A Treatise on Electricity and Magnetism

A Treatise on Electricity and Magnetism is a two-volume treatise on electromagnetism written by James Clerk Maxwell in 1873. Maxwell was revising the Treatise for a second edition when he died in 1879. The revision was completed by William Davidson Niven for publication in 1881. A third edition was prepared by J. J. Thomson for publication in 1892.

According to one historian,

The Treatise was notoriously hard to read; it teemed with ideas but lacked the clear focus and orderly presentation that might have enabled it to win converts more readily. Rather than simply expounding his own system, Maxwell had set out to write a comprehensive treatise on electrical science, and so he had allowed his own new distinctive ideas, notably that of the displacement current, to be almost buried under long accounts of miscellaneous phenomena discussed from several points of view. Except for a fuller treatment of the Faraday effect (in which he again invoked the molecular vortices), Maxwell added little to his earlier work on the electromagnetic theory of light; he said nothing, for example, about how electromagnetic waves might be generated, nor did he attempt to derive laws governing reflection and refraction.[1]

Maxwell introduced the use of vector fields, and his labels have been perpetuated:

A (vector potential), B (magnetic induction), C (electric current), D (displacement), E (electric field – Maxwell's electromotive intensity), F (mechanical force), H (magnetic field – Maxwell's magnetic force).[2]

Maxwell's work is considered an exemplar of rhetoric of science:[3]

Lagrange's equations appear in the Treatise as the culmination of a long series of rhetorical moves, including (among others) Green's theorem, Gauss's potential theory and Faraday's lines of force – all of which have prepared the reader for the Lagrangian vision of a natural world that is whole and connected: a veritable sea change from Newton's vision.


A Treatise on Electricity and Magnetism Volume 1 003
Title, author and publisher page from first volume of Maxwell's masterwork

Preliminary. On the Measurement of Quantities.

PART I. Electrostatics.

  1. Description of Phenomena.
  2. Elementary Mathematical Theory of Electricity.
  3. On Electrical Work and Energy in a System of Conductors.
  4. General Theorems.
  5. Mechanical Action Between Two Electrical Systems.
  6. Points and Lines of Equilibrium.
  7. Forms of Equipotential Surfaces and Lines of Flow.
  8. Simple Cases of Electrification.
  9. Spherical Harmonics.
  10. Confocal Surfaces of the Second Degree.
  11. Theory of Electric Images.
  12. Conjugate Functions in Two Dimensions.
  13. Electrostatic Instruments.

PART II. Electrokinematics.

  1. The Electric Current.
  2. Conduction and Resistance.
  3. Electromotive Force Between Bodies in Contact.
  4. Electrolysis.
  5. Electrolytic Polarization.
  6. Mathematical Theory of the Distribution of Electric Currents.
  7. Conduction in Three Dimensions.
  8. Resistance and Conductivity in Three Dimensions.
  9. Conduction through Heterogeneous Media.
  10. Conduction in Dielectrics.
  11. Measurement of the Electric Resistance of Conductors.
  12. Electric Resistance of Substances.

PART III Magnetism

  1. Elementary Theory of Magnetism.
  2. Magnetic Force and Magnetic Induction.
  3. Particular Forms of Magnets.
  4. Induced Magnetization.
  5. Magnetic Problems.
  6. Weber's Theory of Magnetic Induction.
  7. Magnetic Measurements.
  8. Terrestrial Magnetism.

Part IV. Electromagnetism.

  1. Electromagnetic Force.
  2. Mutual Action of Electric Currents.
  3. Induction of Electric Currents.
  4. Induction of a Current on Itself.
  5. General Equations of Dynamics.
  6. Application of Dynamics to Electromagnetism.
  7. Electrokinetics.
  8. Exploration of the Field by means of the Secondary Circuit.
  9. General Equations.
  10. Dimensions of Electric Units.
  11. Energy and Stress.
  12. Current-Sheets.
  13. Parallel Currents.
  14. Circular Currents.
  15. Electromagnetic Instruments.
  16. Electromagnetic Observations.
  17. Electrical Measurement of Coefficients of Induction.
  18. Determination of Resistance in Electromagnetic Measure.
  19. Comparison of Electrostatic With Electromagnetic Units.
  20. Electromagnetic Theory of Light.
  21. Magnetic Action on Light.
  22. Electric Theory of Magnetism.
  23. Theories of Action at a distance.


On April 24, 1873, Nature announced the publication with an extensive description and much praise.[4] When the second edition was published in 1881, George Chrystal wrote the review for Nature.[5]

Pierre Duhem published a critical essay outlining mistakes he found in Maxwell's Treatise.[6] Duhem's book was reviewed in Nature.[7]


Hermann von Helmholtz (1881): "Now that the mathematical interpretations of Faraday's conceptions regarding the nature of electric and magnetic force has been given by Clerk Maxwell, we see how great a degree of exactness and precision was really hidden behind Faraday's words…it is astonishing in the highest to see what a large number of general theories, the mechanical deduction of which requires the highest powers of mathematical analysis, he has found by a kind of intuition, with the security of instinct, without the help of a single mathematical formula."[8]

Oliver Heaviside (1893):”What is Maxwell's theory? The first approximation is to say: There is Maxwell's book as he wrote it; there is his text, and there are his equations: together they make his theory. But when we come to examine it closely, we find that this answer is unsatisfactory. To begin with, it is sufficient to refer to papers by physicists, written say during the first twelve years following the first publication of Maxwell's treatise to see that there may be much difference of opinion as to what his theory is. It may be, and has been, differently interpreted by different men, which is a sign that is not set forth in a perfectly clear and unmistakable form. There are many obscurities and some inconsistencies. Speaking for myself, it was only by changing its form of presentation that I was able to see it clearly, and so as to avoid the inconsistencies. Now there is no finality in a growing science. It is, therefore, impossible to adhere strictly to Maxwell's theory as he gave it to the world, if only on account of its inconvenient form.[9][10]

Alexander Macfarlane (1902): "This work has served as the starting point of many advances made in recent years. Maxwell is the scientific ancestor of Hertz, Hertz of Marconi and all other workers at wireless telegraphy.[11]

Oliver Lodge (1907) "Then comes Maxwell, with his keen penetration and great grasp of thought, combined with mathematical subtlety and power of expression; he assimilates the facts, sympathizes with the philosophic but untutored modes of expression invented by Faraday, links the theorems of Green and Stokes and Thomson to the facts of Faraday, and from the union rears the young modern science of electricity..."[12]

E. T. Whittaker (1910): "In this celebrated work is comprehended almost every branch of electric and magnetic theory, but the intention of the writer was to discuss the whole from a single point of view, namely, that of Faraday, so that little or no account was given of the hypotheses that had been propounded in the two preceding decades by the great German electricians...The doctrines peculiar to Maxwell ... were not introduced in the first volume, or in the first half of the second."[13]

Albert Einstein (1931): "Before Maxwell people conceived of physical reality – in so far as it is supposed to represent events in nature – as material points, whose changes consist exclusively of motions, which are subject to total differential equations. After Maxwell they conceived physical reality as represented by continuous fields, not mechanically explicable, which are subject to partial differential equations. This change in the conception of reality is the most profound and fruitful one that has come to physics since Newton; but it has at the same time to be admitted that the program has by no means been completely carried out yet."[14]

Richard P. Feynman (1964): "From a long view of the history of mankind—seen from, say, ten thousand years from now—there can be little doubt that the most significant event of the 19th century will be judged as Maxwell's discovery of the laws of electrodynamics. The American Civil War will pale into provincial insignificance in comparison with this important scientific event of the same decade."[15]

L. Pearce Williams (1991): "In 1873, James Clerk Maxwell published a rambling and difficult two-volume Treatise on Electricity and Magnetism that was destined to change the orthodox picture of physical reality. This treatise did for electromagnetism what Newton's Principia had done for classical mechanics. It not only provided the mathematical tools for the investigation and representation of the whole of electromagnetic theory, but it altered the very framework of both theoretical and experimental physics. Although the process had been going on throughout the nineteenth century, it was this work that finally displaced action at a distance physics and substituted the physics of the field."[16]

Mark P. Silverman (1998) "I studied the principles on my own – in this case with Maxwell's Treatise as both my inspiration and textbook. This is not an experience that I would necessarily recommend to others. For all his legendary gentleness, Maxwell is a demanding teacher, and his magnum opus is anything but coffee-table reading...At the same time, the experience was greatly rewarding in that I had come to understand, as I realized much later, aspects of electromagnetism that are rarely taught at any level today and that reflect the unique physical insight of their creator.[2]:202

Andrew Warwick (2003): "In developing the mathematical theory of electricity and magnetism in the Treatise, Maxwell made a number of errors, and for students with only a tenuous grasp of the physical concepts of basic electromagnetic theory and the specific techniques to solve some problems, it was extremely difficult to discriminate between cases where Maxwell made an error and cases where they simply failed to follow the physical or mathematical reasoning."[17]

See also


  1. ^ Bruce J. Hunt (1991) The Maxwellians, page 13
  2. ^ a b Mark P. Silverman (1998) Waves and Grains: reflections on light and learning, pages 205, 6, Princeton University Press ISBN 0-691-00113-8
  3. ^ Thomas K. Simpson (2010) Maxwell's Mathematical Rhetoric: rethinking the Treatise on Electricity and Magnetism, page xiii, Santa Fe, New Mexico: Green Lion Press
  4. ^ "Review: A Treatise on Electricity and Magnetism" from Nature 24 April 1873
  5. ^ George Chrystal (1882) Review: 2nd edition, link from Nature
  6. ^ Pierre Duhem (1902). Les Théories Électriques de J. Clerk Maxwell: Étude Historique et Critique. Paris: A. Hermann
  7. ^ W. McF. Orr (1902) "A French Critic of Maxwell", Nature 17 April 1902
  8. ^ Hermann Helmholtz (1881) "On the modern development of Faraday's conception of electricity", Faraday Lecture at the Royal Society
  9. ^ Oliver Heaviside (1893) Electromagnetic Theory, v. 1, Preface, p. vii, link from Internet Archive
  10. ^ The Maxwellians, page 201
  11. ^ Alexander Macfarlane (1916) Lectures on Ten British Physicists of the Nineteenth Century, link from Internet Archive
  12. ^ Oliver Lodge (1907) Modern Views of Electricity, 3rd edition, page 24, Macmillan & Company
  13. ^ E. T. Whittaker (1910) History of Theories of the Aether and Electricity, page 300, link from Internet Archive
  14. ^ Einstein, Albert (1931). "Maxwell's Influence On The Evolution Of The Idea Of Physical Reality". James Clerk Maxwell: A Commemoration Volume. C.U.P. Archived from the original on 14 July 2014. Retrieved 7 July 2014.
  15. ^ Bruce J. Hunt (1991) The Maxwellians, page 1, Cornell University Press ISBN 0-8014-2641-3. Source The Feynman Lectures on Physics (1964) 2:1.11
  16. ^ L. Pearce Williams (1991) Preface to The Maxwellians
  17. ^ Andrew Warwick (2003) Masters of Theory: Cambridge and the Rise of Mathematical Physics, chapter 6: Making sense of Maxwell's Treatise on Electricity and Magnetism in Mid-Victorian Cambridge, pp. 286–356, quote p. 297, University of Chicago Press ISBN 0-226-87374-9

External links

1873 in science

The year 1873 in science and technology involved some significant events, listed below.

1873 in the United Kingdom

Events from the year 1873 in the United Kingdom.

A Dynamical Theory of the Electromagnetic Field

"A Dynamical Theory of the Electromagnetic Field" is a paper by James Clerk Maxwell on electromagnetism, published in 1865. In the paper, Maxwell derives an electromagnetic wave equation with a velocity for light in close agreement with measurements made by experiment, and deduces that light is an electromagnetic wave.

Action at a distance

In physics, action at a distance is the concept that an object can be moved, changed, or otherwise affected without being physically touched (as in mechanical contact) by another object. That is, it is the nonlocal interaction of objects that are separated in space.

This term was used most often in the context of early theories of gravity and electromagnetism to describe how an object responds to the influence of distant objects. For example, Coulomb's law and Newton's law of universal gravitation are such early theories.

More generally "action at a distance" describes the failure of early atomistic and mechanistic theories which sought to reduce all physical interaction to collision. The exploration and resolution of this problematic phenomenon led to significant developments in physics, from the concept of a field, to descriptions of quantum entanglement and the mediator particles of the Standard Model.

Archibald Smith

Archibald Smith of Jordanhill (10 August 1813, in Greenhead, North Lanarkshire – 26 December 1872, in London) was a Scots-born barrister and amateur mathematician.


Capacitance is the ratio of the change in an electric charge in a system to the corresponding change in its electric potential. There are two closely related notions of capacitance: self capacitance and mutual capacitance. Any object that can be electrically charged exhibits self capacitance. A material with a large self capacitance holds more electric charge at a given voltage than one with low capacitance. The notion of mutual capacitance is particularly important for understanding the operations of the capacitor, one of the three elementary linear electronic components (along with resistors and inductors).

The capacitance is a function only of the geometry of the design (e.g. area of the plates and the distance between them) and the permittivity of the dielectric material between the plates of the capacitor. For many dielectric materials, the permittivity and thus the capacitance, is independent of the potential difference between the conductors and the total charge on them.

The SI unit of capacitance is the farad (symbol: F), named after the English physicist Michael Faraday. A 1 farad capacitor, when charged with 1 coulomb of electrical charge, has a potential difference of 1 volt between its plates. The reciprocal of capacitance is called elastance.

Cavendish Professor of Physics

The Cavendish Professorship is one of the senior faculty positions in physics at the University of Cambridge. It was founded on 9 February 1871 alongside the famous Cavendish Laboratory, which was completed three years later. William Cavendish, 7th Duke of Devonshire endowed both the professorship and laboratory in honor of his relative, chemist and physicist Henry Cavendish.

Charge pump

A charge pump is a kind of DC to DC converter that uses capacitors for energetic charge storage to raise or lower voltage. Charge-pump circuits are capable of high efficiencies, sometimes as high as 90–95%, while being electrically simple circuits.

Classical electromagnetism

Classical electromagnetism or classical electrodynamics is a branch of theoretical physics that studies the interactions between electric charges and currents using an extension of the classical Newtonian model. The theory provides a description of electromagnetic phenomena whenever the relevant length scales and field strengths are large enough that quantum mechanical effects are negligible. For small distances and low field strengths, such interactions are better described by quantum electrodynamics.

Fundamental physical aspects of classical electrodynamics are presented in many texts, such as those by Feynman, Leighton and Sands, Griffiths, Panofsky and Phillips, and Jackson.

Electromagnetic induction

Electromagnetic or magnetic induction is the production of an electromotive force (i.e., voltage) across an electrical conductor in a changing magnetic field.

Michael Faraday is generally credited with the discovery of induction in 1831, and James Clerk Maxwell mathematically described it as Faraday's law of induction. Lenz's law describes the direction of the induced field. Faraday's law was later generalized to become the Maxwell–Faraday equation, one of the four Maxwell equations in his theory of electromagnetism.

Electromagnetic induction has found many applications, including electrical components such as inductors and transformers, and devices such as electric motors and generators.

Electrostatic levitation

Electrostatic levitation is the process of using an electric field to levitate a charged object and counteract the effects of gravity. It was used, for instance, in Robert Millikan's oil drop experiment and is used to suspend the gyroscopes in Gravity Probe B during launch.

Due to Earnshaw's theorem no static arrangement of classical electrostatic fields can be used to stably levitate a point charge. There is an equilibrium point where the two fields cancel, but it is an unstable equilibrium. By using feedback techniques it is possible to adjust the charges to achieve a quasi static levitation.

Faraday's law of induction

Faraday's law of induction (shortly called Faraday's law throughout this document) is a basic law of electromagnetism predicting how a magnetic field will interact with an electric circuit to produce an electromotive force (EMF)—a phenomenon called electromagnetic induction. It is the fundamental operating principle of transformers, inductors, and many types of electrical motors, generators and solenoids.The Maxwell–Faraday equation (listed as one of Maxwell's equations) describes the fact that a spatially varying (and also possibly time-varying, depending on how a magnetic field varies in time) electric field always accompanies a time-varying magnetic field, while Faraday's law states that there is EMF (electromotive force, defined as electromagnetic work done on a unit charge when it has traveled one round of a conductive loop) on the conductive loop when the magnetic flux through the surface enclosed by the loop varies in time.

Historically, Faraday's law had been discovered and one aspect of it (transformer EMF) was formulated as the Maxwell–Faraday equation later. Interestingly, the equation of Faraday's law can be derived by the Maxwell–Faraday equation (describing transformer EMF) and the Lorentz force (describing motional EMF). The integral form of the Maxwell–Faraday equation describes only the transformer EMF, while the equation of Faraday's law describes both the transformer EMF and the motional EMF.

History of Maxwell's equations

In electromagnetism, one of the fundamental fields of physics, the introduction of Maxwell's equations (mainly in "A Dynamical Theory of the Electromagnetic Field") was one of the most important aggregations of empirical facts in the history of physics. It took place in the nineteenth century, starting from basic experimental observations, and leading to the formulations of numerous mathematical equations, notably by Charles-Augustin de Coulomb, Hans Christian Ørsted, Carl Friedrich Gauss, Jean-Baptiste Biot, Félix Savart, André-Marie Ampère, and Michael Faraday. The apparently disparate laws and phenomena of electricity and magnetism were integrated by James Clerk Maxwell, who published an early form of the equations, which modify Ampère's circuital law by introducing a displacement current term. He showed that these equations imply that light propagates as electromagnetic waves. His laws were reformulated by Oliver Heaviside in the more modern and compact vector calculus formalism he independently developed. Increasingly powerful mathematical descriptions of the electromagnetic field were developed, continuing into the twentieth century, enabling the equations to take on simpler forms by advancing more sophisticated mathematics.

James Clerk Maxwell

James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish scientist in the field of mathematical physics. His most notable achievement was to formulate the classical theory of electromagnetic radiation, bringing together for the first time electricity, magnetism, and light as different manifestations of the same phenomenon. Maxwell's equations for electromagnetism have been called the "second great unification in physics" after the first one realised by Isaac Newton.

With the publication of "A Dynamical Theory of the Electromagnetic Field" in 1865, Maxwell demonstrated that electric and magnetic fields travel through space as waves moving at the speed of light. Maxwell proposed that light is an undulation in the same medium that is the cause of electric and magnetic phenomena. The unification of light and electrical phenomena led to the prediction of the existence of radio waves.

Maxwell helped develop the Maxwell–Boltzmann distribution, a statistical means of describing aspects of the kinetic theory of gases. He is also known for presenting the first durable colour photograph in 1861 and for his foundational work on analysing the rigidity of rod-and-joint frameworks (trusses) like those in many bridges.

His discoveries helped usher in the era of modern physics, laying the foundation for such fields as special relativity and quantum mechanics. Many physicists regard Maxwell as the 19th-century scientist having the greatest influence on 20th-century physics. His contributions to the science are considered by many to be of the same magnitude as those of Isaac Newton and Albert Einstein. In the millennium poll—a survey of the 100 most prominent physicists—Maxwell was voted the third greatest physicist of all time, behind only Newton and Einstein. On the centenary of Maxwell's birthday, Einstein described Maxwell's work as the "most profound and the most fruitful that physics has experienced since the time of Newton".

Lenz's law

Lenz's law (pronounced /ˈlɛnts/), named after the physicist Emil Lenz who formulated it in 1834, states that the direction of the current induced in a conductor by a changing magnetic field is such that the magnetic field created by the induced current opposes the initial changing magnetic field. Or as informally, yet concisely summarised by D.J. Griffiths:

Lenz's law is shown by the negative sign in Faraday's law of induction:

which indicates that the induced electromotive force and the rate of change in magnetic flux have opposite signs. It is a qualitative law that specifies the direction of induced current but says nothing about its magnitude. Lenz's law explains the direction of many effects in electromagnetism, such as the direction of voltage induced in an inductor or wire loop by a changing current, or why eddy currents exert a drag force on moving objects in a magnetic field.

Lenz's law can be seen as analogous to Newton's third law in classic mechanics.

For a rigorous mathematical treatment, see electromagnetic induction and Maxwell's equations.

List of textbooks in electromagnetism

Following is a list of notable textbooks in electromagnetism.

The Maxwellians

The Maxwellians is a book by Bruce J. Hunt, published in 1991 by Cornell University Press; a paperback edition appeared in 1994, and the book was reissued in 2005. It chronicles the development of electromagnetic theory in the years after the publication of A Treatise on Electricity and Magnetism by James Clerk Maxwell. The book draws heavily on the correspondence and notebooks as well as the published writings of George Francis FitzGerald, Oliver Lodge, Oliver Heaviside, Heinrich Hertz, and Joseph Larmor.

The book has nine chapters; their titles and section headings are:

FitzGerald and Maxwell’s Theory

FitzGerald and the Dublin School, Maxwell’s Theory, Reflection and Refraction, FitzGerald’s Accomplishment.

FitzGerald, Lodge, and Electromagnetic Waves

Oliver Lodge, Maxwell and Electromagnetic Waves, Lodge and "Electromagnetic Light", FitzGerald and "The Impossibility . . .", The Undetected Waves.

Heaviside the Telegrapher

Oliver Heaviside, Cable Empire, At Newcastle, Cables and Field Theory, Heaviside on Propagation, Turning to Maxwell.

Ether Models and the Vortex Sponge

Models, Wheels and Bands, Charging Displacement, "We Find Ourselves in a Factory", The Vortex Sponge, "Mathematical Machinery".

"Maxwell Redressed"

Energy Paths, Model Research, "When Energy Goes from Place to Place . . .", Heaviside’s Equations.

Waves on Wires

"Beams of Dark Light", Loading and the Distortionless Circuit, Suppression, Campaigning for Recognition, Lightning.

Bath, 1888

Hertz’s Waves, Reception, "The Murder of Ψ", Practice vs Theory.

The Maxwellian Heyday

Strengthening the Links, The Origins of the FitzGerald Contraction, What Is Maxwell’s Theory?

The Advent of the Electron

Joseph Larmor and the Rotational Ether, Inventing Electrons, "Larmor’s Force," Assimilating Electrons, Conclusion.



From Maxwell’s Equations to "Maxwell's Equations".

Abbreviations, Bibliography (10 pages), Index (6 pages).


A treatise is a formal and systematic written discourse on some subject, generally longer and treating it in greater depth than an essay, and more concerned with investigating or exposing the principles of the subject.

Éleuthère Mascart

Éleuthère Élie Nicolas Mascart (20 February 1837 – 24 August 1908) was a noted French physicist, a researcher in optics, electricity, magnetism, and meteorology.

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