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. Physicist Freeman Dyson called the publishing of the paper the "most important event of the nineteenth century in the history of the physical sciences".

The paper was key in establishing the classical theory of electromagnetism. Maxwell derives an electromagnetic wave equation with a velocity for light in close agreement with measurements made by experiment, and also deduces that light is an electromagnetic wave.

Publication

Following standard procedure for the time, the paper was first read to the Royal Society on 8 December 1864, having been sent by Maxwell to the society on 27 October. It then underwent peer review, being sent to William Thomson (later Lord Kelvin) on 24 December 1864. It was then sent to George Gabriel Stokes, the Society's physical sciences secretary, on 23 March 1865. It was approved for publication in the Philosophical Transactions of the Royal Society on 15 June 1865, by the Committee of Papers (essentially the society's governing council) and sent to the printer the following day (16 June). During this period, Philosophical Transactions was only published as a bound volume once a year, and would have been prepared for the society's anniversary day on 30 November (the exact date is not recorded). However, the printer would have prepared and delivered to Maxwell offprints, for the author to distribute as he wished, soon after 16 June.

Maxwell's original equations

In part III of the paper, which is entitled "General Equations of the Electromagnetic Field", Maxwell formulated twenty equations

;(A) The law of total currents

<math>\mathbf{J}_{\rm tot} = </math> <math>\,\mathbf{J}</math> <math> +\,\frac{\partial\mathbf{D{\partial t}</math>

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;(B) Definition of the magnetic potential

<math>\mu \mathbf{H} = \nabla \times \mathbf{A}</math>

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;(C) Ampère's circuital law

<math>\nabla \times \mathbf{H} = \mathbf{J}_{\rm tot}</math>

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;(D) The Lorentz force and Faraday's law of induction

<math>\mathbf{f} = \mu (\mathbf{v} \times \mathbf{H}) - \frac{\partial\mathbf{A{\partial t}-\nabla \phi </math>

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;(E) The electric elasticity equation

<math>\mathbf{f} = \frac{1}{\varepsilon} \mathbf{D}</math>

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;(F) Ohm's law

<math>\mathbf{f} = \frac{1}{\sigma} \mathbf{J}</math>

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;(G) Gauss's law

<math>\nabla \cdot \mathbf{D} = \rho</math>

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;(H) Equation of continuity of charge

<math>\nabla \cdot \mathbf{J} = -\frac{\partial\rho}{\partial t}\,</math>.

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;Notation

: <math>\mathbf{H}</math> is the magnetic field, which Maxwell called the "magnetic intensity".

: <math>\mathbf{J}</math> is the electric current density (with <math>\mathbf{J}_{\rm tot}</math> being the total current density including displacement current).

: <math>\mathbf{D}</math> is the displacement field (called the "electric displacement" by Maxwell).

: <math>\rho</math> is the free charge density (called the "quantity of free electricity" by Maxwell).

: <math>\mathbf{A}</math> is the magnetic potential (called the "angular impulse" by Maxwell).

: <math>\mathbf{f}</math> is the force per unit charge (called the "electromotive force" by Maxwell, not to be confused with the scalar quantity that is now called electromotive force; see below).

: <math>\phi</math> is the electric potential (which Maxwell also called "electric potential").

: <math>\sigma</math> is the electrical conductivity (Maxwell called the inverse of conductivity the "specific resistance", what is now called the resistivity).

: <math>\nabla</math> is the vector operator del.

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<div id="Clarify">Clarifications</div>

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

Gauss's law for magnetism () is not included in the above list, but follows directly from equation (B) by taking divergences (because the divergence of the curl is zero).

Substituting (A) into (C) yields the familiar differential form of the Maxwell-Ampère law.

Equation (D) implicitly contains the Lorentz force law and the differential form of Faraday's law of induction. For a static magnetic field, <math>\partial\mathbf{A}/\partial t</math> vanishes, and the electric field becomes conservative and is given by , so that (D) reduces to

<div class="center"><math>\mathbf{f}=\mathbf{E}+\mathbf{v}\times\mathbf{B}\,</math>.</div>

This is simply the Lorentz force law on a per-unit-charge basis — although Maxwell's equation (D) first appeared at equation (77) in "On Physical Lines of Force" in 1861, concludes that the sign of in (G) is wrong, and observes that this sign is corrected in Maxwell's subsequent Treatise. Arthur speculates that the sign confusion may have arisen from the analogy between momentum and the magnetic vector potential (Maxwell's "electromagnetic momentum"), in which positive mass corresponds to negative charge. Arthur also lists some corresponding equations from Maxwell's earlier paper of 1861-2, and notes that the signs do not always match the later ones. The earlier signs (1861-2) are correct if are the components of while are the components of.

Maxwell – electromagnetic light wave

In part VI of "A Dynamical Theory of the Electromagnetic Field", Maxwell uses the correction to Ampère's Circuital Law made in part III of his 1862 paper, "On Physical Lines of Force", which is defined as displacement current, to derive the electromagnetic wave equation.

He obtained a wave equation with a speed in close agreement to experimental determinations of the speed of light. He commented,

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

Modern equation methods

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

{| style="margin:1em auto;"

|<math>\nabla \cdot \mathbf{E} = 0</math>

|<math> \nabla \times \mathbf{E} = -\mu_o \frac{\partial \mathbf{H {\partial t}</math>

|<math> \nabla \cdot \mathbf{H} = 0</math>

|<math> \nabla \times \mathbf{H} =\varepsilon_o \frac{ \partial \mathbf{E {\partial t}</math>

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If we take the curl of the curl equations we obtain

<math> \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} </math>

<math> \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}

</math>

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If we note the vector identity

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

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

<math> {\partial^2 \mathbf{E} \over \partial t^2} \ - \ c^2 \cdot \nabla^2 \mathbf{E} \ \ = \ \ 0</math>

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<math> {\partial^2 \mathbf{H} \over \partial t^2} \ - \ c^2 \cdot \nabla^2 \mathbf{H} \ \ = \ \ 0</math>

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where

<math>c = { 1 \over \sqrt{ \mu_o \varepsilon_o } } = 2.99792458 \times 10^8 </math> meters per second

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is the speed of light in free space.

Legacy and impact

Of this paper and Maxwell's related works, fellow physicist Richard Feynman said: "From the long view of this history of mankind – seen from, say, 10,000 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 electromagnetism."

Albert Einstein used Maxwell's equations as the starting point for his special theory of relativity, presented in The Electrodynamics of Moving Bodies, one of Einstein's 1905 Annus Mirabilis papers. In it is stated:

: the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good

and

: Any ray of light moves in the "stationary" system of co-ordinates with the determined velocity c, whether the ray be emitted by a stationary or by a moving body.

Maxwell's equations can also be derived by extending general relativity into five physical dimensions.

A Dynamical Theory of the Electromagnetic Field

Publication

Maxwell's original equations

Maxwell – electromagnetic light wave

Modern equation methods

Legacy and impact

See also

References

Further reading