood here” is report by Ivan Tolstoy in his book James Clerk Maxwell, page 80.

  2. This passage and the following one, from Maxwell's inaugural address at Marischal College, are given by R. V. Jones in his essay “The Complete Physicist: James Clerk Maxwell, 1831–79,” The Yearbook of the Royal Society of Edinburgh, 1980.

  3. The quoted words from Maxwell's paper “On Faraday's Lines of Force” can be found in Simpson, page 57.

  4. Faraday wrote this letter to Maxwell on November 13, 1857. It is given in Campbell and Garnett, page 145.

  5. This is the second of four verses of Maxwell's poem “The Song of the Atlantic Telegraph Company.” The complete poem can be found in Campbell and Garnett, page 140.

  6. These are the last four of eight verses of Maxwell's poem “To K. M. D.” The complete poem can be found in Campbell and Garnett, pages 302–303.

  7. Maxwell put the question “If you go at 17 miles a minute…?” in a letter to P. G. Tait that is now in the Cambridge University Archive.

  8. To statisticians, the Maxwell distribution is the square root of a chi-square distribution with three degrees of freedom.

  9. This account of Maxwell as his students at Aberdeen saw him is from George Reith, who became moderator of the Church of Scotland and father of Lord Reith, the first governor of the British Broadcasting Corporation. George Reith's comments are reported by R. V. Jones in his essay “The Complete Physicist: James Clerk Maxwell, 1831–79” in the Yearbook of the Royal Society of Edinburgh, 1980.

  10. David Gill reminisced about Maxwell's classes in the introductory part of his book History and Description of the Royal Observatory Cape of Good Hope, pages xx–xxi.

  11. This farmer's recollection is reported by R. V. Jones in his essay “The Complete Physicist: James Clerk Maxwell, 1831–79” in the Yearbook of the Royal Society of Edinburgh, 1980.

  CHAPTER TWELVE: THE SPEED OF LIGHT

  1. Maxwell's inaugural lecture at King's College, London, is given in Harman, vol. 1, pages 662–74.

  2. Amazingly, no one managed to repeat the feat, and many years passed before the next color photograph appeared. About one hundred years later, a team at Kodak Research Laboratories discovered that the photographic plates Maxwell and Sutton had used were completely insensitive to red light! By coincidence, the plates were sensitive to some ultraviolet light and the solution the experimenters had used as a red filter happened to have a passband in just the right part of the ultraviolet spectrum. Ultraviolet had acted as a surrogate for red. Lucky Maxwell! But perhaps he made his own luck. It was a rule with him never to discourage a man from doing an experiment, however slim the apparent prospect of success. Arthur Schuster, one of Maxwell's students at the Cavendish, recalled Maxwell saying of another student, “if he doesn't find what he is looking for, he may find something else.”

  3. For clarity, the laws are expressed here in a modern way; in 1861, the terms field and flux were not yet in general use. Law 3 is now often called Ampère's Law and law 4 is called Faraday's Law.

  4. Maxwell's medium of cells and idle wheels, like the incompressible fluid in his model from his Cambridge days, had an built-in inverse-square law. This followed from the property of magnetic flux (as modeled by the spinning of the cells) that the amount of flux entering a region enclosed by any closed surface was the same as the amount leaving. The reasoning, though more mathematical, is essentially the same as that given in chapter 10 for Maxwell's fluid model, where the amount of fluid emerging from any sphere with a point source at its center is the same, no matter what the size of the sphere. More generally, any effect that spreads out uniformly in three-dimensional space follows an inverse-square law. The principle is embodied is the divergence theorem, otherwise known as Gauss's theorem, in vector analysis.

  5. Maxwell identified Faraday's electrotonic state with the magnetic vector potential (a quantity whose curl at any point in the field was equal to the magnetic flux density there). Magnetic flux, represented by the rotation of the cells, was, in Maxwell's interpretation, the
electromagnetic momentum of the field.

  6. Thomas K. Simpson gives a guided study of Maxwell's paper “On Physical Lines of Force” in his book Maxwell on the Electromagnetic Field.

  7. This passage from Maxwell's paper “On Physical Lines of Force” is quoted by Richard Glazebrook in his book James Clerk Maxwell and Modern Physics, page 173.

  8. See note 4. The same reasoning about the inverse-square law applies, this time to electric flux, as modeled by the distortion of the cells.

  9. The ratio is the number of electrostatic units in one electromagnetic unit—broadly, the relative strengths of the two types of force. It had the dimensions of velocity because electromagnetic force depends not only on the quantity of charge but also on its velocity. The electromagnetic unit is bigger because it takes more charge to generate a given force by magnetic action when the charge is moving at unit velocity than it does by electrostatic action.

  10. The quoted words from Maxwell's paper “On Physical Lines of Force” can be found in Simpson, page 216.

  11. Monro's letter to Maxwell about the “brilliant result” is given in Campbell and Garnett, page 163.

  12. Newton made these comments in the letter to Richard Bentley quoted in chapter 3—see note 3 to that chapter. The failure of Newton's followers to heed this warning may be due in part to the misplaced zeal of his disciple and evangelist Roger Coates, who wrote in a preface to Newton's Principia Mathematica that action at a distance is one of the primary properties of matter.

  13. Faraday's skepticism of atoms is reported by Simon Blackburn in his book Think: A Compelling Introduction to Philosophy, page 248.

  14. Charles Coulson was professor of theoretical physics at King's College, London, from 1947 to 1952. One of his doctoral students there was Peter Higgs, who later created the theory of the Higgs field and its associated particle, the Higgs Boson.

  CHAPTER THIRTEEN: GREAT GUNS

  1. Maxwell's “great guns” letter to his cousin was written in January 1865. The complete letter can be found in Campbell and Garnett, pages 168–69.

  2. These words from Maxwell's paper “A Dynamical Theory of the Electromagnetic Field” can be found in Simpson, page 255.

  3. These words from Maxwell's paper “A Dynamical Theory of the Electromagnetic Field” can be also found in Simpson, page 255.

  4. Thomson and Tait published their Treatise on Natural Philosophy in 1867 after seven years of collaborative effort, and after a difficult gestation it sold very well. By the usual alphabetical convention, Tait should have been the first-named author, but Thomson's stellar reputation took precedence. It was Tait, however, who did most of the work: he was forever chasing Thomson to comment on drafts he'd sent. Thomson responded to Tait's scolding with good humor and they remained lifelong friends.

  5. Lagrange's set of equations embodied the so-called principle of least action, first formulated by Pierre Louis Moreau de Maupertuis in 1746, and its characteristic function (the difference between the physical system's kinetic energy and its potential energy) became known as the Lagrangian. William Rowan Hamilton extended Lagrange's method to form an alternative system of equations now almost universally used to describe the dynamics of systems. His characteristic function, the Hamiltonian, represents the total energy of a system.

  6. Maxwell included the “belfry” passage in a review for Nature of the second edition of Thomson and Tait's Treatise on Natural Philosophy in 1879. He was making the point that it might never be possible to give an explanatory model of electrodynamics, and he had probably also used the belfry analogy in his lectures to students.

  7. A guided study of Maxwell's paper “A Dynamical Theory of the Electromagnetic Field” is given by Thomas K. Simpson in his book Maxwell on the Electromagnetic Field.

  8. Maxwell explained why he did not condense his equations in his Treatise on Electricity and Magnetism, vol. 2, page 254.

  9. Thomson's view that Maxwell had “lapsed into mysticism” is reported by Templeton and Herrmann in their book The God Who Would Be Known: Revelations of Divine Contemporary Science, page 161.

  10. Gill made these comments about Maxwell's teaching in the introductory part of his book History and Description of the Royal Observatory Cape of Good Hope, pages xx–xi.

  CHAPTER FOURTEEN: COUNTRY LIFE

  1. This is the first of four verses of Maxwell's poem “Will You Come along with Me?” The complete poem can be found in Campbell and Garnett, page 301.

  2. This and the following passage are quoted by Campbell and Garnett on page 180, though they do not identify the observer.

  3. These passages from Maxwell's essay “Is Autobiography Possible?” are given in Campbell and Garnett, page 125.

  4. The quotation, given by Campbell and Garnett on page 196, is from a draft found after Maxwell's death. The draft ends at “…continually,” but the final “changing” is clearly implied.

  5. Maxwell made this comment about his decision not to apply for the St. Andrews post in an October 30, 1868, letter to William Thomson. The letter is held in the Glasgow University Library.

  6. Maxwell's letter seeking advice on political matters was to W. R. Grove, who was vice president of the Royal Institution. The complete letter can be found in Harman, vol. 2, page 461. Grove was not only a scientist but also a successful barrister who became a judge of the Queen's Bench. As far as he was political at all, Maxwell was a Conservative, and the Conservatives lost the election held shortly before the appointment was made.

  CHAPTER FIFTEEN: THE CAVENDISH

  * Quoted passages in this chapter for which the source is not evident from the text or given a numbered note marker are from Campbell and Garnett.

  1. Maxwell spoke of Alexander Graham Bell's father when giving the Rede lecture at Cambridge in 1878. His title for the lecture was “On the Telephone”; a longer extract from it can be found in Campbell and Garnett, pages 177–78.

  2. Examples may be found in Campbell and Garnett on pages 16 and 18.

  3. Maxwell gave this premonition of chaos theory in 1873 in an essay for the Eranus Club: “Does the Progress of Physical Science Give Any Advantage to the Opinion of Necessity (or Determinism) over That of the Contingency of Events and the Freedom of the Will?” For a wider audience he would no doubt have shortened the title to something like “Science and Free Will.” The complete essay can be found in Campbell and Garnett, pages 209 to 213.

  4. The current owner of Glenlair, a great admirer of Maxwell, has gone to great lengths to have the remains of the house made safe for visitors, and has established a visitor center in the original porch. At the time of this writing, the house was temporarily closed for further restoration.

  5. Heaviside did not include this eulogy of Maxwell in any of his publications, but one of his American followers, Ernst J. Berg, recorded it in “Oliver Heaviside, a Sketch of His Work and Some Reminiscences of His Later Years,” published in the Journal of the American Academy of Sciences in 1930, and a fuller version can also be found in Rollo Appleyard's Pioneers of Electrical Communication, page 257.

  CHAPTER SIXTEEN: THE MAXWELLIANS

  Our search for Einstein's statement “One scientific epoch ended and another began with James Clerk Maxwell” has been unfruitful, but the quote can be found, for example, on the websites of King's College, London, and the National High Magnetic Field Laboratory, Florida State University.

  Feynman's observation on the historical importance of Maxwell's discovery of the laws of electrodynamics is from The Feynman Lectures on Physics, by Richard Feynman, Robert Leighton, and Matthew Sands, vol. 2, chap. 1, p. 11 (1964).

  1. Heaviside wrote of his first sight of Maxwell's Treatise on Electricity and Magnetism to a French admirer, Joseph Bethenod, in 1918, and Bethenod included it (translated into French) in an obituary to Heaviside published in Annales des postes télégraphes et téléphones in 1925, pages 521 to 538. The original letter has not survived, and we are greatly indebted to Paul J. Nahin for translating the French v
ersion back into English and including it in his book Oliver Heaviside: Sage in Solitude. Both Bethenod and Nahin were careful to translate as literally as possible, so we can be confident that the words are close to Heaviside's own.

  2. Heaviside compared himself to “Old Teufelsdröckh” in a July 1908 letter to Joseph Larmor. Diogenes Teufelsdröckh was Thomas Carlyle's alter ego in his satirical book Sartor Resartus. Teufelsdröckh was an eccentric professor whose “philosophy of clothes” provided metaphors for the author's own thoughts on life.

  3. Heaviside made this reference to the sea serpent in his article “The Earth as a Return Conductor,” first published in the Electrician and reprinted in as article 23 of his Electrical Papers, vol. 1, pages 190–95.

  4. Fitzgerald wrote of Heaviside's achievement in condensing Maxwell's theory in an 1893 Electrician review of Heaviside's collected Electrical Papers.

  5. The (scalar) electric or magnetic potential at any point in the field is the energy needed to move a unit electric charge or a unit magnetic pole there from an infinite distance away. Maxwell used fluid pressure as an analogy for these potentials in his fluid model. There is also the magnetic vector potential that Maxwell identified with Faraday's electrotonic state; its curl gives the magnetic flux density at any point in the field.

  6. Heaviside explained his view of the potentials in an article that can be found in his Electrical Papers, vol. 2, pages 483–85.

  7. μ and ε are the ratio of the electric and magnetic flux densities to their respective field forces. In empty space they are usually given the subscript 0, omitted here for simplicity.

  8. In the presence of electric charge density ρ and current density J, the equations become:

  div E = ρ/ε

  div H = 0

  curl E = − μ∂H/∂t

  curl H = ε∂E/∂t + J

  The electric flux density D (= εE)—Maxwell called it displacement—may be used rather than the electric field intensity vector E, and today the magnetic flux density B (= μH) is now generally used rather than the magnetic field intensity vector H. The constants ε and μ depend on the medium. If the medium is nonisotropic, they are replaced by tensors and the equations need to be adjusted accordingly.