A Dynamical Theory of the Electromagnetic Field

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A Dynamical Theory of the Electromagnetic Field which was written in the year 1864, is the third of James Clerk Maxwell's papers concerned with electromagnetism. It is the paper in which the original set of eight Maxwell's equations first appeared. The concept of displacement current that he had introduced in his 1861 paper On Physical Lines of Force was utilized for the first time, in order to derive the electromagnetic wave equation.

Contents

[edit] The Original Eight 'Maxwell's Equations'

In PART III of 'A Dynamical Theory of the Electromagnetic Field' which is entitled "GENERAL EQUATIONS OF THE ELECTROMAGNETIC FIELD' [1] (page 480 of the article and page 2 of the pdf link), Maxwell formulated eight equations labelled (A) to (H). These eight equations were to become known as Maxwell's equations, until this term became applied instead to a set of four equations selected in 1884 by Oliver Heaviside, and which had all appeared in Maxwell's 1861 paper On Physical Lines of Force'.

Heaviside's versions of Maxwell's equations are distinct by virtue of the fact that that are written in modern vector notation. They actually only contain one of the original eight ie. equation (G) Gauss's Law. Another of Heaviside's four equations is an amalgamation of Maxwell's Law of Total Currents (equation (A)) with Ampère's Circuital Law (equation (C)). This amalgamation, which Maxwell himself had actually originally made at equation (112) in his 1861 paper 'On Physical Lines of Force', is the one that modifies Ampère's Circuital Law to include Maxwell's Displacement current.

All eight of the original Maxwell's equations will now be listed below in modern vector notation,


(A) The Law of Total Currents


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


(B) Definition of the Magnetic Vector Potential


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


(C) Ampère's Circuital Law


\nabla \times \mathbf{H} = \mu_0 \mathbf{J}_{tot}


(D) The Lorentz Force. Electric fields created by convection, induction, and by charges.


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


(E) The Electric Elasticity Equation


\mathbf{E} = \frac{1}{\epsilon} \mathbf{D}


(F) Ohm's Law


\mathbf{E} = \frac{1}{\sigma} \mathbf{J}


(G) Gauss's Law


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


(H) Equation of Continuity of Charge


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


Notation
\mathbf{H} is the magnetic field, which Maxwell called the "magnetic intensity".
\mathbf{J} is the electric current density (with \mathbf{J}_{tot} being the total current including displacement current).
\mathbf{D} is the displacement field (called the "electric displacement" by Maxwell).
ρ is the free charge density (called the "quantity of free electricity" by Maxwell).
\mathbf{A} is the magnetic vector potential (called the "angular impulse" by Maxwell).
\mathbf{E} is the electric field (called the "electromotive force" by Maxwell, not to be confused with the scalar quantity that is now called electromotive force).
φ is the electric potential (which Maxwell also called "electric potential").
σ is the electrical conductivity (Maxwell called the inverse of conductivity the "specific resistance", what is now called the resistivity).

Maxwell did not consider completely general materials; his initial formulation used linear, isotropic, nondispersive permittivity ε and permeability μ, although he also discussed the possibility of anisotropic materials.

It is of particular interest to note that Maxwell includes a \mu \mathbf{v} \times \mathbf{H} term in his expression for the "electromotive force" at equation (D) , which corresponds to the magnetic force per unit charge on a moving conductor with velocity \mathbf{v}. This means that equation (D) is effectively the Lorentz force. This equation first appeared at equation (77) in Maxwell's 1861 paper 'On Physical Lines of Force' quite some time before Lorentz thought of it. Nowadays, the Lorentz force sits alongside Maxwell's Equations as an additional electromagnetic equation that is not included as part of the set.

When Maxwell derives the electromagnetic wave equation in his 1864 paper, he uses equation (D) as opposed to using Faraday's law of electromagnetic induction as in modern textbooks. Maxwell however drops the \mu \mathbf{v} \times \mathbf{H} term from equation (D) when he is deriving the electromagnetic wave equation, and he considers the situation only from the rest frame.

[edit] Maxwell - First to propose that light is an electromagnetic wave

Father of Electromagnetic Theory
Father of Electromagnetic Theory
A postcard from Maxwell to Peter Tait.
A postcard from Maxwell to Peter Tait.

In his 1864 paper entitled A Dynamical Theory of the Electromagnetic Field, Maxwell utilized the correction to Ampère's Circuital Law that he had made in part III of his 1861 paper On Physical Lines of Force. In PART VI of his 1864 paper which is entitled 'ELECTROMAGNETIC THEORY OF LIGHT' [2] (page 497 of the article and page 9 of the pdf link), Maxwell combined displacement current with some of the other equations of electromagnetism and he obtained a wave equation with a speed equal to the speed of light. He commented,

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.

(see [3], page 499 of the article and page 1 of the pdf link)

Maxwell's derivation of the electromagnetic wave equation has been replaced in modern physics by a much less cumbersome method involving combining the corrected version of Ampère's Circuital Law with Faraday's law of electromagnetic induction.

To obtain the electromagnetic wave equation in a vacuum using the modern method, we begin with the modern 'Heaviside' form of Maxwell's equations. Using (SI units) in a vacuum, these equations are


\nabla \cdot \mathbf{E} = 0
\nabla \times \mathbf{E} = -\mu_o \frac{\partial \mathbf{H}} {\partial t}
\nabla \cdot \mathbf{H} = 0
\nabla \times \mathbf{H} =\varepsilon_o \frac{ \partial \mathbf{E}} {\partial t}

If we take the curl of the curl equations we obtain

\nabla \times \nabla \times \mathbf{E} = -\mu_o \frac{\partial } {\partial t} \nabla \times \mathbf{H} = -\mu_o \varepsilon_o \frac{\partial^2 \mathbf{E} } {\partial t^2}
\nabla \times \nabla \times \mathbf{H} = \varepsilon_o \frac{\partial } {\partial t} \nabla \times \mathbf{E} = -\mu_o \varepsilon_o \frac{\partial^2 \mathbf{H} } {\partial t^2}

If we note the vector identity

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

where \mathbf{V} is any vector function of space, we recover the wave equations

{\partial^2 \mathbf{E} \over \partial t^2} \ - \ c^2 \cdot \nabla^2 \mathbf{E} \ \ = \ \ 0
{\partial^2 \mathbf{H} \over \partial t^2} \ - \ c^2 \cdot \nabla^2 \mathbf{H} \ \ = \ \ 0

where

c = { 1 \over \sqrt{ \mu_o \varepsilon_o } } = 2.998 \times 10^8 meters per second

is the speed of light in free space.

[edit] See also

[edit] Further reading

  • Maxwell, James Clerk, "A Dynamical Theory of the Electromagnetic Field", Philosophical Transactions of the Royal Society of London 155, 459-512 (1865). (This article accompanied a December 8, 1864 presentation by Maxwell to the Royal Society.)
  • Maxwell, James Clerk, "A Dynamical Theory of the Electromagnetic Field", Vol. CLV, 1865.

http://www.zpenergy.com/downloads/Maxwell_1864_1.pdf http://www.zpenergy.com/downloads/Maxwell_1864_2.pdf http://www.zpenergy.com/downloads/Maxwell_1864_3.pdf http://www.zpenergy.com/downloads/Maxwell_1864_4.pdf http://www.zpenergy.com/downloads/Maxwell_1864_5.pdf http://www.zpenergy.com/downloads/Maxwell_1864_6.pdf

  • James C. Maxwell, Thomas F. Torrance, "A Dynamical Theory of the Electromagnetic Field". March, 1996. ISBN 1-57910-015-5
  • Niven, W. D., "The Scientific Papers of James Clerk Maxwell", 2 vols. Dover, New York, 1952, Vol. 1.

[edit] External links and references

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