"A Dynamical Theory of the Electromagnetic
Field" is a paper by James Clerk Maxwell on
electromagnetism, published in 1865. 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 ==
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 Thompson
(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
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 "On physical
lines of force".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 Ampère'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had actually originally made at equation (112)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 Ampère's Circuital Law to include
Maxwell's displacement current.
For his original text on force, see: On Physical
Lines of Force. Wikisource.
For his original text on dynamics, see: A
Dynamical Theory of the Electromagnetic Field.
Wikisource.
=== Heaviside's equations ===
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.
(A) The law of total currents
J
t
o
t
=
{\displaystyle \mathbf {J} _{tot}=}
J
{\displaystyle \,\mathbf {J} }
+
∂
D
∂
t
{\displaystyle +\,{\frac {\partial \mathbf
{D} }{\partial t}}}
(B) Definition of the magnetic potential
μ
H
=
∇
×
A
{\displaystyle \mu \mathbf {H} =\nabla \times
\mathbf {A} }
(C) Ampère's circuital law
∇
×
H
=
J
t
o
t
{\displaystyle \nabla \times \mathbf {H} =\mathbf
{J} _{tot}}
(D) The Lorentz force and Faraday's law of
induction
f
=
μ
(
v
×
H
)
−
∂
A
∂
t
−
∇
ϕ
{\displaystyle \mathbf {f} =\mu (\mathbf {v}
\times \mathbf {H} )-{\frac {\partial \mathbf
{A} }{\partial t}}-\nabla \phi }
(E) The electric elasticity equation
f
=
1
ϵ
D
{\displaystyle \mathbf {f} ={\frac {1}{\epsilon
}}\mathbf {D} }
(F) Ohm's law
f
=
1
σ
J
{\displaystyle \mathbf {f} ={\frac {1}{\sigma
}}\mathbf {J} }
(G) Gauss's law
∇
⋅
D
=
ρ
{\displaystyle \nabla \cdot \mathbf {D} =\rho
}
(H) Equation of continuity of charge
∇
⋅
J
=
−
∂
ρ
∂
t
{\displaystyle \nabla \cdot \mathbf {J} =-{\frac
{\partial \rho }{\partial t}}\,}
.
Notation
H
{\displaystyle \mathbf {H} }
is the magnetic field, which Maxwell called
the "magnetic intensity".
J
{\displaystyle \mathbf {J} }
is the electric current density (with
J
t
o
t
{\displaystyle \mathbf {J} _{tot}}
being the total current density including
displacement current).
D
{\displaystyle \mathbf {D} }
is the displacement field (called the "electric
displacement" by Maxwell).
ρ
{\displaystyle \rho }
is the free charge density (called the "quantity
of free electricity" by Maxwell).
A
{\displaystyle \mathbf {A} }
is the magnetic potential (called the "angular
impulse" by Maxwell).
f
{\displaystyle \mathbf {f} }
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).
ϕ
{\displaystyle \phi }
is the electric potential (which Maxwell also
called "electric potential").
σ
{\displaystyle \sigma }
is the electrical conductivity (Maxwell called
the inverse of conductivity the "specific
resistance", what is now called the resistivity).
∇
{\displaystyle \nabla }
is the vector operator del.
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 (∇⋅ 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-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,
∂
A
/
∂
t
{\displaystyle \partial \mathbf {A} /\partial
t}
vanishes, and the electric field E becomes
conservative and is given by −∇ϕ, so
that (D) reduces to
f
=
E
+
v
×
B
{\displaystyle \mathbf {f} =\mathbf {E} +\mathbf
{v} \times \mathbf {B} \,}
.
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, 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
E
=
−
∂
A
∂
t
−
∇
ϕ
{\displaystyle \mathbf {E} =-{\frac {\partial
\mathbf {A} }{\partial t}}-\nabla \phi \,}
.
Taking curls, noting that the curl of a gradient
is zero, we obtain
∇
×
E
=
−
∇
×
∂
A
∂
t
=
−
∂
∂
t
(
∇
×
A
)
=
−
∂
B
∂
t
,
{\displaystyle \nabla \times \mathbf {E} \,=\,-\nabla
\times {\frac {\partial \mathbf {A} }{\partial
t}}\,=\,-{\frac {\partial }{\partial t}}{\big
(}\nabla \times \mathbf {A} {\big )}\,=\,-{\frac
{\partial \mathbf {B} }{\partial t}}\,,}
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 = ϵE and J = σE.
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.
== Maxwell – electromagnetic light wave
==
In part VI of "A Dynamical Theory of the Electromagnetic
Field", subtitled "Electromagnetic theory
of light", 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,
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 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
If we take the curl of the curl equations
we obtain
∇
×
∇
×
E
=
−
μ
o
∂
∂
t
∇
×
H
=
−
μ
o
ε
o
∂
2
E
∂
t
2
{\displaystyle \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}}}}
∇
×
∇
×
H
=
ε
o
∂
∂
t
∇
×
E
=
−
μ
o
ε
o
∂
2
H
∂
t
2
{\displaystyle \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
∇
×
(
∇
×
V
)
=
∇
(
∇
⋅
V
)
−
∇
2
V
{\displaystyle \nabla \times \left(\nabla
\times \mathbf {V} \right)=\nabla \left(\nabla
\cdot \mathbf {V} \right)-\nabla ^{2}\mathbf
{V} }
where
V
{\displaystyle \mathbf {V} }
is any vector function of space, we recover
the wave equations
∂
2
E
∂
t
2
−
c
2
⋅
∇
2
E
=
0
{\displaystyle {\partial ^{2}\mathbf {E} \over
\partial t^{2}}\ -\ c^{2}\cdot \nabla ^{2}\mathbf
{E} \ \ =\ \ 0}
∂
2
H
∂
t
2
−
c
2
⋅
∇
2
H
=
0
{\displaystyle {\partial ^{2}\mathbf {H} \over
\partial t^{2}}\ -\ c^{2}\cdot \nabla ^{2}\mathbf
{H} \ \ =\ \ 0}
where
c
=
1
μ
o
ε
o
=
2.99792458
×
10
8
{\displaystyle c={1 \over {\sqrt {\mu _{o}\varepsilon
_{o}}}}=2.99792458\times 10^{8}}
meters per second
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, a paper produced during
his 1905 Annus Mirabilis. 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 ==
A Treatise on Electricity and Magnetism
"On Physical Lines of Force"
Gauge theory
