A Treatise on Electricity and Magnetism

Title page of Volume 1, 1873

A Treatise on Electricity and Magnetism from 1873 is a textbook on electricity and magnetism and the major work of James Clerk Maxwell on his theory of electromagnetism .

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Maxwell had his theory of electromagnetism, which is based on the concept of field and lines of force by Michael Faraday , in his essays On Faradays lines of force (1856), On physical lines of force (1861/62) and in A dynamical theory of the electromagnetic field published in 1865, but it was not well known or recognized even in England. The work of William Thomson from the 1840s and 1850s, who at that time already brought Faraday's considerations into mathematical form and used the mathematical methods of the French and continental European schools, of Carl Friedrich Gauss and George Green , which Maxwell continued , also had a great influence on Maxwell .

Further influences, which also strongly influenced the structure of the book, were the fact that Maxwell was involved in a committee for the establishment of a resistance standard from 1862, which brought him into close contact with the experiment. The committee was founded by Thomson to cope with the rapidly developing telegraph and electrical industries. A second influence was the renewal of curricula at Cambridge and other universities to include electricity and magnetism, which required a new textbook. Maxwell himself was one of the examiners of the Tripos exams from 1866 , which he expanded to include new physical areas (electricity, magnetism, heat). This was reflected in the detailed treatment of exercise examples in the Treatise. Maxwell became head of the newly established Cavendish Laboratory in Cambridge and professor of experimental physics in 1871, and was regarded in Great Britain as the leading authority on the new field of electricity alongside Thomson. Maxwell did not deal with the actual innovations of his own theory of electromagnetism until the second half of the second volume of his Treatise.

As a textbook author, he agreed with Thomson and Peter Guthrie Tait , who in 1867 published their then innovative textbook on theoretical physics (Treatise on Natural Philosophy), which dealt with kinematics and dynamics, but left electromagnetism to Maxwell's treatment. Thomson and Tait systematically considered the representation with extremal principles according to Lagrange and Hamilton, which was also reflected in Maxwell's Treatise, which also deals with the Lagrange formalism (Volume 2, Chapter 5). Maxwell's emphasis is on theoretical treatment, but he also goes into instruments and experiments. He grouped the relevant sections in such a way that they were accessible to experimenters and engineers without having to work through the more theoretical chapters. Maxwell covers a variety of subjects such as geomagnetism, electrolysis, measuring instruments and procedures (he described e.g. the Maxwell bridge ) or systems of units of measure and a separate chapter is the electromagnetic theory of light (Volume 2, Chapter 20) and the Effect of magnetic fields on light (Volume 2, Chapter 21). The last chapter (Volume 2, Chapter 23) deals with the long-range theories of electromagnetism, which were especially developed in Germany (Gauß, Wilhelm Weber, Carl Neumann, Bernhard Riemann).

When Maxwell died, he was working on the second edition, which he was able to get ready for printing up to the first nine chapters (they were a substantially rewritten version). The remainder, edited by William Davidson Niven , a collaborator at Maxwell in Cambridge, was more or less a reprint of the first edition and appeared in 1881. A third edition was published by J. J. Thomson in 1891, since Niven was unable to do so at the time. In addition to explanations and references to more recent developments, he also printed the treatment of the self-induction of a coil from the essay from 1865, as this was missing in the Treatise.

In 1872, Cambridge University Press proposed Maxwell to write an abridged version of the Treatise, which he was unable to complete. The book was published posthumously in 1881 by Maxwell's former collaborator William Garnett William Davidson Niven ( Elementary treatise on electricity ). They used material from the Treatise for the unfinished parts.

According to Achard, the commercial success of the Treatise was primarily based on the detailed treatment of the phenomena of electricity and magnetism, but Maxwell's electromagnetic theory was not even by contemporaries such as William Thomson (who pursued his own mechanical etheric ideas) and Peter Guthrie Tait accepted. Tait did not even recognize all innovations like the introduction of the displacement current and was against a connection of electricity and light. Only the further research of the Maxwellians , which also included students from Niven (JJ Thomson, John Henry Poynting ) and those who were not at all connected with the Maxwell School, such as Oliver Lodge , George Francis FitzGerald and the autodidact Oliver Heaviside led into Great Britain to gradually accept the theory. The detection of electromagnetic waves by Heinrich Hertz from 1886 onwards was of particular importance. Hermann von Helmholtz had given him the task of verifying Maxwell's theory in 1879, but Hertz initially failed so that he let the problem rest. In 1884 he had already supported Maxwell's theory in a theoretical essay and presented his simplified form of Maxwell's equations in 1890. Helmholtz himself was one of the few who received Maxwell's theory on the continent early on (1870).

Maxwell's equations

The Maxwell equations in the form peeled off by Oliver Heaviside and others consist of four equations, which Maxwell has under a series of other equations (or can be easily derived from), which Maxwell as basic equations in his Treatise and in his essay of 1865 turned out. Maxwell specified them in the form of coordinates and in a pre-form of the vector notation, the quaternion formalism, whereby he treated the vector and scalar part of the quaternions separately and thus came close to the later vector form. In summary, the equations are listed in Chapter 9 ( General equations of the electromagnetic field ) of Part 4 of the second volume (the letter assignment corresponds to that of Maxwell).

${\displaystyle \mathbf {B} }$is in the following the magnetic induction (today magnetic flux density ), the magnetic force according to Maxwell (today magnetic field strength), the electric field, the displacement (today electrical flux density ), the current density (without displacement current) and the mechanical force. Many of Maxwell's notations and symbols later became established. ${\displaystyle \mathbf {H} }$ ${\displaystyle \mathbf {E} }$ ${\displaystyle \mathbf {D} }$ ${\displaystyle \mathbf {j_{0}} }$ ${\displaystyle \mathbf {F} }$

• (A) Definition of the vector potential (with Maxwell electromagnetic impulse, electromagnetic momentum ): This means that the magnetic field is free of sources, one of Maxwell's laws. ${\displaystyle \mathbf {A} }$ ${\displaystyle \mathbf {B} =\nabla \times \mathbf {A} }$

• (B) The electromotive force on a conductor moving in a magnetic field (speed ) according to Maxwell: According to Maxwell, the last term can in a certain sense be called electrical potential. The law of induction is contained in this equation . ${\displaystyle \mathbf {v} }$ ${\displaystyle \mathbf {E} =\mathbf {v} \times \mathbf {B} -{\frac {\partial \mathbf {A} }{\partial t}}-\nabla \phi }$

• (C) Equation for the electromagnetic force: ${\displaystyle \mathbf {F} =\mathbf {j} \times \mathbf {B} }$

He also gives a more general form for this:

${\displaystyle \mathbf {F} =\mathbf {j} \times \mathbf {B} -e\,\nabla \phi -m\,\nabla \Omega }$

with the electrical potential and the magnetic potential (according to Maxwell with the restriction that it exists) and the electrical charges and magnetic pole strengths as pre-factors. ${\displaystyle \phi }$ ${\displaystyle \Omega }$

• (D) with the magnetization (at Maxwell intensity of the magnetization and marked with ). ${\displaystyle \mathbf {B} =\mathbf {H} +4\pi \,\mathbf {M} }$ ${\displaystyle \mathbf {M} }$ ${\displaystyle \mathbf {I} }$

• (E) Ampère's law : ${\displaystyle \nabla \times \mathbf {H} =4\pi \,\mathbf {j} }$

• (F) The equation of electrical displacement: (using instead of the variable ). ${\displaystyle \mathbf {D} =\varepsilon \,\mathbf {E} }$ ${\displaystyle \varepsilon }$ ${\displaystyle {\frac {K}{4\pi }}}$

• (G) Ohm's law: ${\displaystyle \sigma \,\mathbf {E} =\mathbf {j_{0}} }$

• (H) Consideration of the displacement current: ${\displaystyle \mathbf {j} =\mathbf {j_{0}} +{\frac {\partial \mathbf {D} }{\partial t}}}$

• (I) (which follows immediately by inserting (F), (G) into (H)). ${\displaystyle \mathbf {j} =\left(\sigma +\varepsilon \,{\frac {\partial }{\partial t}}\right)\,\mathbf {E} }$

• (J) Gauss's Law: This is another of Maxwell's Laws . ${\displaystyle \nabla \cdot \mathbf {D} =\rho }$

• (K) is used to define an areal density of electricity at the interface between two media

• (L) ${\displaystyle \mathbf {B} =\mu \,\mathbf {H} }$

The continuity equation, which he added to his system of equations in 1865, is missing (it is dealt with in the first volume, paragraph 295).

He had already dealt with the integral forms of the equations now called after Maxwell in various chapters of the Treatise.

Contents of volume 1

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.

Contents of volume 2

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.

expenditure

• Treatise on Electricity and Magnetism . 2 volumes. Clarendon Press, Oxford 1873. The first volume had 425 pages, the second 444 pages, the circulation was 1500 copies. ( Wikisource )
  • The second edition of 1881 was published by WD Niven, the third edition of 1891 by JJ Thomson. A reprint of the third edition was published by Dover in 1954 and by Oxford University Press in 1998. Dover Reprint: Volume 1  - Internet Archive , Volume 2  - Internet Archive
• German translation of the 2nd edition by Max Bernhard Weinstein : Textbook of Electricity and Magnetism . 2 volumes. Julius Springer, Berlin 1883, Volume 1  - Internet Archive , Volume 2  - Internet Archive . In the German translation, Weinstein also added interim invoices and corrected errors in the original.
• French translation by G. Séligman-Lui (with notes by A. Cornu, A. Poitier, E. Sarrau): Traité d'électricité et de magnétisme , 2 volumes, Gauthier-Villars 1885, 1887. Reprinted by Jacques Gabay 1989

There were also Italian translations (1973, Turin: UTET) and a Russian one (Moscow, Nauka 1989).

An abridged version of Elementary treatise on electricity, edited by William Garnett, appeared in Clarendon Press in 1881 (2nd edition, 1888); archive.org . A German translation of this also appeared (The electricity in elementary treatment, Vieweg 1883, translator L. Graetz).

literature

• Franck Achard: James Clerk Maxwell, A Treatise on Electricity and Magnetism, First Edition (1873) . In: Ivor Grattan-Guinness (Ed.): Landmark writings in Western Mathematics, 1640-1940 . Elsevier 2005, Chapter 44, pp. 564-587
• F. Achard: La publication du Treatise on electricity and magnetism de James Clerk Maxwell , La Revue de Synthèse, Volume 119, 1998, pp. 511-544
• Jed Z. Buchwald : From Maxwell to microphysics: aspects of electromagnetic theory in the last quarter of the nineteenth century . University of Chicago Press 1985
• George Chrystal , Review of Treatise, 2nd Edition and Elementary Treatise in Nature, Jan. 12, 1882, pp. 237-240
• Olivier Darrigol : Electrodynamics from Ampère to Einstein . Oxford UP, 2000
• Bruce Hunt: The Maxwellians . Cornell University Press, 1991
• DM Siegel: Innovation in Maxwell's electromagnetic theory: Molecular vortices, displacement current, and light . Cambridge University Press, Cambridge 1991

References and comments

1. ↑ Niven writes in the foreword that he added some interim calculations and clarifying footnotes and corrected errors (although J. J. Thomson already supported him). According to Niven, Maxwell intended significant changes in the derivation of the coefficients for calculating the mutual induction of coils and in the treatment of electrical conduction in networks.
2. ↑ The footnotes on this turned out to be too extensive and were published separately by J. J. Thomson: Notes on recent researches in electricity and magnetism: intended as a sequel to Professor Clerk-Maxwell's Treatise on Electricity and Magnetism . Clarendon Press, Oxford 1893, archive.org
3. ^ Achard in: Grattan-Guinness (Ed.), Landmark writings in Western Mathematics, p. 583
4. ↑ Hertz, About the relationships between Maxwell's basic electrodynamic equations and the basic equations of opposing electrodynamics, Annalen der Physik, Volume 259, 1884, pp. 84-103
5. ↑ Contained in Hertz, Gesammelte Werke, Volume 2: Investigations into the propagation of electric force, Barth 1894
6. ↑ Maxwell, Treatise, 1873, Volume 2, p. 227ff, where equation (A) is already on p. 215, (B) p. 221, (C) p. 226, and the quaternionic representation from p. 236.
7. ^ Maxwell, Treatise, 1873, Vol. 2, p. 237