Maxwell's equations
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For thermodynamic relations, see Maxwell relations.
Maxwell's equations, or Maxwell–Heaviside equations, are a set of
coupled partial differential equations that, together with the Lorentz force law,
form the foundation of classical electromagnetism, classical optics,
and electric circuits. The equations provide a mathematical model for electric,
optical, and radio technologies, such as power generation, electric
motors, wireless communication, lenses, radar etc. They describe
how electric and magnetic fields are generated by charges, currents, and
changes of the fields. The equations are named after the physicist and
[note 1]
mathematician James Clerk Maxwell, who, in 1861 and 1862, published an
early form of the equations that included the Lorentz force law. Maxwell first
used the equations to propose that light is an electromagnetic phenomenon.
The modern form of the equations in their most common formulation is
credited to Oliver Heaviside. [1]
Maxwell's equations may be combined to demonstrate how fluctuations in
electromagnetic fields (waves) propagate at a constant
speed, c (299792458 m/s in vacuum). Known as electromagnetic radiation,
[2]
these waves occur at various wavelengths to produce a spectrum of radiation
from radio waves to gamma rays.
The equations have two major variants. The microscopic equations have
universal applicability but are unwieldy for common calculations. They relate
the electric and magnetic fields to total charge and total current, including the
complicated charges and currents in materials at the atomic scale.
The macroscopic equations define two new auxiliary fields that describe the
large-scale behaviour of matter without having to consider atomic-scale
charges and quantum phenomena like spins. However, their use requires
experimentally determined parameters for a phenomenological description of
the electromagnetic response of materials. The term "Maxwell's equations" is
often also used for equivalent alternative formulations. Versions of Maxwell's
equations based on the electric and magnetic scalar potentials are preferred
for explicitly solving the equations as a boundary value problem, analytical
mechanics, or for use in quantum mechanics. The covariant
formulation (on spacetime rather than space and time separately) makes the
compatibility of Maxwell's equations with special relativity manifest. Maxwell's
equations in curved spacetime, commonly used in high-
energy and gravitational physics, are compatible with general relativity. In
[note 2]
fact, Albert Einstein developed special and general relativity to accommodate
the invariant speed of light, a consequence of Maxwell's equations, with the
principle that only relative movement has physical consequences.
The publication of the equations marked the unification of a theory for
previously separately described phenomena: magnetism, electricity, light, and
,associated radiation. Since the mid-20th century, it has been understood that
Maxwell's equations do not give an exact description of electromagnetic
phenomena, but are instead a classical limit of the more precise theory
of quantum electrodynamics.
History of the equations[edit]
Main article: History of Maxwell's equations
Conceptual descriptions[edit]
Gauss's law[edit]
Main article: Gauss's law
Gauss's law describes the relationship between a static electric
field and electric charges: a static electric field points away from positive
charges and towards negative charges, and the net outflow of the electric field
through a closed surface is proportional to the enclosed charge, including
bound charge due to polarization of material. The coefficient of the proportion
is the permittivity of free space.
Gauss's law for magnetism: magnetic field lines never begin nor end but form loops or extend to infinity as shown here with the
magnetic field due to a ring of current.
Gauss's law for magnetism[edit]
Main article: Gauss's law for magnetism
Gauss's law for magnetism states that electric charges have no magnetic
analogues, called magnetic monopoles, i.e no single pole exists. Instead, the [3]
magnetic field of a material is attributed to a dipole, and the net outflow of the
magnetic field through a closed surface is zero. Magnetic dipoles may be
represented as loops of current or inseparable pairs of equal and opposite
"magnetic charges". Precisely, the total magnetic flux through a Gaussian
surface is zero, and the magnetic field is a solenoidal vector field. [note 3]
, Faraday's law[edit]
Main article: Faraday's law of induction
In a geomagnetic storm, a surge in the flux of charged particles temporarily alters Earth's magnetic field, which induces electric fields
in Earth's atmosphere, thus causing surges in electrical power grids. (Not to scale.)
The Maxwell–Faraday version of Faraday's law of induction describes how a
time-varying magnetic field corresponds to curl of an electric field. In integral [3]
form, it states that the work per unit charge required to move a charge around
a closed loop equals the rate of change of the magnetic flux through the
enclosed surface.
The electromagnetic induction is the operating principle behind many electric
generators: for example, a rotating bar magnet creates a changing magnetic
field and generates an electric field in a nearby wire.
Ampère's law with Maxwell's addition[edit]
Main article: Ampère's circuital law
Magnetic-core memory (1954) is an application of Ampère's law. Each core stores one bit of data.
The original law of Ampère states that magnetic fields relate to electric
current. Maxwell's addition states that they also relate to changing electric
fields, which Maxwell called displacement current. The integral form states
that electric and displacement currents are associated with a proportional
magnetic field along any enclosing curve.
Jump to navigationJump to search
For thermodynamic relations, see Maxwell relations.
Maxwell's equations, or Maxwell–Heaviside equations, are a set of
coupled partial differential equations that, together with the Lorentz force law,
form the foundation of classical electromagnetism, classical optics,
and electric circuits. The equations provide a mathematical model for electric,
optical, and radio technologies, such as power generation, electric
motors, wireless communication, lenses, radar etc. They describe
how electric and magnetic fields are generated by charges, currents, and
changes of the fields. The equations are named after the physicist and
[note 1]
mathematician James Clerk Maxwell, who, in 1861 and 1862, published an
early form of the equations that included the Lorentz force law. Maxwell first
used the equations to propose that light is an electromagnetic phenomenon.
The modern form of the equations in their most common formulation is
credited to Oliver Heaviside. [1]
Maxwell's equations may be combined to demonstrate how fluctuations in
electromagnetic fields (waves) propagate at a constant
speed, c (299792458 m/s in vacuum). Known as electromagnetic radiation,
[2]
these waves occur at various wavelengths to produce a spectrum of radiation
from radio waves to gamma rays.
The equations have two major variants. The microscopic equations have
universal applicability but are unwieldy for common calculations. They relate
the electric and magnetic fields to total charge and total current, including the
complicated charges and currents in materials at the atomic scale.
The macroscopic equations define two new auxiliary fields that describe the
large-scale behaviour of matter without having to consider atomic-scale
charges and quantum phenomena like spins. However, their use requires
experimentally determined parameters for a phenomenological description of
the electromagnetic response of materials. The term "Maxwell's equations" is
often also used for equivalent alternative formulations. Versions of Maxwell's
equations based on the electric and magnetic scalar potentials are preferred
for explicitly solving the equations as a boundary value problem, analytical
mechanics, or for use in quantum mechanics. The covariant
formulation (on spacetime rather than space and time separately) makes the
compatibility of Maxwell's equations with special relativity manifest. Maxwell's
equations in curved spacetime, commonly used in high-
energy and gravitational physics, are compatible with general relativity. In
[note 2]
fact, Albert Einstein developed special and general relativity to accommodate
the invariant speed of light, a consequence of Maxwell's equations, with the
principle that only relative movement has physical consequences.
The publication of the equations marked the unification of a theory for
previously separately described phenomena: magnetism, electricity, light, and
,associated radiation. Since the mid-20th century, it has been understood that
Maxwell's equations do not give an exact description of electromagnetic
phenomena, but are instead a classical limit of the more precise theory
of quantum electrodynamics.
History of the equations[edit]
Main article: History of Maxwell's equations
Conceptual descriptions[edit]
Gauss's law[edit]
Main article: Gauss's law
Gauss's law describes the relationship between a static electric
field and electric charges: a static electric field points away from positive
charges and towards negative charges, and the net outflow of the electric field
through a closed surface is proportional to the enclosed charge, including
bound charge due to polarization of material. The coefficient of the proportion
is the permittivity of free space.
Gauss's law for magnetism: magnetic field lines never begin nor end but form loops or extend to infinity as shown here with the
magnetic field due to a ring of current.
Gauss's law for magnetism[edit]
Main article: Gauss's law for magnetism
Gauss's law for magnetism states that electric charges have no magnetic
analogues, called magnetic monopoles, i.e no single pole exists. Instead, the [3]
magnetic field of a material is attributed to a dipole, and the net outflow of the
magnetic field through a closed surface is zero. Magnetic dipoles may be
represented as loops of current or inseparable pairs of equal and opposite
"magnetic charges". Precisely, the total magnetic flux through a Gaussian
surface is zero, and the magnetic field is a solenoidal vector field. [note 3]
, Faraday's law[edit]
Main article: Faraday's law of induction
In a geomagnetic storm, a surge in the flux of charged particles temporarily alters Earth's magnetic field, which induces electric fields
in Earth's atmosphere, thus causing surges in electrical power grids. (Not to scale.)
The Maxwell–Faraday version of Faraday's law of induction describes how a
time-varying magnetic field corresponds to curl of an electric field. In integral [3]
form, it states that the work per unit charge required to move a charge around
a closed loop equals the rate of change of the magnetic flux through the
enclosed surface.
The electromagnetic induction is the operating principle behind many electric
generators: for example, a rotating bar magnet creates a changing magnetic
field and generates an electric field in a nearby wire.
Ampère's law with Maxwell's addition[edit]
Main article: Ampère's circuital law
Magnetic-core memory (1954) is an application of Ampère's law. Each core stores one bit of data.
The original law of Ampère states that magnetic fields relate to electric
current. Maxwell's addition states that they also relate to changing electric
fields, which Maxwell called displacement current. The integral form states
that electric and displacement currents are associated with a proportional
magnetic field along any enclosing curve.
