1865 physics paper by James Maxwell
"
A Dynamical Theory of the Electromagnetic Field
" is a paper by
James Clerk Maxwell
on
electromagnetism
, published in 1865.
[1]
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.
Publication
[
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]
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.
[2]
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,
[3]
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
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In part III of the paper, which is entitled "General Equations of the Electromagnetic Field", Maxwell formulated twenty equations
[1]
which were to become known as
Maxwell's equations
, until this term became applied instead to a vectorized set of four equations selected in 1884, which had all appeared in his 1861 paper "
On Physical Lines of Force
".
[4]
Heaviside's versions of Maxwell's equations are distinct by virtue of the fact that they are written in modern
vector notation
. They actually only contain one of the original eight?equation "G" (
Gauss's Law
). Another of Heaviside's four equations is an amalgamation of Maxwell's law of total currents (equation "A") with
Ampere's circuital law
(equation "C"). This amalgamation, which Maxwell himself had actually originally made at equation (112) in "On Physical Lines of Force", is the one that modifies Ampere's Circuital Law to include Maxwell's
displacement current
.
[4]
Heaviside's equations
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Eighteen of Maxwell's twenty original equations can be
vectorized
into six equations, labeled
(A)
to
(F)
below, each of which represents a group of three original equations in
component form
. The 19th and 20th of Maxwell's component equations appear as
(G)
and
(H)
below, making a total of eight vector equations. These are listed below in Maxwell's original order, designated by the letters that Maxwell assigned to them in his 1864 paper.
[5]
- (A)
The law of total currents
- (B)
Definition of the
magnetic potential
- (C)
Ampere's circuital law
- (D)
The
Lorentz force
and
Faraday's law of induction
- (E)
The electric elasticity equation
- (F)
Ohm's law
- (G)
Gauss's law
- (H)
Equation of
continuity of charge
.
- Notation
- is the
magnetic field
, which Maxwell called the "
magnetic intensity
".
- is the
electric current
density (with
being the total current density including
displacement current
).
- is the
displacement field
(called the "
electric displacement
" by Maxwell).
- is the
free charge
density (called the "
quantity of free electricity
" by Maxwell).
- is the
magnetic potential
(called the "
angular impulse
" by Maxwell).
- 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
).
- is the
electric potential
(which Maxwell also called "
electric potential
").
- is the
electrical conductivity
(Maxwell called the inverse of conductivity the "
specific resistance
", what is now called the
resistivity
).
- is the vector operator
del
.
Clarifications
Maxwell did not consider completely general materials; his initial formulation used
linear
,
isotropic
,
nondispersive
media with
permittivity
?
and
permeability
μ
, although he also discussed the possibility of
anisotropic
materials.
Gauss's law for magnetism
(
∇?
B
= 0
) 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-Ampere law
.
Equation
(D)
implicitly contains the
Lorentz force law
and the differential form of
Faraday's law of induction
. For a
static
magnetic field,
vanishes, and the
electric field
E
becomes
conservative
and is given by
?∇
?
, so that
(D)
reduces to
.
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,
[4]
34 years before Lorentz derived his force law, which is now usually presented as a supplement to the four "
Maxwell's equations
". The cross-product term in the Lorentz force law is the source of the so-called
motional emf
in electric generators (see also
Moving magnet and conductor problem
). Where there is no motion through the magnetic field ? e.g., in
transformers
? we can drop the cross-product term, and the force per unit charge (called
f
) reduces to the electric field
E
, so that Maxwell's equation
(D)
reduces to
.
Taking curls, noting that the curl of a
gradient
is zero, we obtain
which is the
differential form of Faraday's law
. Thus the three terms on the right side of equation
(D)
may be described, from left to right, as the motional term, the transformer term, and the conservative term.
In deriving the
electromagnetic wave equation
, Maxwell considers the situation only from the
rest frame
of the medium, and accordingly drops the cross-product term. But he still works from equation
(D)
, in contrast to modern textbooks which tend to work from Faraday's law (see
below
).
The
constitutive equations
(E)
and
(F)
are now usually written in the rest frame of the medium as
D
=
?
E
and
J
=
σ
E
.
Maxwell's equation
(G)
, viewed in isolation as printed in the 1864 paper, at first seems to say that
ρ
+ ∇?
D
= 0
. However, if we trace the signs through the previous two triplets of equations, we see that what seem to be the components of
D
are in fact the components of
?
D
. The notation used in Maxwell's later
Treatise on Electricity and Magnetism
is different, and avoids the misleading first impression.
[6]
Maxwell ? electromagnetic light wave
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In part VI of "A Dynamical Theory of the Electromagnetic Field",
[1]
subtitled "Electromagnetic theory of light",
[7]
Maxwell uses the correction to Ampere's Circuital Law made in part III of his 1862 paper, "On Physical Lines of Force",
[4]
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,
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'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 Ampere's Circuital Law with Faraday's law of electromagnetic induction.
Modern equation methods
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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
|
|
|
|
If we take the
curl
of the curl equations we obtain
If we note the vector identity
where
is any vector function of space, we recover the wave equations
where
meters per second
is the speed of light in free space.
Legacy and impact
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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
.
See also
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]
Wikisource
has original text related to this article:
Wikisource
has original text related to this article:
References
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Further reading
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- Maxwell, James C.; Torrance, Thomas F. (March 1996).
A Dynamical Theory of the Electromagnetic Field
. Eugene, OR: Wipf and Stock.
ISBN
1-57910-015-5
.
- Niven, W. D. (1952).
The Scientific Papers of James Clerk Maxwell
. Vol. 1. New York: Dover.
- Johnson, Kevin (May 2002).
"The electromagnetic field"
.
James Clerk Maxwell ? The Great Unknown
. Archived from
the original
on September 15, 2008
. Retrieved
September 7,
2009
.
- Darrigol, Olivier (2000).
Electromagnetism from Ampere to Einstein.
Oxford University Press. ISBN 978-0198505945
- Katz, Randy H. (February 22, 1997).
"
'Look Ma, No Wires': Marconi and the Invention of Radio"
.
History of Communications Infrastructures
. Retrieved
Sep 7,
2009
.