A Dynamical Theory of the Electromagnetic Field
Encyclopedia
"A Dynamical Theory of the Electromagnetic Field" is the third of James Clerk Maxwell
's papers regarding electromagnetism
, published in 1865. It is the paper in which the original set of four Maxwell's equations
first appeared. The concept of displacement current
, which he had introduced in his 1861 paper "On Physical Lines of Force
", was utilized for the first time, to derive the electromagnetic wave equation
.
, until this term became applied instead to a set of four vectorized equations selected in 1884 by Oliver Heaviside
, 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 This amalgamation, which Maxwell himself had actually originally made at equation (112) in "On Physical Lines of Force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
.
Eighteen of the twenty original Maxwell's equations can be vectorized into 6 equations. Each vectorized equation represents 3 original equations in component form. Including the other two equations, in modern vector notation, they can form a set of eight equations. They are listed below:
(A) The law of total currents:
(B) Definition of the magnetic potential
:
(C) Ampère's circuital law:
(D) The Lorentz force
:
(E) The electric elasticity equation:
(F) Ohm's law
:
(G) Gauss's law
:
(H) Equation of continuity of charge:
Notation
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
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
. This means that equation "D" is effectively the Lorentz force. This equation first appeared at equation (77) in "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
term from equation "D" when he is deriving the electromagnetic wave equation, and he considers the situation only from the rest frame.
In "A dynamical theory of the electromagnetic field", Maxwell utilized the correction to Ampère's Circuital Law that he had made in part III of "On physical lines of force". In part VI of his 1864 paper "Electromagnetic theory of light", Maxwell combined displacement current with some of the other equations of electromagnetism and obtained a wave equation with a speed equal to 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.
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.
James Clerk Maxwell
James Clerk Maxwell of Glenlair was a Scottish physicist and mathematician. His most prominent achievement was formulating classical electromagnetic theory. This united all previously unrelated observations, experiments and equations of electricity, magnetism and optics into a consistent theory...
's papers regarding electromagnetism
Electromagnetism
Electromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...
, published in 1865. It is the paper in which the original set of four Maxwell's equations
Maxwell's equations
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...
first appeared. The concept of displacement current
Displacement current
In electromagnetism, displacement current is a quantity that is defined in terms of the rate of change of electric displacement field. Displacement current has the units of electric current density, and it has an associated magnetic field just as actual currents do. However it is not an electric...
, which he had introduced in his 1861 paper "On Physical Lines of Force
On Physical Lines of Force
On Physical Lines of Force is a famous four-part paper written by James Clerk Maxwell published between 1861 and 1862. In it, Maxwell derived the equations of electromagnetism in conjunction with a "sea" of "molecular vortices" which he used to model Faraday's lines of force...
", was utilized for the first time, to derive the electromagnetic wave equation
Electromagnetic wave equation
The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum...
.
Maxwell's original equations
In part III of "A Dynamical Theory of the Electromagnetic Field", which is entitled "General Equations of the Electromagnetic Field", Maxwell formulated twenty equations which were to become known as Maxwell's equationsMaxwell's equations
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...
, until this term became applied instead to a set of four vectorized equations selected in 1884 by Oliver Heaviside
Oliver Heaviside
Oliver Heaviside was a self-taught English electrical engineer, mathematician, and physicist who adapted complex numbers to the study of electrical circuits, invented mathematical techniques to the solution of differential equations , reformulated Maxwell's field equations in terms of electric and...
, 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
Gauss's law
In physics, Gauss's law, also known as Gauss's flux theorem, is a law relating the distribution of electric charge to the resulting electric field. Gauss's law states that:...
). 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
On Physical Lines of Force
On Physical Lines of Force is a famous four-part paper written by James Clerk Maxwell published between 1861 and 1862. In it, Maxwell derived the equations of electromagnetism in conjunction with a "sea" of "molecular vortices" which he used to model Faraday's lines of force...
", is the one that modifies Ampère's Circuital Law to include Maxwell's displacement current
Displacement current
In electromagnetism, displacement current is a quantity that is defined in terms of the rate of change of electric displacement field. Displacement current has the units of electric current density, and it has an associated magnetic field just as actual currents do. However it is not an electric...
.
Eighteen of the twenty original Maxwell's equations can be vectorized into 6 equations. Each vectorized equation represents 3 original equations in component form. Including the other two equations, in modern vector notation, they can form a set of eight equations. They are listed below:
(A) The law of total currents:
(B) Definition of the magnetic potential
Magnetic potential
The term magnetic potential can be used for either of two quantities in classical electromagnetism: the magnetic vector potential, A, and the magnetic scalar potential, ψ...
:
(C) Ampère's circuital law:
(D) The Lorentz force
Lorentz force
In physics, the Lorentz force is the force on a point charge due to electromagnetic fields. It is given by the following equation in terms of the electric and magnetic fields:...
:
- This force represents the effect of electric fields created by convection, induction, and by charges.
(E) The electric elasticity equation:
(F) Ohm's law
Ohm's law
Ohm's law states that the current through a conductor between two points is directly proportional to the potential difference across the two points...
:
(G) Gauss's law
Gauss's law
In physics, Gauss's law, also known as Gauss's flux theorem, is a law relating the distribution of electric charge to the resulting electric field. Gauss's law states that:...
:
(H) Equation of continuity of charge:
Notation
-
is the magnetic fieldMagnetic fieldA magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
, which Maxwell called the "magnetic intensity". -
is the electric currentElectric currentElectric current is a flow of electric charge through a medium.This charge is typically carried by moving electrons in a conductor such as wire...
density (with
being the total current 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 potentialMagnetic potentialThe term magnetic potential can be used for either of two quantities in classical electromagnetism: the magnetic vector potential, A, and the magnetic scalar potential, ψ...
(called the "angular impulse" by Maxwell). -
is the electric fieldElectric fieldIn physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
(called the "electromotive force" by Maxwell, not to be confused with the scalar quantity that is now called electromotive forceElectromotive forceIn physics, electromotive force, emf , or electromotance refers to voltage generated by a battery or by the magnetic force according to Faraday's Law, which states that a time varying magnetic field will induce an electric current.It is important to note that the electromotive "force" is not a...
). -
is the electric potentialElectric potentialIn classical electromagnetism, the electric potential at a point within a defined space is equal to the electric potential energy at that location divided by the charge there...
(which Maxwell also called "electric potential"). -
is the electrical conductivity (Maxwell called the inverse of conductivity the "specific resistance", what is now called the resistivityResistivityElectrical resistivity is a measure of how strongly a material opposes the flow of electric current. A low resistivity indicates a material that readily allows the movement of electric charge. The SI unit of electrical resistivity is the ohm metre...
).
Maxwell did not consider completely general materials; his initial formulation used linear, isotropic, nondispersive permittivity
Permittivity
In electromagnetism, absolute permittivity is the measure of the resistance that is encountered when forming an electric field in a medium. In other words, permittivity is a measure of how an electric field affects, and is affected by, a dielectric medium. The permittivity of a medium describes how...
ε and permeability
Permeability (electromagnetism)
In electromagnetism, permeability is the measure of the ability of a material to support the formation of a magnetic field within itself. In other words, it is the degree of magnetization that a material obtains in response to an applied magnetic field. Magnetic permeability is typically...
μ, although he also discussed the possibility of anisotropic materials.
It is of particular interest to note that Maxwell includes a
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 velocityVelocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...
. This means that equation "D" is effectively the Lorentz force. This equation first appeared at equation (77) in "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
Electromagnetic wave equation
The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum...
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
term from equation "D" when he is deriving the electromagnetic wave equation, and he considers the situation only from the rest frame.Maxwell – First to propose that light is an electromagnetic wave
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.
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 equationswhere
meters per secondis the speed of light in free space.
See also
- A Treatise on Electricity and MagnetismA Treatise on Electricity and MagnetismA Treatise on Electricity and Magnetism is a two volume treatise on electromagnetism written by James Clerk Maxwell in 1873.-See also:* On Physical Lines of Force* A Dynamical Theory of the Electromagnetic Field-External links:...
- On Physical Lines of ForceOn Physical Lines of ForceOn Physical Lines of Force is a famous four-part paper written by James Clerk Maxwell published between 1861 and 1862. In it, Maxwell derived the equations of electromagnetism in conjunction with a "sea" of "molecular vortices" which he used to model Faraday's lines of force...