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In electromagnetism, Maxwell's equations are a set of four equations that were first presented as a distinct group in 1884 by Oliver Heaviside in conjunction with Willard Gibbs. These equations had appeared in substance throughout James Clerk Maxwell's 1861 paper entitled On Physical Lines of Force.

They describe the interrelationship between electric field, magnetic field, electric charge, and electric current. Maxwell himself was not actually the originator of all four of these equations. The importance of his role in these equations lies in the correction he made to Ampère's circuital law. He added the displacement current term to Ampère's circuital law and this enabled him to derive the electromagnetic wave equation and demonstrate the fact that light is an electromagnetic wave. This fact was then later confirmed experimentally by Heinrich Hertz in 1887.

Although Maxwell was not the originator of all four of these equations, he nevertheless derived them all again independently in conjunction with his molecular vortex model of Faraday's "lines of force".

These equations are ( definitions in Table 3 below ):

Gauss' law	${\displaystyle \nabla \cdot \mathbf {E} ={\frac {\rho }{\epsilon _{0}}}}$
Gauss' law for magnetic fields (no magnetic monopoles)	${\displaystyle \nabla \cdot \mathbf {B} =0}$
Maxwell-Faraday equation Faraday's law of induction	${\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}$
Ampère's circuital law with Maxwell's correction	${\displaystyle \nabla \times \mathbf {B} =\mu _{0}\mathbf {J} +\mu _{0}\epsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}}$ ${\displaystyle }$

There is one more equation, not in the box above, yet essential to complete the laws of classical electromagnetism: the Lorentz force law. This equation, derived by Maxwell, nowadays is classified as supplementary to Maxwell's equations, and not as one of them.

History

The term Maxwell's equations

There always has been controversy surrounding the term Maxwell's equations concerning the extent to which Maxwell himself was actually involved in these equations. The term Maxwell's equations nowadays applies to a set of four equations that were grouped together as a distinct set in 1884 by Oliver Heaviside, in conjunction with Willard Gibbs. The reason that these equations are called Maxwell's equations is disputed. Some say that these equations were originally called the Heaviside-Hertz equations but that Einstein for whatever reason later referred to them as the Maxwell-Hertz equations. see pages 110-112 of Nahin's book

At any rate, these equations are based on the works of James Clerk-Maxwell, and Heaviside made no secret of the fact that he was working from Maxwell's papers. Heaviside aimed to produce a symmetrical set of equations that were crucial as regards deriving the telegrapher's equations. The net result was a set of four equations, three of which had appeared in substance throughout Maxwell's previous papers, in particular Maxwell's 1861 paper On Physical Lines of Force and 1865 paper A Dynamical Theory of the Electromagnetic Field. The fourth was a partial time derivative version of Faraday's law of induction that doesn't include motionally induced EMF.

Of Heaviside's equations, the most important one as regards deriving the telegrapher's equations was the version of Ampère's circuital law that had been amended by Maxwell in this 1861 paper to include what is termed the displacement current. For this reason alone, the term Maxwell's equations as applied to Heaviside's group of four, contains a considerable degree of merit.

Maxwell's On Physical Lines of Force (1861)

(Alternate source.)

Three of Heaviside's four equations appeared throughout Maxwell's 1861 paper On Physical Lines of Force:

(i) At equation (56) of Maxwell's 1861 paper we see div B = 0.

(ii) At equation (112) we see Ampère's circuital law with Maxwell's correction. It is this correction called displacement current which is the most significant aspect of Maxwell's work in electromagnetism as it enabled him to later derive the electromagnetic wave equation in his 1865 paper A Dynamical Theory of the Electromagnetic Field, and hence show that light is an electromagnetic wave. It is therefore this aspect of Maxwell's work which gives Heaviside's equations their full significance. (Interestingly, Kirchhoff derived the telegrapher's equations in 1857 without using displacement current. But he did use Poisson's equation and the equation of continuity which are the mathematical ingredients of the displacement current. Nevertheless, Kirchhoff believed his equations to be applicable only inside an electric wire and so he is not credited with having discovered that light is an electromagnetic wave).

(iii) At equation (113) we see Gauss's law.

(iv) Heaviside's fourth equation introduced a restricted partial time derivative version of Faraday's law of induction. (A full version of Faraday's law of induction had appeared at equation (54) of Maxwell's 1861 paper). It is important however to note that Heaviside's partial time derivative notation, as opposed to the total time derivative notation used by Maxwell at equations (54) and (112), resulted in the loss of the v × B term that appeared in Maxwell's equation (77). Nowadays, the v × B term appears in the force law F = q ( E + v × B ) which sits adjacent to Maxwell's equations and bears the name Lorentz force. The Lorentz Force corresponds in effect to Maxwell's equation (77), but it appeared in this paper when Lorentz was still a young boy.

Maxwell's A Dynamical Theory of the Electromagnetic Field (1865)

Confusion over the term "Maxwell's equations" is further increased because it is also sometimes used for a set of eight equations that appeared in Part III of Maxwell's 1865 paper A Dynamical Theory of the Electromagnetic Field, entitled "General Equations of the Electromagnetic Field" (page 480 of the article and page 2 of the pdf link), a confusion compounded by the writing of six of those eight equations as three separate equations (one for each of the Cartesian axes), resulting in twenty equations in twenty unknowns. (As noted above, this terminology is not common: Modern references to the term "Maxwell's equations" usually refer to the Heaviside restatements.)

These original eight equations are nearly identical to the Heaviside versions in substance, but they have some superficial differences. In fact, only one of the Heaviside versions is completely unchanged from these original equations, and that is Gauss's law (Maxwell's equation G below). Another of Heaviside's four equations is an amalgamation of Maxwell's law of total currents (equation A below) with Ampère's circuital law (equation C below). This amalgamation, which Maxwell himself originally made at equation (112) in his "On Physical Lines of Force" (see above), is the one that modifies Ampère's circuital law to include Maxwell's displacement current.

The eight original Maxwell's equations can be written in modern vector notation as follows:

(A) The law of total currents

${\displaystyle \mathbf {j} _{tot}=\mathbf {j} +{\frac {\partial \mathbf {D} }{\partial t}}}$

(B) Definition of the magnetic vector potential

${\displaystyle \mu \mathbf {H} =\nabla \times \mathbf {A} }$

(C) Ampère's circuital law

${\displaystyle \nabla \times \mathbf {H} =\mathbf {j} _{tot}}$

(D) Electromotive force created by convection, induction, and by static electricity. (This is in effect the Lorentz force)

${\displaystyle \mathbf {E} =\mu \mathbf {v} \times \mathbf {H} -{\frac {\partial \mathbf {A} }{\partial t}}-\nabla \phi }$

(E) The electric elasticity equation

${\displaystyle \mathbf {E} ={\frac {1}{\epsilon }}\mathbf {D} }$

(F) Ohm's law

${\displaystyle \mathbf {E} ={\frac {1}{\sigma }}\mathbf {j} }$

(G) Gauss's law

${\displaystyle \nabla \cdot \mathbf {D} =\rho }$

(H) Equation of continuity

${\displaystyle \nabla \cdot \mathbf {j} =-{\frac {\partial \rho }{\partial t}}}$

Notation

${\displaystyle \mathbf {H} }$ is the magnetizing field, which Maxwell called the "magnetic intensity".

${\displaystyle \mathbf {j} }$ is the electric current density (with ${\displaystyle \mathbf {j} _{tot}}$ being the total current including displacement current).

${\displaystyle \mathbf {D} }$ is the displacement field (called the "electric displacement" by Maxwell).

${\displaystyle \rho }$ is the free charge density (called the "quantity of free electricity" by Maxwell).

${\displaystyle \mathbf {A} }$ is the magnetic vector potential (called the "angular impulse" by Maxwell).

${\displaystyle \mathbf {E} }$ is called the "electromotive force" by Maxwell. The term electromotive force is nowadays used for voltage, but it is clear from the context that Maxwell's meaning corresponded more to the modern term electric field.

${\displaystyle \Phi }$ is the electric potential (which Maxwell also called "electric potential").

${\displaystyle \sigma }$ is the electrical conductivity (Maxwell called the inverse of conductivity the "specific resistance", what is now called the resistivity).

It is interesting to note the ${\displaystyle \mu \mathbf {v} \times \mathbf {H} }$ term that appears in equation D. Equation D is therefore effectively the Lorentz force, similarly to equation (77) of his 1861 paper (see above).

When Maxwell derives the electromagnetic wave equation in his 1865 paper, he uses equation D to cater for electromagnetic induction rather than Faraday's law of induction which is used in modern textbooks. (Faraday's law itself does not appear among his equations.) However, Maxwell drops the ${\displaystyle \mu \mathbf {v} \times \mathbf {H} }$ term from equation D when he is deriving the electromagnetic wave equation, as he considers the situation only from the rest frame.

Formulation of Maxwell's equations

Symbols in bold represent vector quantities, whereas symbols in italics represent scalar quantities. The equations in this section, unless otherwise stated, are given in SI units. Unlike the equations of mechanics (for example), Maxwell's equations are not unchanged in other unit systems. Though the general form remains the same, various definitions get changed and different constants appear at different places. Other than SI (used in engineering), the units commonly used are Gaussian units (based on the cgs system and considered to have some theoretical advantages over SI), Lorentz-Heaviside units (used mainly in particle physics) and Planck units (used in theoretical physics). See below for CGS-Gaussian units.

General formulations

Two equivalent, general formulations of Maxwell's equations follow. The first separates free charge and free current from bound charge and bound current. This separation is useful for calculations involving dielectric and/or magnetized materials. The second formulation treats all charge equally, combining free and bound charge into total charge (and likewise with current). Of course, such an approach applies where no dielectric or magnetic material is present, and therefore no bound charge or current, but it also is a more fundamental or microscopic point of view. For more detail, and a proof of equivalence, see below. The definitions of terms used in the two tables of equations are given in another table immediately following.

Table 1: Formulation in terms of free charge and current

Name	Differential form	Integral form
Gauss's law:	${\displaystyle \nabla \cdot \mathbf {D} =\rho _{f}}$	${\displaystyle \oint _{S}\mathbf {D} \cdot \mathrm {d} \mathbf {A} =Q_{f,S}}$
Gauss's law for magnetism (absence of magnetic monopoles):	${\displaystyle \nabla \cdot \mathbf {B} =0}$	${\displaystyle \oint _{S}\mathbf {B} \cdot \mathrm {d} \mathbf {A} =0}$
Maxwell-Faraday equation (Faraday's law of induction):	${\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}$	${\displaystyle \oint _{\partial S}\mathbf {E} \cdot \mathrm {d} \mathbf {l} =-{\frac {d\Phi _{B,S}}{dt}}}$
Ampère's Circuital Law (with Maxwell's correction):	${\displaystyle \nabla \times \mathbf {H} =\mathbf {J} _{f}+{\frac {\partial \mathbf {D} }{\partial t}}}$ ${\displaystyle }$	${\displaystyle \oint _{\partial S}\mathbf {H} \cdot \mathrm {d} \mathbf {l} =I_{f,S}+{\frac {d\Phi _{E,S}}{dt}}}$

Table 2: Formulation in terms of total charge and current

Name	Differential form	Integral form
Gauss's law:	${\displaystyle \nabla \cdot \mathbf {E} ={\frac {\rho }{\epsilon _{0}}}}$	${\displaystyle \oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} ={\frac {Q_{S}}{\epsilon _{0}}}}$
Gauss's law for magnetism (absence of magnetic monopoles):	${\displaystyle \nabla \cdot \mathbf {B} =0}$	${\displaystyle \oint _{S}\mathbf {B} \cdot \mathrm {d} \mathbf {A} =0}$
Maxwell-Faraday equation (Faraday's law of induction):	${\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}$	${\displaystyle \oint _{\partial S}\mathbf {E} \cdot \mathrm {d} \mathbf {l} =-{\frac {d\Phi _{B,S}}{dt}}}$
Ampère's Circuital Law (with Maxwell's correction):	${\displaystyle \nabla \times \mathbf {B} =\mu _{0}\mathbf {J} +\mu _{0}\epsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\ }$ ${\displaystyle \ }$	${\displaystyle \oint _{\partial S}\mathbf {B} \cdot \mathrm {d} \mathbf {l} =\mu _{0}I_{S}+\mu _{0}\epsilon _{0}{\frac {d\Phi _{E,S}}{dt}}}$

The following table provides the meaning of each symbol and the SI unit of measure:

Table 3: Definitions and units

Symbol	Meaning (first term is the most common)	SI Unit of Measure
${\displaystyle \mathbf {\nabla \cdot } }$	the divergence operator	per meter (factor contributed by applying either operator)
${\displaystyle \mathbf {\nabla \times } }$	the curl operator
${\displaystyle {\frac {\partial }{\partial t}}}$	partial derivative with respect to time	per second (factor contributed by applying the operator)
${\displaystyle \mathbf {E} \ }$	electric field also called the electric flux density	volt per meter or, equivalently, newton per coulomb
${\displaystyle \mathbf {B} \ }$	magnetic field also called the magnetic induction also called the magnetic field density also called the magnetic flux density	tesla, or equivalently, weber per square meter volt•second per square meter
${\displaystyle \mathbf {D} \ }$	electric displacement field	coulombs per square meter or, equivalently, newton per volt-meter
${\displaystyle \mathbf {H} \ }$	magnetizing field also called auxiliary magnetic field also called magnetic field intensity also called magnetic field	ampere per meter
${\displaystyle \epsilon _{0}\ }$	permittivity of free space, officially the electric constant, a universal constant	farads per meter
${\displaystyle \mu _{0}\ }$	permeability of free space, officially the magnetic constant, a universal constant	henries per meter, or newtons per ampere squared
${\displaystyle \ \rho _{f}\ }$	free charge density (not including bound charge)	coulomb per cubic meter
${\displaystyle \ \rho \ }$	total charge density (including both free and bound charge)	coulomb per cubic meter
${\displaystyle \oint _{S}\mathbf {E\cdot \mathrm {d} A} }$	the flux of the electric field over any closed gaussian surface S	joule-meter per coulomb
${\displaystyle Q_{f,S}\ }$	net unbalanced free electric charge enclosed by the Gaussian surface S (not including bound charge)	coulombs
${\displaystyle Q_{S}\ }$	net unbalanced electric charge enclosed by the Gaussian surface S (including both free and bound charge)	coulombs
${\displaystyle \oint _{S}\mathbf {B\cdot \mathrm {d} A} }$	the flux of the magnetic field over any closed surface S	tesla meter-squared or weber
${\displaystyle \oint _{\partial S}\mathbf {E} \cdot \mathrm {d} \mathbf {l} }$	line integral of the electric field along the boundary ∂S (therefore necessarily a closed curve) of the surface S	joule per coulomb
${\displaystyle \Phi _{B,S}=\int _{S}\mathbf {B} \cdot \mathrm {d} \mathbf {A} }$	magnetic flux over any surface S (not necessarily closed)	weber
${\displaystyle \mathbf {J} _{f}}$	free current density (not including bound current)	ampere per square meter
${\displaystyle \mathbf {J} }$	total current density (including both free and bound current)	ampere per square meter
${\displaystyle \oint _{\partial S}\mathbf {B} \cdot \mathrm {d} \mathbf {l} }$	line integral of the magnetic field over the closed boundary ∂S of the surface S	tesla-meter
${\displaystyle I_{f,S}=\int _{S}\mathbf {J} _{f}\cdot \mathrm {d} \mathbf {A} }$	net free electrical current passing through the surface S (not including bound current)	amperes
${\displaystyle I_{S}=\int _{S}\mathbf {J} _{f}\cdot \mathrm {d} \mathbf {A} }$	net electrical current passing through the surface S (including both free and bound current)	amperes
${\displaystyle \Phi _{E,S}=\int _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} }$	electric flux over any surface S, not necessarily closed
${\displaystyle \mathrm {d} \mathbf {A} }$	differential vector element of surface area A, with infinitesimally small magnitude and direction normal to surface S	square meters
${\displaystyle \mathrm {d} \mathbf {l} }$	differential vector element of path length tangential to contour	meters

Maxwell's equations are generally applied to macroscopic averages of the fields, which vary wildly on a microscopic scale in the vicinity of individual atoms (where they undergo quantum mechanical effects as well). It is only in this averaged sense that one can define quantities such as the permittivity and permeability of a material. At the microscopic level, Maxwell's equations, ignoring quantum effects, describe fields, charges and currents in free space — but at this level of detail one must include all charges, even those at an atomic level, generally an intractable problem.

Bound charge, and proof that formulations are equivalent

If an electric field is applied to a dielectric material, each of the molecules responds by forming a microscopic dipole -- its atomic nucleus will move a tiny distance in the direction of the field, while its electrons will move a tiny distance in the opposite direction. This is called polarization of the material. The distribution of charge that results from these tiny movements turn out to be identical to having a layer of positive charge on one side of the material, and a layer of negative charge on the other side -- a macroscopic separation of charge, even though all of the charges involved are "bound" to a single molecule. This is called bound charge. Likewise, in a magnetized material, there is effectively a "bound current" circulating around the material, despite the fact that no individual charge is travelling a distance larger than a single molecule. The relation between polarization, magnetization, bound charge, and bound current is as follows:

${\displaystyle \rho _{b}=-\nabla \cdot \mathbf {P} }$

${\displaystyle \mathbf {J} _{b}=\nabla \times \mathbf {M} +{\frac {\partial \mathbf {P} }{\partial t}}}$

${\displaystyle \mathbf {D} =\epsilon _{0}\mathbf {E} +\mathbf {P} }$

${\displaystyle \mathbf {B} =\mu _{0}(\mathbf {H} +\mathbf {M} )}$

${\displaystyle \rho =\rho _{b}+\rho _{f}\ }$

${\displaystyle \mathbf {J} =\mathbf {J} _{b}+\mathbf {J} _{f}}$

where P and M are polarization and magnetization, and ρ_b and J_b are bound charge and current, respectively. Plugging in these relations, it can be easily demonstrated that the two formulations of Maxwell's equations given above are precisely equivalent.

Constitutive relations

In order to apply Maxwell's equations (the formulation in terms of free charge and current, and D and H), it is necessary to specify the relations between D and E, and B and H. These are called constitutive relations, and correspond physically to specifying the response of bound charge and current to the field, or equivalently, how much polarization and magnetization a material acquires in the presence of electromagnetic fields.

Case without magnetic or dielectric materials

In the absence of magnetic or dielectric materials, the relations are simple:

${\displaystyle \mathbf {D} =\epsilon _{0}\mathbf {E} ,\;\;\;\mathbf {H} =\mathbf {B} /\mu _{0}}$

where ε₀ and μ₀ are two universal constants, called the permittivity of free space and permeability of free space, respectively.

Case of linear materials

In a "linear", isotropic, nondispersive, uniform material, the relations are also straightforward:

${\displaystyle \mathbf {D} =\epsilon \mathbf {E} ,\;\;\;\mathbf {H} =\mathbf {B} /\mu }$

where ε and μ are constants (which depend on the material), called the permittivity and permeability, respectively, of the material.

General case

For real-world materials, the constitutive relations are not simple proportionalities, except approximately. The relations can usually still be written:

${\displaystyle \mathbf {D} =\epsilon \mathbf {E} ,\;\;\;\mathbf {H} =\mathbf {B} /\mu }$

but ε and μ are not, in general, simple constants, but rather functions. For example, ε and μ can depend upon:

• The strength of the fields (the case of nonlinearity, which occurs when ε and μ are functions of E and B; see, for example, Kerr and Pockels effects),
• The direction of the fields (the case of anisotropy, birefringence, or dichroism; which occurs when ε and μ are second-rank tensors),
• The frequency with which the fields vary (the case of dispersion, which occurs when ε and μ are functions of frequency; see, for example, Kramers-Kronig relations),
• The position inside the material (the case of a nonuniform material, which occurs when ε and μ vary from point to point within the material; for example in a domained structure, heterostructure or a liquid crystal,
• The history of the fields (the case of hysteresis, which occurs when ε and μ are functions of both present and past values of the fields).

Equations in terms of E and B for linear materials

Substituting in the constitutive relations above, Maxwell's equations in a linear material (differential form only) are:

${\displaystyle \nabla \cdot \mathbf {E} ={\frac {\rho _{f}}{\epsilon }}}$

${\displaystyle \nabla \cdot \mathbf {B} =0}$

${\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}$

${\displaystyle \nabla \times \mathbf {B} =\mu \mathbf {J} _{f}+\mu \epsilon {\frac {\partial \mathbf {E} }{\partial t}}}$

These are formally identical to the general formulation in terms of E and B (given above), except that the permittivity of free space was replaced with the permittivity of the material (see also displacement field, electric susceptibility and polarization density), the permeability of free space was replaced with the permeability of the material (see also magnetization, magnetic susceptibility and magnetic field), and only free charges and currents are included (instead of all charges and currents).

Maxwell's equations in vacuum

Starting with the equations appropriate in the case without dielectric or magnetic materials, and assuming that there is no current or electric charge present in the vacuum, we obtain the Maxwell equations in free space:

${\displaystyle \nabla \cdot \mathbf {E} =0}$

${\displaystyle \nabla \cdot \mathbf {B} =0}$

${\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}$

${\displaystyle \nabla \times \mathbf {B} =\ \ \mu _{0}\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}}$

These equations have a solution in terms of travelling sinusoidal plane waves, with the electric and magnetic field directions orthogonal to one another and the direction of travel, and with the two fields in phase, travelling at the speed

${\displaystyle c_{0}={\frac {1}{\sqrt {\mu _{0}\varepsilon _{0}}}}\ .}$

The travelling wave solution is found by substitution of one of the curl equations into the other, producing:

${\displaystyle \nabla \times \left(\nabla \times \mathbf {E} \right)=\nabla \times \left(-{\frac {\partial \mathbf {B} }{\partial t}}\right)=\ \ -\mu _{0}\varepsilon _{0}{\frac {\partial ^{2}\mathbf {E} }{\partial t^{2}}}\ ,}$

which reduces to the electromagnetic wave equation due to an identity in vector calculus. The equation is satisfied in one dimension, for example, by a solution of the form E = E( x − c₀t ), that is, by a solution that is unchanged when t advances to t + Δt at a position x that advances to x + c₀ Δt.

Maxwell discovered that this quantity c₀ is the speed of light in vacuum, and thus that light is a form of electromagnetic radiation. The current SI values for the speed of light, the electric and the magnetic constant are summarized in the following table (values from NIST:Latest value of the constants):

Symbol	Name	Numerical Value	SI Unit of Measure	Type
${\displaystyle c_{0}\ }$	Speed of light in vacuum	${\displaystyle 2.99792458\times 10^{8}}$	meters per second	defined
${\displaystyle \ \varepsilon _{0}}$	Electric constant	${\displaystyle 8.854187817\ldots \times 10^{-12}}$	Farads per meter	derived   ${\displaystyle \ {\stackrel {\mathrm {def} }{=}}\ {\frac {1}{\mu _{0}{c_{0}}^{2}}}}$
${\displaystyle \ \mu _{0}\ }$	Magnetic constant	${\displaystyle 4\pi \times 10^{-7}}$	Henries per meter	defined

Nondimensionalization and unobservability of the speed of light

Because c₀ and μ₀ have defined values (they are properties of the ideal reference state of free space), they are not subject to alteration due to experimental observation. For example, if length is measured in units λ and time in units τ, the distance x in units of λ becomes x = λ ζ and the time t becomes t = τ η, where ζ is the number of length units in x and η is the number of time units in t. The above curl equation for the travelling wave becomes (see nondimensionalization):

${\displaystyle \nabla _{\xi }\times \left(\nabla _{\xi }\times \mathbf {E} \right)=\ \ -\left({\frac {\lambda }{c_{0}\tau }}\right)^{2}{\frac {\partial ^{2}\mathbf {E} }{\partial \eta ^{2}}}\ ,}$

and because the SI units are related by λ = c₀τ this equation does not depend any longer on the speed of light. Experiment could in principle, however, alter the standard meter, for example, as a result of greater measurement accuracy.

With magnetic monopoles

Maxwell's equations of electromagnetism relate the electric and magnetic fields to the motions of electric charges. The standard form of the equations provide for an electric charge, but posit no magnetic charge. Except for this, the equations are symmetric under interchange of electric and magnetic field. In fact, symmetric equations can be written when all charges are zero, and this is how the wave equation is derived (see immediately above).

Fully symmetric equations can also be written if one allows for the possibility of "magnetic charges" analogous to electric charges. With the inclusion of a variable for these magnetic charges, say ${\displaystyle \rho _{m}\,}$, there will also be "magnetic current" variable in the equations, ${\displaystyle {\vec {J}}_{m}\,}$. The extended Maxwell's equations, simplified by nondimensionalization, are as follows:

Name	Without Magnetic Monopoles	With Magnetic Monopoles (hypothetical)
Gauss's law:	${\displaystyle {\vec {\nabla }}\cdot {\vec {E}}=4\pi \rho _{e}}$	${\displaystyle {\vec {\nabla }}\cdot {\vec {E}}=4\pi \rho _{e}}$
Gauss' law for magnetism:	${\displaystyle {\vec {\nabla }}\cdot {\vec {B}}=0}$	${\displaystyle {\vec {\nabla }}\cdot {\vec {B}}=4\pi \rho _{m}}$
Maxwell-Faraday equation (Faraday's law of induction):	${\displaystyle -{\vec {\nabla }}\times {\vec {E}}={\frac {\partial {\vec {B}}}{\partial t}}}$	${\displaystyle -{\vec {\nabla }}\times {\vec {E}}={\frac {\partial {\vec {B}}}{\partial t}}+4\pi {\vec {j}}_{m}}$
Ampère's law (with Maxwell's extension):	${\displaystyle {\vec {\nabla }}\times {\vec {B}}={\frac {\partial {\vec {E}}}{\partial t}}+4\pi {\vec {j}}_{e}}$	${\displaystyle {\vec {\nabla }}\times {\vec {B}}={\frac {\partial {\vec {E}}}{\partial t}}+4\pi {\vec {j}}_{e}}$
Note: the Bivector notation embodies the sign swap, and these four equations can be written as only one equation.

If magnetic charges do not exist, or if they exist but where they are not present in a region, then the new variables are zero, and the symmetric equations reduce to the conventional equations of electromagnetism such as ${\displaystyle {\vec {\nabla }}\cdot {\vec {B}}=0}$. Classically, the question is "Why does the magnetic charge always seem to be zero?"

Solving for the dynamics

The fields in Maxwell's equations are generated by charges and currents. Conversely, the charges and currents are affected by the fields through the Lorentz force equation:

${\displaystyle \mathbf {F} =q(\mathbf {E} +\mathbf {v} \times \mathbf {B} ),}$

where ${\displaystyle q\ }$ is the charge on the particle and ${\displaystyle \mathbf {v} \ }$ is the particle velocity. (It also should be remembered that the Lorenz force is not the only force exerted upon charged bodies, which also may be subject to gravitational, nuclear, etc. forces.) Therefore, in both classical and quantum physics, the precise dynamics of a system form a set of coupled differential equations, which are almost always too complicated to be solved exactly, even at the level of statistical mechanics. This remark applies to not only the dynamics of free charges and currents (which enter Maxwell's equations directly), but also the dynamics of bound charges and currents (which enter Maxwell's equations through the constitutive relations, which can be quite complicated, as described above). As a result, various approximation schemes are typically used.

For example, in real materials, complex transport equations must be solved to determine the time and spatial response of charges, for example, the Boltzmann equation or the Fokker–Planck equation or the Navier-Stokes equations. For example, see magnetohydrodynamics, fluid dynamics, electrohydrodynamics, superconductivity, plasma modeling. An entire physical apparatus for dealing with these matters has developed. See for example, Green–Kubo relations and Green's function (many-body theory).

Of course, the charge-bearing particles may respond to forces that are not electromagnetic interactions; like gravity, or nuclear forces; or be subject to boundary conditions that either are not electromagnetic in origin, or must be approximated for tractability. See for example: boundary layer, boundary condition, Casimir effect. For example, a metallic boundary might be approximated as having infinite conductivity, or collisions between particles and a wall might be treated approximating the wall as a rigid impenetrable barrier.

The connection of Maxwell's equations to the rest of the physical world is via the fundamental charge and current sources, on one hand, and upon the boundary conditions imposed upon the fields themselves, on the other.

Role of boundary conditions

Although Maxwell's equations apply throughout space and time, practical problems are finite and require excising the region to be analyzed from the rest of the universe. To do that, the solutions to Maxwell's equations inside the solution region are joined to the remainder of the universe through boundary conditions and started in time using initial conditions. In addition, the solution region often is broken up into subregions with their own simplified properties, and the solutions in each subregion must be joined to each other across the subregion interfaces using boundary conditions. The links examples of boundary value problems, Sturm-Liouville theory, Dirichlet boundary condition, Neumann boundary condition, mixed boundary condition, Cauchy boundary condition, Sommerfeld radiation condition describe some of the possibilities.

Example:dielectric jump condition

Consider the case where the relative permittivity is a function of x,

${\displaystyle \epsilon _{r}(x)=\kappa _{1}\ }$   for x < 0

${\displaystyle \epsilon _{r}(x)=\kappa _{2}\ }$   for x > 0.

Consider solving the problem:

${\displaystyle {\frac {dD(x)}{dx}}=\rho (x)\ ,}$

with the constitutive relation

${\displaystyle D(x)=\epsilon _{r}(x)\epsilon _{0}E(x)\ .}$

Define the potential φ ( x ) by:

${\displaystyle E(x)=-{\frac {d\phi (x)}{dx}}\ .}$

Then φ ( x ) satisfies the equation:

${\displaystyle {\frac {d}{dx}}\left(\epsilon _{r}(x){\frac {d\phi }{dx}}\right)=-{\frac {\rho }{\epsilon _{0}}}\ .}$

Using this equation, we find a jump condition or discontinuity condition at x = 0 where the step in ε_r occurs by integrating over a small range of x that spans the point x = 0:

${\displaystyle \int _{-\delta }^{\delta }dx{\frac {d}{dx}}\left(\epsilon _{r}(x){\frac {d\phi }{dx}}\right)=\left.\epsilon _{r}(\delta ){\frac {d\phi }{dx}}\right|_{\delta }-\left.\epsilon _{r}(-\delta ){\frac {d\phi }{dx}}\right|_{-\delta }}$

${\displaystyle =\left.\kappa _{2}{\frac {d\phi }{dx}}\right|_{\delta }-\left.\kappa _{1}{\frac {d\phi }{dx}}\right|_{-\delta }}$

${\displaystyle =-\int _{-\delta }^{\delta }dx{\frac {\rho (x)}{\epsilon _{0}}}\ .}$

Assuming the charge density is a continuous function, we let δ → 0 and find the required condition on the derivative of the potential:

${\displaystyle \left.\kappa _{2}{\frac {d\phi }{dx}}\right|_{+}-\left.\kappa _{1}{\frac {d\phi }{dx}}\right|_{-}=0\ ,}$

where the subscripts + and − refer to the two sides of the interface where the jump in ε_r occurs. So the field dφ / dx is not continuous. In words, the jump condition says that at a step in the dielectric constant the product of the relative dielectric constant and the derivative of the potential is continuous.

The above derivation assumed that the charge density ρ ( x ) was continuous. However, that is not always a good model of an interface, and the concept of surface charge sometimes is useful. Surface charge is an idealization of a charge density confined within a small distance of an interface. A practical example is the MOSFET, where such charges occur at the interface between the gate oxide and the silicon substrate. To model such situations, the charge density is taken as

${\displaystyle \rho (x)=\sigma \delta (x)\ ,}$

where the symbol δ ( x ) represents the Dirac delta function, an idealized function that, despite have a value of zero everywhere except at the point x = 0, has an integral of unity over any interval including zero. The symbol σ is the surface charge density. With such a surface charge:

${\displaystyle \int _{-\delta }^{\delta }dx\rho (x)\ =\sigma \ ,}$

making the jump condition:

${\displaystyle \left.\kappa _{2}{\frac {d\phi }{dx}}\right|_{+}-\left.\kappa _{1}{\frac {d\phi }{dx}}\right|_{-}={\frac {\sigma }{\epsilon _{0}}}\ ,}$

In more general terms, this rule is stated as

At a jump discontinuity in dielectric constant, the component of D normal to the interface at a point of discontinuity exhibits a jump in value equal to the surface charge density (if there is any) located at the interface.

Transformation of fields from Maxwell's equations

Consider two inertial frames. As notation, the field variables in one frame are unprimed, and in a frame moving relative to the unprimed frame at velocity v, the fields are denoted with primes. In addition, the fields parallel to the velocity v are denoted by ${\displaystyle {\stackrel {{\vec {E}}_{\parallel }}{}}}$ while the fields perpendicular to v are denoted as ${\displaystyle {\stackrel {{\vec {E}}_{\bot }}{}}}$. In these two frames moving at relative velocity v, the E-fields and B-fields are related according to Maxwell's equations as:

${\displaystyle {\vec {{E}_{\parallel }}}'={\vec {{E}_{\parallel }}}}$            ${\displaystyle {\vec {{B}_{\parallel }}}'={\vec {{B}_{\parallel }}}}$

${\displaystyle {\vec {{E}_{\bot }}}'=\gamma \left({\vec {E}}+{\vec {v}}\times {\vec {B}}\right)_{\parallel }}$    ${\displaystyle {\vec {{B}_{\bot }}}'=\gamma \left({\vec {B}}-{\frac {1}{c_{0}^{2}}}{\vec {v}}\times {\vec {E}}\right)_{\bot }\ ,}$

where

${\displaystyle \gamma \equiv {\frac {1}{\sqrt {1-v^{2}/{c_{0}}^{2}}}}}$

is called the Lorentz factor and ${\displaystyle c_{0}}$ is the speed of light in free space. The inverse transformations are the same except v → −v.

The Heaviside versions in detail

Gauss's law

Gauss's law describes the relation between the electric field and the distribution of electric charge, as follows:

${\displaystyle \nabla \cdot \mathbf {D} =\rho }$

where ${\displaystyle {\rho }}$ is the "free" electric charge density (in units of C/m³), not including bound charge from the polarization of a material, and ${\displaystyle \mathbf {D} }$ is the electric displacement field (in units of C/m²). For stationary charges in vacuum, the solution to Gauss's Law is Coulomb's law.

The equivalent integral form (by the divergence theorem) of Gauss' law is:

${\displaystyle \oint _{S}\mathbf {D} \cdot \mathrm {d} \mathbf {A} =Q_{\mathrm {enclosed} }}$

where:

S is any fixed, closed surface,

The integral is a surface integral, i.e. ${\displaystyle \mathrm {d} \mathbf {A} }$ is a vector whose magnitude is the area of a differential square on the closed surface A, and whose direction is an outward-facing normal vector, and

${\displaystyle Q_{\mathrm {enclosed} }}$ is the free charge enclosed within the surface S. (If the surface itself is charged, that gives an extra contribution weighted by a factor 1/2.)

In a linear, isotropic material, ${\displaystyle \mathbf {D} }$ is directly related to the electric field ${\displaystyle \mathbf {E} }$ via a material-dependent constant called the permittivity, ${\displaystyle \epsilon }$:

${\displaystyle \mathbf {D} =\varepsilon \mathbf {E} }$.

No material (except free space) is precisely linear and isotropic, but many materials are approximately so; and the approximation tends to improve as the electric field weakens. The permittivity of free space is referred to as ${\displaystyle \epsilon _{0}}$, and appears in:

${\displaystyle \nabla \cdot \mathbf {E} ={\frac {\rho _{t}}{\varepsilon _{0}}}}$

where, again, ${\displaystyle \mathbf {E} }$ is the electric field (in units of V/m), ${\displaystyle \rho _{t}}$ is the total charge density (including bound charges), and ${\displaystyle \epsilon _{0}}$ (approximately 8.854 pF/m) is the permittivity of free space. ${\displaystyle \epsilon }$ can also be written as ${\displaystyle \varepsilon _{0}\varepsilon _{r}}$, where ${\displaystyle \epsilon _{r}}$ is the material's relative permittivity or its dielectric constant.

In electrostatics (i.e., when the system is unchanging in time), the expression of the electrostatic potential in terms of the charge density turns out to be mathematically equivalent to Poisson's equation.

Gauss's law for magnetism

"Gauss's law for magnetism" states that the divergence of the magnetic field is always zero (in other words, the magnetic field is a solenoidal vector field):

${\displaystyle \nabla \cdot \mathbf {B} =0}$

where ${\displaystyle \mathbf {B} }$ is the magnetic B-field (in units of Tesla, denoted "T"), also called "magnetic flux density", "magnetic induction", or simply "magnetic field".

By the divergence theorem, it has an equivalent integral form:

${\displaystyle \oint _{S}\mathbf {B} \cdot \mathrm {d} \mathbf {A} =0}$

${\displaystyle \mathrm {d} \mathbf {A} }$ is an infinitesimal vector corresponding t the area of a differential square on the surface ${\displaystyle A}$ with an outward facing surface normal defining its direction.

Like the electric field's integral form, this equation only works if the integral is done over a closed surface.

This equation is related to the magnetic field's structure because it states that given any volume element, the net magnitude of the vector components that point outward from the surface must be equal to the net magnitude of the vector components that point inward. Structurally, this means that the magnetic field lines must be closed loops. Another way of putting it is that the field lines cannot originate from somewhere; attempting to follow the lines backwards to their source or forward to their terminus ultimately leads back to the starting position. Hence, this is a mathematical formulation of the statement that there are no magnetic monopoles.

The Maxwell-Faraday equation

The Maxwell-Faraday equation states:

${\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}\ .}$

This equation is usually referred to as "Faraday's law of induction", but in fact it is only a restricted form of Faraday's law since it doesn't apply to situations involving motionally induced EMF.

The Maxwell-Ampère equation

Ampère's circuital law describes the source of the magnetic field,

${\displaystyle \nabla \times \mathbf {H} =\mathbf {j} +{\frac {\partial \mathbf {D} }{\partial t}}}$

where ${\displaystyle \mathbf {H} }$ is the magnetic field strength (in units of A/m), related to the magnetic flux density ${\displaystyle \mathbf {B} }$ by a constant called the permeability, μ (${\displaystyle \mathbf {B} =\mu \mathbf {H} }$), and ${\displaystyle \mathbf {j} }$ is the current density, defined by: ${\displaystyle \mathbf {j} =\rho _{q}\mathbf {v} }$ where ${\displaystyle \mathbf {v} }$ is a vector field called the drift velocity that describes the velocities of the charge carriers which have a density described by the scalar function ρ_q. The second term on the right hand side of Ampère's Circuital Law is known as the displacement current.

It was Maxwell who added the displacement current term to Ampère's Circuital Law at equation (112) in his 1861 paper On Physical Lines of Force.

Maxwell used the displacement current in conjunction with the original eight equations in his 1865 paper A Dynamical Theory of the Electromagnetic Field to derive a wave equation that has the velocity of light. Most modern textbooks derive this electromagnetic wave equation using the 'Heaviside Four'.

In free space, the permeability μ is the magnetic constant, μ₀, which is defined to be exactly 4π×10 Wb/A•m. Also, the permittivity becomes the electric constant ε₀, also a defined quantity. Thus, in free space, the equation becomes:

${\displaystyle \nabla \times \mathbf {B} =\mu _{0}\mathbf {j} +\mu _{0}\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}}$

Using Stokes theorem the equivalent integral form can be found:

${\displaystyle \oint _{C}\mathbf {B} \cdot \mathrm {d} \mathbf {l} }$ ${\displaystyle =\mu _{0}\int _{S}\mathbf {j} \cdot \mathrm {d} \mathbf {A} +\mu _{0}\varepsilon _{0}\int _{S}{\frac {\partial \mathbf {E} }{\partial t}}\cdot \mathrm {d} \mathbf {A} }$ ${\displaystyle =\mu _{0}I_{\mathrm {encircled} }+\mu _{0}\varepsilon _{0}\int _{S}{\frac {\partial \mathbf {E} }{\partial t}}\cdot \mathrm {d} \mathbf {A} }$

C is the edge of the open surface A (any surface with the curve C as its edge will do), and I_encircled is the current encircled by the curve C (the current through any surface is defined by the equation: ${\displaystyle {\begin{matrix}I_{\mathrm {through} \ A}=\int _{S}\mathbf {j} \cdot \mathrm {d} \mathbf {A} \end{matrix}}}$). Sometimes this integral form of Ampere-Maxwell Law is written as:

${\displaystyle \oint _{C}\mathbf {B} \cdot \mathrm {d} \mathbf {l} =\mu _{0}(I_{\mathrm {enc} }+I_{\mathrm {d,enc} })}$     because the term     ${\displaystyle \varepsilon _{0}\int _{S}{\frac {\partial \mathbf {E} }{\partial t}}\cdot \mathrm {d} \mathbf {A} }$

is displacement current. The displacement current concept was Maxwell's greatest innovation in electromagnetic theory. It implies that a magnetic field appears during the charge or discharge of a capacitor. If the electric flux density does not vary rapidly, the second term on the right hand side (the displacement flux) is negligible, and the equation reduces to Ampere's law.

Due to relativistic invariance, the expression ${\displaystyle \int _{S}{\frac {\partial }{\partial t}}...}$ can be replaced by ${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\left(\int _{S}...\right)}$ provided the surface S rigidly translates in time without deformation, and with some restrictions on the integrand. See Leibniz integral rule.

Maxwell's equations in CGS units

The above equations are given in the International System of Units, or SI for short. In a related unit system, called cgs (short for centimeter-gram-second), the equations take the following form:

${\displaystyle \nabla \cdot \mathbf {D} =4\pi \rho }$

${\displaystyle \nabla \cdot \mathbf {B} =0}$

${\displaystyle \nabla \times \mathbf {E} =-{\frac {1}{c}}{\frac {\partial \mathbf {B} }{\partial t}}}$

${\displaystyle \nabla \times \mathbf {H} ={\frac {1}{c}}{\frac {\partial \mathbf {D} }{\partial t}}+{\frac {4\pi }{c}}\mathbf {j} }$

Where c is the speed of light in a vacuum. For the electromagnetic field in a vacuum, the equations become:

${\displaystyle \nabla \cdot \mathbf {E} =0}$

${\displaystyle \nabla \cdot \mathbf {B} =0}$

${\displaystyle \nabla \times \mathbf {E} =-{\frac {1}{c}}{\frac {\partial \mathbf {B} }{\partial t}}}$

${\displaystyle \nabla \times \mathbf {B} ={\frac {1}{c}}{\frac {\partial \mathbf {E} }{\partial t}}}$

In this system of units the relation between magnetic induction, magnetic field and total magnetization take the form:

${\displaystyle \mathbf {B} =\mathbf {H} +4\pi \mathbf {M} }$

With the linear approximation:

${\displaystyle \mathbf {B} =(\ 1+4\pi \chi _{m}\ )\mathbf {H} }$

${\displaystyle \chi _{m}}$ for vacuum is zero and therefore:

${\displaystyle \mathbf {B} =\mathbf {H} }$

and in the ferro or ferri magnetic materials where ${\displaystyle \chi _{m}}$ is much bigger than 1:

${\displaystyle \mathbf {B} =4\pi \chi _{m}\mathbf {H} }$

The force exerted upon a charged particle by the electric field and magnetic field is given by the Lorentz force equation:

${\displaystyle \mathbf {F} =q\left(\mathbf {E} +{\frac {\mathbf {v} }{c}}\times \mathbf {B} \right),}$

where ${\displaystyle q\ }$ is the charge on the particle and ${\displaystyle \mathbf {v} \ }$ is the particle velocity. This is slightly different from the SI-unit expression above. For example, here the magnetic field ${\displaystyle \mathbf {B} \ }$ has the same units as the electric field ${\displaystyle \mathbf {E} \ }$.

Maxwell's equations and special relativity

Maxwell's equations have a close relation to special relativity: Not only were Maxwell's equations a crucial part of the historical development of special relativity, but also, special relativity has shed new light into Maxwell's equations.

Historical developments

In the late 19th century, because of the appearance of a velocity,

${\displaystyle c_{0}={\frac {1}{\sqrt {\varepsilon _{0}\mu _{0}}}}}$

in the equations, Maxwell's equations were only thought to express electromagnetism in the rest frame of the luminiferous aether (the postulated medium for light, whose interpretation was considerably debated). The symbols represent the permittivity and permeability of free space. The prevailing theory of the aether was that it was a medium that supported electromagnetic waves. Maxwell's work suggested to the American scientist A.A. Michelson that the velocity of the earth through the stationary aether could be detected by a light wave interferometer that he had invented. When the Michelson-Morley experiment was conducted by Edward Morley and Albert Abraham Michelson in 1887, it produced a null result for the change of the velocity of light due to the Earth's motion through the hypothesized aether. Two alternative explanations for this result were investigated. Michelson conducted experiments which sought to prove that the aether was dragged by the earth according to the Stokes aether theory. Another solution was suggested by George FitzGerald, Joseph Larmor and Hendrik Lorentz. Both Larmor (1897) and Lorentz (1899, 1904) derived the Lorentz transformation (so named by Henri Poincaré) as one under which Maxwell's equations were invariant. Poincaré (1900) analyzed the coordination of moving clocks by exchanging light signals. He also established mathematically the group property of the Lorentz transformation (Poincaré 1905).

This culminated in Albert Einstein's revolutionary theory of special relativity, which postulated the absence of any absolute rest frame, dismissed the aether as unnecessary (a bold idea, which did not come to Lorentz nor to Poincaré), and established the invariance of Maxwell's equations in all inertial frames of reference, in contrast to the famous Newtonian equations for classical mechanics. But the transformations between two different inertial frames had to correspond to Lorentz' equations and not - as former believed - to those of Galileo (called Galilean transformations).

The electromagnetic field equations have an intimate link with special relativity, because the equations of special relativity are derived from Maxwell's equations by the Lorentz invariance requirement. Einstein motivated the special theory by noting that a description of a conductor moving with respect to a magnet must generate a consistent set of fields irrespective of whether the force is calculated in the rest frame of the magnet or that of the conductor.

General relativity has also had a close relationship with Maxwell's equations. For example, Kaluza and Klein showed in the 1920s that Maxwell's equations can be derived by extending general relativity into five dimensions. This strategy of using higher dimensions to unify different forces continues to be an active area of research in particle physics.

Electromagnetic tensor

The insights of special relativity motivated the combination of the electric and magnetic fields into a single object, a rank-2 antisymmetric tensor called the electromagnetic tensor:

${\displaystyle F=\left({\begin{matrix}0&{\frac {-E_{x}}{c}}&{\frac {-E_{y}}{c}}&{\frac {-E_{z}}{c}}\\{\frac {E_{x}}{c}}&0&-B_{z}&B_{y}\\{\frac {E_{y}}{c}}&B_{z}&0&-B_{x}\\{\frac {E_{z}}{c}}&-B_{y}&B_{x}&0\end{matrix}}\right)}$

(Here SI units are used; in cgs units, one would have to replace c by 1.)

Two related reasons for this construction are first, this is a covariant tensor -- i.e., when changing to a different inertial frame of reference, this tensor transforms in a certain, specific, mathematically-convenient way -- and second, Maxwell's equations can be written in a compact, "manifestly covariant" form in terms of this tensor, as shown below.

B and E interrelated

The fact that both electric and magnetic fields are combined into a single tensor expresses the fact that, according to relativity, these two fields are different aspects of the same thing—by changing frames of reference, what seemed to be an electric field in one frame can appear as a magnetic field in another frame, and vice versa. By different "frames of reference" is meant the different viewpoints of observers moving with different constant velocities, different inertial frames.

Correspondingly, various authors have attempted to derive various laws of magnetism, starting by assuming various laws of electricity, and also assuming that special relativity is true. For example, it has been surmised that the v×B component of the Lorentz force might be derived from Coulomb's law and special relativity if one assumes invariance of electric charge. See Haskell, Landau and Field. For more examples, see relativistic electromagnetism.

Formulation of Maxwell's equations in special relativity

In special relativity, in order to more clearly express the fact that Maxwell's equations (in vacuum) take the same form in any inertial coordinate system, Maxwell's equations are written in terms of four-vectors and tensors in the "manifestly covariant" form (cgs units):

${\displaystyle {4\pi \over c}j^{\beta }={\partial F^{\alpha \beta } \over {\partial x^{\alpha }}}\ {\stackrel {\mathrm {def} }{=}}\ \partial _{\alpha }F^{\alpha \beta }\ {\stackrel {\mathrm {def} }{=}}\ {F^{\alpha \beta }}_{,\alpha }\,\!}$,

and

${\displaystyle 0=\partial _{\gamma }F_{\alpha \beta }+\partial _{\beta }F_{\gamma \alpha }+\partial _{\alpha }F_{\beta \gamma }\ {\stackrel {\mathrm {def} }{=}}\ {F_{\alpha \beta }}_{,\gamma }+{F_{\gamma \alpha }}_{,\beta }+{F_{\beta \gamma }}_{,\alpha }\ {\stackrel {\mathrm {def} }{=}}\ \epsilon _{\delta \alpha \beta \gamma }{F^{\beta \gamma }}_{,\alpha }}$

where ${\displaystyle \,j^{\alpha }}$ is the 4-current, ${\displaystyle \,F^{\alpha \beta }}$ is the electromagnetic tensor (see above section), ${\displaystyle \,\epsilon _{\alpha \beta \gamma \delta }}$ is the Levi-Civita symbol, and

${\displaystyle {\partial \over {\partial x^{\alpha }}}\ {\stackrel {\mathrm {def} }{=}}\ \partial _{\alpha }\ {\stackrel {\mathrm {def} }{=}}\ {}_{,\alpha }\ {\stackrel {\mathrm {def} }{=}}\ \left({\frac {\partial }{\partial ct}},\nabla \right)}$

is the 4-gradient. Repeated indices are summed over according to Einstein summation convention. We have displayed the results in several common notations. Upper and lower components of a vector, ${\displaystyle v^{\alpha }}$ and ${\displaystyle v_{\alpha }}$ respectively, are interchanged with the fundamental matrix g, e.g., g=diag(+1,-1,-1,-1).

The first tensor equation is an expression of the two inhomogeneous Maxwell's equations, Gauss' law and Ampere's law with Maxwell's correction. The second equation is an expression of the two homogeneous equations, Faraday's law of induction and the absence of magnetic monopoles.

Maxwell's equations in terms of differential forms

In a vacuum, where ε and μ are constant everywhere, Maxwell's equations simplify considerably once the language of differential geometry and differential forms is used. The electric and magnetic fields are now jointly described by a 2-form F in a 4-dimensional spacetime manifold. Maxwell's equations then reduce to the Bianchi identity

${\displaystyle \mathrm {d} {\mathbf {F}}=0}$

where d denotes the exterior derivative — a natural coordinate and metric independent differential operator acting on forms — and the source equation

${\displaystyle \mathrm {d} *{\mathbf {F}}={\mathbf {J}}}$

where the (dual) Hodge star operator * is a linear transformation from the space of 2-forms to the space of (4-2)-forms defined by the metric in Minkowski space (in four dimensions even by any metric conformal to this metric), and the fields are in natural units where ${\displaystyle 1/4\pi \epsilon _{0}=1}$. Here, the 3-form J is called the "electric current form" or "current 3-form" satisfying the continuity equation

${\displaystyle \mathrm {d} {\mathbf {J}}=0}$.

The current 3-form can be integrated over a 3-dimensional space-time region. The physical interpretation of this integral is the charge in that region if it is spacelike, or the amount of charge that flows through a surface in a certain amount of time if that region is a spacelike surface cross a timelike interval. As the exterior derivative is defined on any manifold, the differential form version of the Bianchi identity makes sense for any 4-dimensional manifold, whereas the source equation is defined if the manifold is oriented and has a Lorentz metric. In particular the differential form version of the Maxwell equations are a convenient and intuitive formulation of the Maxwell equations in general relativity.

In a linear, macroscopic theory, the influence of matter on the electromagnetic field is described through more general linear transformation in the space of 2-forms. We call

${\displaystyle C:\Lambda ^{2}\ni {\mathbf {F}}\mapsto {\mathbf {G}}\in \Lambda ^{(4-2)}}$

the constitutive transformation. The role of this transformation is comparable to the Hodge duality transformation. The Maxwell equations in the presence of matter then become:

${\displaystyle \mathrm {d} {\mathbf {F}}=0}$

${\displaystyle \mathrm {d} {\mathbf {G}}={\mathbf {J}}}$

where the current 3-form J still satisfies the continuity equation dJ= 0.

When the fields are expressed as linear combinations (of exterior products) of basis forms ${\displaystyle {\mathbf {\theta }}^{p}}$,

${\displaystyle {\mathbf {F}}={\frac {1}{2}}F_{pq}{\mathbf {\theta }}^{p}\wedge {\mathbf {\theta }}^{q}}$.

the constitutive relation takes the form

${\displaystyle G_{pq}=C_{pq}^{mn}F_{mn}}$

where the field coefficient functions are antisymmetric in the indices and the constitutive coefficients are antisymmetric in the corresponding pairs. In particular, the Hodge duality transformation leading to the vacuum equations discussed above are obtained by taking

${\displaystyle C_{pq}^{mn}=g^{ma}g^{nb}\epsilon _{abpq}{\sqrt {-g}}}$

which up to scaling is the only invariant tensor of this type that can be defined with the metric.

In this formulation, electromagnetism generalises immediately to any 4-dimensional oriented manifold or with small adaptations any manifold, requiring not even a metric. Thus the expression of Maxwell's equations in terms of differential forms leads to a further notational and conceptual simplification. Whereas Maxwell's Equations could be written as two tensor equations instead of eight scalar equations, from which the propagation of electromagnetic disturbances and the continuity equation could be derived with a little effort, using differential forms leads to an even simpler derivation of these results.

Conceptual insight from this formulation

On the conceptual side, from the point of view of physics, this shows that the second and third Maxwell equations should be grouped together, be called the homogeneous ones, and be seen as geometric identities expressing nothing else than: the field F derives from a more "fundamental" potential A. While the first and last one should be seen as the dynamical equations of motion, obtained via the Lagrangian principle of least action, from the "interaction term" A J (introduced through gauge covariant derivatives), coupling the field to matter.

Often, the time derivative in the third law motivates calling this equation "dynamical", which is somewhat misleading; in the sense of the preceding analysis, this is rather an artifact of breaking relativistic covariance by choosing a preferred time direction. To have physical degrees of freedom propagated by these field equations, one must include a kinetic term F *F for A; and take into account the non-physical degrees of freedom which can be removed by gauge transformation A→A' = A-dα: see also gauge fixing and Fadeev-Popov ghosts.

Classical electrodynamics as the curvature of a line bundle

An elegant and intuitive way to formulate Maxwell's equations is to use complex line bundles or principal bundles with fibre U(1). The connection ${\displaystyle \nabla }$ on the line bundle has a curvature ${\displaystyle {\mathbf {F}}=\nabla ^{2}}$ which is a two-form that automatically satisfies ${\displaystyle \mathrm {d} {\mathbf {F}}=0}$ and can be interpreted as a field-strength. If the line bundle is trivial with flat reference connection d we can write ${\displaystyle \nabla =\mathrm {d} +{\mathbf {A}}}$ and F = dA with A the 1-form composed of the electric potential and the magnetic vector potential.

In quantum mechanics, the connection itself is used to define the dynamics of the system. This formulation allows a natural description of the Aharonov-Bohm effect. In this experiment, a static magnetic field runs through a long magnetic wire (e.g. an Fe wire magnetized longitudinally). Outside of this wire the magnetic induction is zero, in contrast to the vector potential, which essentially depends on the magnetic flux through the cross-section of the wire and does not vanish outside. Since there is no electric field either, the Maxwell tensor F = 0 throughout the space-time region outside the tube, during the experiment. This means by definition that the connection ${\displaystyle \nabla }$ is flat there.

However, as mentioned, the connection depends on the magnetic field through the tube since the holonomy along a non-contractible curve encircling the tube is the magnetic flux through the tube in the proper units. This can be detected quantum-mechanically with a double-slit electron diffraction experiment on an electron wave traveling around the tube. The holonomy corresponds to an extra phase shift, which leads to a shift in the diffraction pattern. (See Michael Murray, Line Bundles, 2002 (PDF web link) for a simple mathematical review of this formulation. See also R. Bott, On some recent interactions between mathematics and physics, Canadian Mathematical Bulletin, 28 (1985) no. 2 pp 129-164.)

Maxwell's equations in curved spacetime

Traditional formulation

Matter and energy generate curvature of spacetime. This is the subject of general relativity. Curvature of spacetime affects electrodynamics. An electromagnetic field having energy and momentum will also generate curvature in spacetime. Maxwell's equations in curved spacetime can be obtained by replacing the derivatives in the equations in flat spacetime with covariant derivatives. (Whether this is the appropriate generalization requires separate investigation.) The sourced and source-free equations become (cgs units):

${\displaystyle {4\pi \over c}j^{\beta }=\partial _{\alpha }F^{\alpha \beta }+{\Gamma ^{\alpha }}_{\mu \alpha }F^{\mu \beta }+{\Gamma ^{\beta }}_{\mu \alpha }F^{\alpha \mu }\ {\stackrel {\mathrm {def} }{=}}\ D_{\alpha }F^{\alpha \beta }\ {\stackrel {\mathrm {def} }{=}}\ {F^{\alpha \beta }}_{;\alpha }\,\!}$,

and

${\displaystyle 0=\partial _{\gamma }F_{\alpha \beta }+\partial _{\beta }F_{\gamma \alpha }+\partial _{\alpha }F_{\beta \gamma }=D_{\gamma }F_{\alpha \beta }+D_{\beta }F_{\gamma \alpha }+D_{\alpha }F_{\beta \gamma }}$.

Here,

${\displaystyle {\Gamma ^{\alpha }}_{\mu \beta }\!}$

is a Christoffel symbol that characterizes the curvature of spacetime and ${\displaystyle D_{\gamma }}$ is the covariant derivative.

Formulation in terms of differential forms

The formulation of the Maxwell equations in terms of differential forms can be used without change in general relativity. The equivalence of the more traditional general relativistic formulation using the covariant derivative with the differential form formulation can be seen as follows. Choose local coordinates ${\displaystyle x^{\alpha }}$ which gives a basis of 1-forms ${\displaystyle dx^{\alpha }}$ in every point of the open set where the coordinates are defined. Using this basis and cgs units we define

• The antisymmetric infinitesimal field tensor ${\displaystyle F_{\alpha \beta }}$, corresponding to the field 2-form F

${\displaystyle {\mathbf {F}}:={\frac {1}{2}}F_{\alpha \beta }\,\mathrm {d} \,x^{\alpha }\wedge \mathrm {d} \,x^{\beta }}$

• The current-vector infinitesimal 3-form J

${\displaystyle {\mathbf {J}}:={4\pi \over c}j^{\alpha }{\sqrt {-g}}\,\epsilon _{\alpha \beta \gamma \delta }\mathrm {d} \,x^{\beta }\wedge \mathrm {d} \,x^{\gamma }\wedge \mathrm {d} \,x^{\delta }}$

Here g is as usual the determinant of the metric tensor ${\displaystyle g_{\alpha \beta }}$. A small computation that uses the symmetry of the Christoffel symbols (i.e. the torsion-freeness of the Levi Civita connection) and the covariant constantness of the Hodge star operator then shows that in this coordinate neighborhood we have:

• the Bianchi identity

${\displaystyle \mathrm {d} {\mathbf {F}}=2(\partial _{\gamma }F_{\alpha \beta }+\partial _{\beta }F_{\gamma \alpha }+\partial _{\alpha }F_{\beta \gamma })\mathrm {d} \,x^{\alpha }\wedge \mathrm {d} \,x^{\beta }\wedge \mathrm {d} \,x^{\gamma }=0}$

• the source equation

${\displaystyle \mathrm {d} *{\mathbf {F}}={F^{\alpha \beta }}_{;\alpha }{\sqrt {-g}}\,\epsilon _{\beta \gamma \delta \eta }\mathrm {d} \,x^{\gamma }\wedge \mathrm {d} \,x^{\delta }\wedge \mathrm {d} \,x^{\eta }={\mathbf {J}}}$

• the continuity equation

${\displaystyle \mathrm {d} {\mathbf {J}}={4\pi \over c}{j^{\alpha }}_{;\alpha }{\sqrt {-g}}\,\epsilon _{\alpha \beta \gamma \delta }\mathrm {d} \,x^{\alpha }\wedge \mathrm {d} \,x^{\beta }\wedge \mathrm {d} \,x^{\gamma }\wedge \mathrm {d} \,x^{\delta }=0}$

Footnotes and references

1. Ironically it is an equation which Maxwell himself was absolutely responsible for even though it doesn't count as a "Maxwell's equation". This extra equation appeared in an original list of eight Maxwell's equations in his 1865 paper entitled A Dynamical Theory of the Electromagnetic Field. Maxwell derived it from Faraday's law when Lorentz was still a young boy.
2. Oliver Heaviside ((2001) Facsimile of 1893 Edition). Electromagnetic theory. Adamant Media Corporation. p. Vol. 1. ISBN 1402172982. {{cite book}}: Check date values in: |year= (help)CS1 maint: year (link)
3. Oliver Heaviside ((2007) Facsimile of 1912 Edition). Electromagnetic theory. Cosimo Classics. p. Vol. 3. ISBN 1602062625. {{cite book}}: Check date values in: |year= (help)CS1 maint: year (link)
4. In this article, this version is termed the Maxwell-Faraday equation to keep clear the distinction from Faraday's law of induction.
5. There is merit, however, in the Heaviside decision to separate the Lorentz force from the main four Maxwell equations. Heaviside's four equations express the fields' dependence upon current and charge, making calculation of these currents and charges an undertaking apart. In fact, this separate calculation is very complicated, and may involve the Lorentz force law directly, or only indirectly via constitutive relations, as described later in the article. Placing this complication outside the Maxwell equations is a tidy separation.
6. David J Griffiths (1999). Introduction to electrodynamics (Third Edition ed.). Prentice Hall. p. pp. 559-562. ISBN 013805326X. {{cite book}}: |edition= has extra text (help); |page= has extra text (help)
7. See NIST Special Publication 330, Appendix 2, p. 45 : "Current practice is to use c₀ to denote the speed of light in vacuum (ISO 31)."
8. Tai L. Chow (2006). Electromagnetic theory. Sudbury MA: Jones and Bartlett. p. Chapter 10.21; p. 402-403 ff. ISBN 0-7637-3827-1.
9. The term Maxwell-Faraday equation frequently is replaced by Faraday's law of induction or even Faraday's law. These last two terms have multiple meanings, so Maxwell-Faraday equation is used here to avoid confusion.
10. See ISO 31-5
11. Details can be found in U. Krey, A. Owen, Basic Theoretical Physics - A Concise Overview, Springer, Berlin and elsewhere, 2007, ISBN 978-3-540-36804-5
12. Tai L. Chow (2006). Electromagnetic theory. Sudbury MA: Jones and Bartlett. p. p. 395. ISBN 0-7637-3827-1. {{cite book}}: |page= has extra text (help)
13. E M Lifshitz, L D Landau (1980). The classical theory of fields: Vol. 2 (Course of theoretical physics) (Fourth Edition ed.). Oxford UK: Butterworth-Heinemann. ISBN 0750627689. {{cite book}}: |edition= has extra text (help)
14. J H Field (2006) "Classical electromagnetism as a consequence of Coulomb's law, special relativity and Hamilton's principle and its relationship to quantum electrodynamics". Phys. Scr. 74 702-717

See also

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• Electromagnetic wave equation
• Sinusoidal plane-wave solutions of the electromagnetic wave equation
• Nonhomogeneous electromagnetic wave equation
• Interface conditions for electromagnetic fields
• Photon dynamics in the double-slit experiment
• Photon polarization
• Computational electromagnetics
• Finite-difference time-domain method

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• Jefimenko's equations
• Moving magnet and conductor problem
• Abraham-Lorentz force
• Theoretical and experimental justification for the Schrödinger equation
• Mathematical descriptions of the electromagnetic field
• Lorentz force
• Ampere's law

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• Electrical generator
• Transformer
• Waveguide
• Antenna (radio)
• Photonic crystal
• Fresnel equations
• Laser
• Bremsstrahlung
• Linear response function
• Kramers-Kronig relation
• Green–Kubo relations
• Green's function (many-body theory)

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• Mathematical aspects of Maxwell's equation are discussed on the Dispersive PDE Wiki.

Further reading

Journal articles

• James Clerk Maxwell, "A Dynamical Theory of the Electromagnetic Field", Philosophical Transactions of the Royal Society of London 155, 459-512 (1865). (This article accompanied a December 8, 1864 presentation by Maxwell to the Royal Society.)

The developments before relativity

• Joseph Larmor (1897) "On a dynamical theory of the electric and luminiferous medium", Phil. Trans. Roy. Soc. 190, 205-300 (third and last in a series of papers with the same name).
• Hendrik Lorentz (1899) "Simplified theory of electrical and optical phenomena in moving systems", Proc. Acad. Science Amsterdam, I, 427-43.
• Hendrik Lorentz (1904) "Electromagnetic phenomena in a system moving with any velocity less than that of light", Proc. Acad. Science Amsterdam, IV, 669-78.
• Henri Poincaré (1900) "La theorie de Lorentz et la Principe de Reaction", Archives Neerlandaies, V, 253-78.
• Henri Poincaré (1901) Science and Hypothesis
• Henri Poincaré (1905) "Sur la dynamique de l'electron", Comptes Rendues, 140, 1504-8.

see

• Macrossan, M. N. (1986) "A note on relativity before Einstein", Brit. J. Phil. Sci., 37, 232-234

University level textbooks

Undergraduate

• Sadiku, Matthew N. O. (2006). Elements of Electromagnetics (4th ed.). Oxford University Press. ISBN 0-19-5300483.
• Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.). Prentice Hall. ISBN 0-13-805326-X.
• Hoffman, Banesh, 1983. Relativity and Its Roots. W. H. Freeman.
• Lounesto, Pertti, 1997. Clifford Algebras and Spinors. Cambridge Univ. Press. Chpt. 8 sets out several variants of the equations, using exterior algebra and differential forms.
• Edward Mills Purcell (1985). Electricity and Magnetism. McGraw-Hill. ISBN 0-07-004908-4.
• Stevens, Charles F., 1995. The Six Core Theories of Modern Physics. MIT Press. ISBN 0-262-69188-4.
• Tipler, Paul (2004). Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics (5th ed.). W. H. Freeman. ISBN 0-7167-0810-8.
• Schwarz, Melvin (1987). Principles of Electrodynamics. Dover Publications. ISBN 0-486-65493-1.
• Ulaby, Fawwaz T. (2007). Fundamentals of Applied Electromagnetics (5th ed.). Pearson Education, Inc. ISBN 0-13-241326-4.
• Krey, U., Owen, A. (2007), Basic Theoretical Physics - A Concise Overview, esp. part II, Springer, ISBN 978-3-540-36804-5

Graduate

• J. D. Jackson, 1999. Classical Electrodynamics (3rd ed.). Wiley. ISBN 0-471-30932-X.

Older classics

• E M Lifshitz, L D Landau (1980). The classical theory of fields: Vol. 2 (Course of theoretical physics) (Fourth Edition ed.). Oxford UK: Butterworth-Heinemann. ISBN 0750627689. {{cite book}}: |edition= has extra text (help)
• E M Lifshitz, L D Landau, & L. P. Pitaevskii (1984). Electrodynamics of continuous media: Vol. 8 (Course of theoretical physics) (Second Edition ed.). Oxford UK: Butterworth-Heinemann. ISBN 0750626348. {{cite book}}: |edition= has extra text (help)CS1 maint: multiple names: authors list (link)
• James Clerk Maxwell, 1873. A Treatise on Electricity and Magnetism. Dover. ISBN 0-486-60637-6.
• Charles W. Misner, Kip Thorne, John Archibald Wheeler, 1973. Gravitation. W.H. Freeman. ISBN 0-7167-0344-0. Sets out the equations using differential forms.

Computational techniques

• R. F. Harrington (1993). Field Computation by Moment Methods. Wiley-IEEE Press. ISBN 0-78031-014-4.
• W. C. Chew, J.-M. Jin, E. Michielssen, and J. Song (2001). Fast and Efficient Algorithms in Computational Electromagnetics. Artech House Publishers. ISBN 1-58053-152-0.{{cite book}}: CS1 maint: multiple names: authors list (link)
• J. Jin (2002). The Finite Element Method in Electromagnetics, 2nd. ed. Wiley-IEEE Press. ISBN 0-47143-818-9.
• Allen Taflove and Susan C. Hagness (2005). Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. Artech House Publishers. ISBN 1-58053-832-0.

External links

Modern treatments

• Electromagnetism, B. Crowell, Fullerton College
• Lecture series: Relativity and electromagnetism, R. Fitzpatrick, University of Texas at Austin
• Electromagnetic waves from Maxwell's equations on Project PHYSNET.

Historical

• A Treatise on Electricity And Magnetism - Volume 1 - 1873 - Posner Memorial Collection - Carnegie Mellon University
• A Treatise on Electricity And Magnetism - Volume 2 - 1873 - Posner Memorial Collection - Carnegie Mellon University
• Original Maxwell Equations - Maxwell's 20 Equations in 20 Unknowns - PDF
• On Faraday's Lines of Force - 1855/56 Maxwell's first paper (Part 1 & 2) - Compiled by Blaze Labs Research (PDF)
• On Physical Lines of Force - 1861 Maxwell's 1861 paper describing magnetic lines of Force - Predecessor to 1873 Treatise
• A Dynamical Theory Of The Electromagnetic Field - 1865 Maxwell's 1865 paper describing his 20 Equations in 20 Unknowns - Predecessor to the 1873 Treatise
• http://www.zpenergy.com/downloads/Maxwell_1864_1.pdf
• http://www.zpenergy.com/downloads/Maxwell_1864_2.pdf
• http://www.zpenergy.com/downloads/Maxwell_1864_3.pdf
• http://www.zpenergy.com/downloads/Maxwell_1864_4.pdf
• http://www.zpenergy.com/downloads/Maxwell_1864_5.pdf
• http://www.zpenergy.com/downloads/Maxwell_1864_6.pdf
• Catt, Walton and Davidson. "The History of Displacement Current". Wireless World, March 1979.

Feynman’s derivation of Maxwell equations

• Feynman's derivation of Maxwell equations and extra dimensions

Other

• Simple explanation of the equations and their physical implications - David Morgan-Mar - Irregular Webcomic!
• According to an article in Physicsweb, the Maxwell equations rate as "The greatest equations ever".

• Electrodynamics
• Electromagnetism
• Equations
• Partial differential equations
• Fundamental physics concepts