A dynamic theory of the electromagnetic field

" A dynamic theory of the electromagnetic field " (original title: " A dynamical theory of the electromagnetic field ") is the third book published in 1864 by James Clerk Maxwell on electrodynamics . It is the publication in which the original four formulas of Maxwell's equations appeared for the first time. He used the concept of the displacement current , which he had introduced in his publication On physical lines of force in 1861 , to derive the electromagnetic wave equation .

Maxwell's original equations

In Part III of A Dynamic Theory of the Electromagnetic Field , entitled "General equations of the electromagnetic field" (Orig .: "General equations of the electromagnetic field"), Maxwell formulated twenty equations. These were known as Maxwell's equations until the term was applied by Oliver Heaviside to the set of four vectorized equations that Maxwell published in On physical lines of force in 1884 .

Of the twenty equations, only Gaussian law (G) can be translated directly into modern form. The flux law is a fusion of Maxwell's laws of displacement current (A) and Ampère's law (C) and was carried out by Maxwell himself in equation 112 in On physical lines of force . Another of the later Maxwell equations, the freedom from sources of the magnetic field, is a direct consequence of (B). The law of induction as the last of Maxwell's equations is contained in (D), which also contains a component that is intended to describe the forces acting on a conductor moving in a magnetic field. The law is not correctly formulated by Maxwell if the last term is supposed to be the electrostatic potential, as assumed by Maxwell. The usual form is obtained if one assumes that the conductor is at rest and one forms the rotation on both sides of equation (D) . Other laws listed by Maxwell, such as the law of continuity, which describes the conservation of charge, or Ohm's law are no longer included in the equations known today as Maxwell's equations. ${\displaystyle \phi }$

18 of the 20 original Maxwell's equations can be summarized into six equations by vectorization. Each vectorized equation corresponds to three original ones in component form. Together with the other two equations in modern vector notation, they form a set of eight equations:

(A) Consideration of the displacement current

${\displaystyle \mathbf {J} _{\mathrm {tot} }=\mathbf {J} +{\frac {\partial \mathbf {D} }{\partial t}}}$

(B) Definition of the magnetic potential

${\displaystyle \mu \mathbf {H} =\nabla \times \mathbf {A} }$

(C) Ampère's law

${\displaystyle \nabla \times \mathbf {H} =\mathbf {J} _{\mathrm {tot} }}$

(D) The electromotive force on a moving conductor according to Maxwell

${\displaystyle \mathbf {E} =\mu \mathbf {v} \times \mathbf {H} -{\frac {\partial \mathbf {A} }{\partial t}}-\nabla \phi }$

(E) The equation of electrical elasticity

${\displaystyle \mathbf {E} ={\frac {1}{\varepsilon }}\,\mathbf {D} }$

(F) Ohm's law

${\displaystyle \mathbf {E} ={\frac {1}{\sigma }}\,\mathbf {J} }$

(G) Gaussian law

${\displaystyle \nabla \cdot \mathbf {D} =\rho }$

(H) Equation of charge conservation (Maxwell's continuity equation)

${\displaystyle \nabla \cdot \mathbf {J} =-{\frac {\partial \rho }{\partial t}}}$

notation

${\displaystyle \mathbf {H} }$is the magnetic field (called "magnetic intensity" by Maxwell).

${\displaystyle \mathbf {J} }$is the electrical current density (with as the total current including the displacement current). ${\displaystyle \mathbf {J} _{\mathrm {tot} }}$

${\displaystyle \mathbf {D} }$is the electrical flux density (called “electrical displacement” by Maxwell).

${\displaystyle \rho }$ is the free charge density ("amount of free electricity" according to Maxwell).

${\displaystyle \mathbf {A} }$is the magnetic vector potential ("angular momentum" for Maxwell).

${\displaystyle \mathbf {E} }$is the electric field (at Maxwell " electromotive force ").

${\displaystyle \phi }$is the electrical potential .

${\displaystyle \sigma }$ is the electrical conductivity (Maxwell called the reciprocal value of the conductivity "specific resistance").

Maxwell did not involve general material properties; its original formulation assumed linear, isotropic and non-dispersive ε ( permittivity ) and μ ( permeability ). However, he discussed the possibility of anisotropic materials.

It is of particular interest that Maxwell inserted the term in his equation (D) for "electromotive force". This corresponds to the magnetic force per unit of charge that acts on a conductor moving with speed . The equation (D) could be understood as a description of the Lorentz force . It occurs for the first time in equation (77) in the publication On physical lines of force some time before Lorentz found this equation. Usually, however, one ascribes the first consideration of the forces on a moving point charge in a magnetic field to JJ Thomson (1881). He also derived an incorrect prefactor; Oliver Heaviside (1889) and Lorentz (1895) found the correct formula. Today the Lorentz force is not treated as a component of the Maxwell equations. ${\displaystyle \mu \mathbf {v} \times \mathbf {H} }$ ${\displaystyle \mathbf {v} }$

When Maxwell derived the electromagnetic wave equation in his 1864 paper, he used equation (D) instead of Faraday's law of electromagnetic induction, as found in textbooks today. However, Maxwell dropped the term in the derivation in equation (D) . ${\displaystyle \mu \mathbf {v} \times \mathbf {H} }$

Maxwell used a slightly modified list in the second volume of his Treatise on electricity and magnetism from 1873 (Chapter 9), but the equations from 1865 are essentially found there again. He also gives treatment in quaternion form, an alternative to vector notation that was popular in England at the time.

Light as an electromagnetic wave

James Clerk Maxwell: father of the theory of electromagnetism

A postcard from Maxwell to Peter Tait .

In A dynamical theory of the electromagnetic field , Maxwell uses the correction to Ampère's law from Part III of On physical lines of force . In Part VI of his publication (Chapter Electromagnetic Theory of Light ) from 1864, he combined the displacement current with other equations of electromagnetism and obtained a wave equation with a speed corresponding to the speed of light. He commented:

“The agreement of the results suggests that light and magnetism are caused by the same substance and that light moves through the field as an electromagnetic disturbance according to electromagnetic laws. (“The agreement of the results seems to show that light and magnetism are affections of the same substance, and that light is an electromagnetic disturbance propagated through the field according to electromagnetic laws.”) “

- Maxwell

Maxwell's derivation of the electromagnetic wave equation has been replaced in modern physics with a less laborious method, with a corrected version of Ampère's law and Faraday's law of electromagnetic induction.

The modern derivation of the electromagnetic wave equation in vacuum begins with the Heaviside form of Maxwell's equation. These are written in Si units:

${\displaystyle \nabla \cdot \mathbf {E} =0}$

${\displaystyle \nabla \times \mathbf {E} =-\mu _{0}{\frac {\partial \mathbf {H} }{\partial t}}}$

${\displaystyle \nabla \cdot \mathbf {H} =0}$

${\displaystyle \nabla \times \mathbf {H} =\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}}$

If we take the rotation of the rotational equations we get:

${\displaystyle \nabla \times \nabla \times \mathbf {E} =-\mu _{0}{\frac {\partial }{\partial t}}\nabla \times \mathbf {H} =-\mu _{0}\varepsilon _{0}{\frac {\partial ^{2}\mathbf {E} }{\partial t^{2}}}}$

${\displaystyle \nabla \times \nabla \times \mathbf {H} =\varepsilon _{0}{\frac {\partial }{\partial t}}\nabla \times \mathbf {E} =-\mu _{0}\varepsilon _{0}{\frac {\partial ^{2}\mathbf {H} }{\partial t^{2}}}}$

With the identity of the vector equations

${\displaystyle \nabla \times \left(\nabla \times \mathbf {V} \right)=\nabla \left(\nabla \cdot \mathbf {V} \right)-\nabla ^{2}\mathbf {V} }$

with as each of the spatial vector functions, we get the wave equations ${\displaystyle \mathbf {V} }$

${\displaystyle {\frac {\partial ^{2}\mathbf {E} }{\partial t^{2}}}\ -\ c^{2}\cdot \nabla ^{2}\mathbf {E} \ =\ 0}$

${\displaystyle {\frac {\partial ^{2}\mathbf {H} }{\partial t^{2}}}\ -\ c^{2}\cdot \nabla ^{2}\mathbf {H} \ =\ 0}$

With

${\displaystyle c={\frac {1}{\sqrt {\mu _{0}\varepsilon _{0}}}}=2{,}99792458\times 10^{8}}$ Meters per second

as the speed of light in a vacuum.

literature

• James Clerk Maxwell: A Dynamical Theory of the Electromagnetic Field , Philosophical Transactions of the Royal Society of London, Volume 155, 1865, pp. 459-512
  • Reprint: Thomas F. Torrance (Ed.): Maxwell, A Dynamical Theory of the Electromagnetic Field , Wipf and Stock, Eugene (Oregon) 1996
  • Reprinted in: WD Niven: The Scientific Papers of James Clerk Maxwell , Volume Vol. 1. Dover, New York 1952.
• Kevin Johnson: The electromagnetic field . In: James Clerk Maxwell - The Great Unknown . May 2002. Retrieved Sept. 7, 2009.

Individual evidence

1. ^ ^{A b c} James Clerk Maxwell: A dynamical theory of the electromagnetic field. in: Philosophical Transactions of the Royal Society of London , Vol. 155, pp. 459-512, 1865, doi : 10.1098 / rstl.1865.0008 (This article was included in a presentation by Maxwell to the Royal Society on December 8, 1864.)
2. ↑ ^{a b c d} James Clerk Maxwell: On physical lines of force. (PDF) In: Philosophical Magazine , 1861
3. ^ Whittaker, History of the theories of ether and electricity, Nelson, 1951, Volume 1, p. 259
4. ↑ Darrigol, Electrodynamics from Ampère to Einstein, Oxford UP, 2000, pp. 429ff
5. ^ Gerhard Bruhn, The Maxwell equations - from the original to the modern notation , TU Darmstadt