Maxwell's equations
Maxwell's equations (sometimes called the Maxwell equations) are the set of four equations, attributed to James Clerk Maxwell, that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter.
Maxwell's four equations express, respectively, how electric charges produce electric fields (Gauss' law), the experimental absence of magnetic charges, how currents produce magnetic fields (Ampere's law), and how changing magnetic fields produce electric fields (Faraday's law of induction). Maxwell, in 1864, was the first to put all four equations together and to notice that a correction was required to Ampere's law: changing electric fields act like currents, likewise producing magnetic fields. (This additional term is called the displacement current.) The most common modern notation for these equations was developed by Oliver Heaviside.
Furthermore, Maxwell showed that waves of oscillating electric and magnetic fields travel through empty space at a speed that could be predicted from simple electrical experiments—using the data available at the time, Maxwell obtained a velocity of 310,740,000 m/s. Maxwell (1865) wrote:
- This velocity is so nearly that of light, that it seems we have strong reason to conclude that light itself (including radiant heat, and other radiations if any) is an electromagnetic disturbance in the form of waves propagated through the electromagnetic field according to electromagnetic laws.
Maxwell was correct in this conjecture, though he did not live to see its vindication by Heinrich Hertz in 1888. Maxwell's quantitative explanation of light as an electromagnetic wave is considered one of the great triumphs of 19th-century physics. (Actually, Michael Faraday had postulated a similar picture of light in 1846, but had not been able to give a quantitative description or predict the velocity.) Moreover, it laid the foundation for many future developments in physics, such as special relativity and its unification of electric and magnetic fields as a single tensor quantity, and Kaluza and Klein's unification of electromagnetism with gravity and general relativity.
Contents
- 1 Historical developments of Maxwell's equations and relativity
- 2 Summary of the equations
- 3 Detail
- 4 Maxwell's equations in CGS units
- 5 Formulation of Maxwell's equations in special relativity
- 6 Maxwell's equations in terms of differential forms
- 7 Classical electrodynamics as a line bundle
- 8 See also
- 9 References
- 10 External links
Historical developments of Maxwell's equations and relativity
Maxwell's 1865 formulation was in terms of 20 equations in 20 variables, which included several equations now considered to be auxiliary to what are now called "Maxwell's equations" — the corrected Ampere's law (three component equations), Gauss' law for charge (one equation), the relationship between total and displacement current densities (three component equations), the relationship between magnetic field and the vector potential (three component equations, which imply the absence of magnetic charge), the relationship between electric field and the scalar and vector potentials (three component equations, which imply Faraday's law), the relationship between the electric and displacement fields (three component equations), Ohm's law relating current density and electric field (three component equations), and the continuity equation relating current density and charge density (one equation).
The modern mathematical formulation of Maxwell's equations is due to Oliver Heaviside and Willard Gibbs, who in 1884 reformulated Maxwell's original system of equations to a far simpler representation using vector calculus. (In 1873 Maxwell also published a quaternion-based notation that ultimately proved unpopular.) The change to the vector notation produced a symmetric mathematical representation that reinforced the perception of physical symmetries between the various fields. This highly symmetrical formulation would directly inspire later developments in fundamental physics.
In the late 19th century, because of the appearance of a velocity,
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c=\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 (electromagnetism) of empty space. When the Michelson-Morley experiment, conducted by Edward Morley and Albert Abraham Michelson, produced a null result for the change of the velocity of light due to the Earth's hypothesized motion through the aether, however, alternative explanations were sought 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) analysed the coordination of moving clocks by exchanging light signals. He also established the group property of the Lorentz transformation (Poincaré 1905). This culminated in Einstein's theory of special relativity, which postulated the absence of any absolute rest frame (or aether) and the invariance of Maxwell's equations in all frames of reference.
The electromagnetic field equations have an intimate link with special relativity: the magnetic field equations can be derived from consideration of the transformation of the electric field equations under relativistic transformations at low velocities. (In relativity, the equations are written in an even more compact, "manifestly covariant" form, in terms of the rank-2 antisymmetric field-strength 4-tensor that unifies the electric and magnetic fields into a single object.)
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 is an active area of research in particle physics.
Summary of the equations
All variables that are in bold represent vector quantities.
General case
Name | Differential form | Integral form |
---|---|---|
Gauss' law: | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \cdot \mathbf{D} = \rho } | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \oint_S \mathbf{D} \cdot d\mathbf{A} = \int_V \rho \cdot dV} |
Gauss' law for magnetism (absence of magnetic monopoles): |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \cdot \mathbf{B} = 0} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \oint_S \mathbf{B} \cdot d\mathbf{A} = 0} |
Faraday's law of induction: | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \oint_C \mathbf{E} \cdot d\mathbf{l} = - \ { d \over dt } \int_S \mathbf{B} \cdot d\mathbf{A}} |
Ampère's law (with Maxwell's extension): |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}} {\partial t}} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \oint_C \mathbf{H} \cdot d\mathbf{l} = \int_S \mathbf{J} \cdot d \mathbf{A} + {d \over dt} \int_S \mathbf{D} \cdot d \mathbf{A}} |
where the following table provides the meaning of each symbol and the SI unit of measure:
Symbol | Meaning | SI Unit of Measure |
---|---|---|
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{E}} | electric field | volt per metre |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{H}} | magnetic field strength | ampere per metre |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{D}} | electric displacement field | coulomb per square metre |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{B}} | magnetic flux density also called the magnetic induction. |
tesla, or equivalently, weber per square metre |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \rho \ } | free electric charge density, not including dipole charges bound in a material |
coulomb per cubic metre |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{J}} | free current density, not including polarization or magnetization currents bound in a material |
ampere per square metre |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d\mathbf{A}} | differential vector element of surface area A, with infinitesimally small magnitude and direction normal to surface S |
square meters |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dV \ } | differential element of volume V enclosed by surface S | cubic meters |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d \mathbf{l} } | differential vector element of path length tangential to contour C enclosing surface S | meters |
and
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \cdot} is the divergence operator (SI unit: 1 per metre),
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \times} is the curl operator (SI unit: 1 per metre).
Although SI units are given here for the various symbols, Maxwell's equations will hold unchanged in many different unit systems (and with only minor modifications in all others). The most commonly used systems of units are SI units, used for engineering, electronics and most practical physics experiments, and Planck units (also known as "natural units"), used in theoretical physics, quantum physics and cosmology. An older system of units, the cgs system, is sometimes also used.
The second equation is equivalent to the statement that magnetic monopoles do not exist. The force exerted upon a charged particle by the electric field and magnetic field is given by the Lorentz force equation:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{F} = q (\mathbf{E} + \mathbf{v} \times \mathbf{B}),}
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q \ } is the charge on the particle and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{v} \ } is the particle velocity. This is slightly different when expressed in the cgs system of units below.
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, below (the microscopic Maxwell's equations, ignoring quantum effects, are simply those of a vacuum — but one must include all atomic charges and so on, which is normally an intractable problem).
In linear materials
In linear materials, the polarization density P (in coulombs per square meter) and magnetization density M (in amperes per meter) are given by:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{P} = \chi_e \varepsilon_0 \mathbf{E} }
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{M} = \chi_m \mathbf{H} }
and the D and B fields are related to E and H by:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{D} \ \ = \ \ \varepsilon_0 \mathbf{E} + \mathbf{P} \ \ = \ \ (1 + \chi_e) \varepsilon_0 \mathbf{E} \ \ = \ \ \varepsilon \mathbf{E} }
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{B} \ \ = \ \ \mu_0 ( \mathbf{H} + \mathbf{M} ) \ \ = \ \ (1 + \chi_m) \mu_0 \mathbf{H} \ \ = \ \ \mu \mathbf{H} }
where:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi_e } is the electrical susceptibility of the material,
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi_m } is the magnetic susceptibility of the material,
ε is the electrical permittivity of the material, and
μ is the magnetic permeability of the material
(This can actually be extended to handle nonlinear materials as well, by making ε and μ depend upon the field strength; see e.g. the Kerr and Pockels effects.)
In non-dispersive, isotropic media, ε and μ are time-independent scalars, and Maxwell's equations reduce to
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \cdot \varepsilon \mathbf{E} = \rho }
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \cdot \mu \mathbf{H} = 0}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \times \mathbf{E} = - \mu \frac{\partial \mathbf{H}} {\partial t}}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \times \mathbf{H} = \mathbf{J} + \varepsilon \frac{\partial \mathbf{E}} {\partial t}}
In a uniform (homogeneous) medium, ε and μ are constants independent of position, and can thus be furthermore interchanged with the spatial derivatives.
More generally, ε and μ can be rank-2 tensors (3×3 matrices) describing birefringent (anisotropic) materials. Also, although for many purposes the time/frequency-dependence of these constants can be neglected, every real material exhibits some material dispersion by which ε and/or μ depend upon frequency (and causality constrains this dependence to obey the Kramers-Kronig relations).
In vacuum, without charges or currents
The vacuum is a linear, homogeneous, isotropic, dispersionless medium, and the proportionality constants in the vacuum are denoted by ε_{0} and μ_{0} (neglecting very slight nonlinearities due to quantum effects).
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{D} = \varepsilon_0 \mathbf{E} }
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{B} = \mu_0 \mathbf{H} }
Since there is no current or electric charge present in the vacuum, we obtain the Maxwell's equations in free space:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \cdot \mathbf{E} = 0}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \cdot \mathbf{H} = 0}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \times \mathbf{E} = - \mu_0 \frac{\partial\mathbf{H}} {\partial t}}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \times \mathbf{H} = \ \ \varepsilon_0 \frac{\partial \mathbf{E}} {\partial t}}
These equations have a simple 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
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}} }
Maxwell discovered that this quantity c is simply the speed of light in vacuum, and thus that light is a form of electromagnetic radiation. The currently accepted values for the speed of light, the permittivity,and the permeability are summarized in the following table:
Symbol | Name | Numerical Value | SI Unit of Measure | Type |
---|---|---|---|---|
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c \ } | Speed of light | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2.998 \times 10^{8} } | meters per second | defined |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \varepsilon_0 } | Permittivity | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 8.854 \times 10^{-12} } | farads per meter | derived |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \mu_0 \ } | Permeability | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4 \pi \times 10^{-7} } | henries per meter | defined |
Detail
Charge density and the electric field
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \cdot \mathbf{D} = \rho} ,
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\rho}} is the free electric charge density (in units of C/m^{3}), not including dipole charges bound in a material, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{D}} is the electric displacement field (in units of C/m^{2}). This equation corresponds to Coulomb's law for stationary charges in vacuum.
The equivalent integral form (by the divergence theorem), also known as Gauss' law, is:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \oint_A \mathbf{D} \cdot d\mathbf{A} = Q_\mbox{enclosed}}
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d\mathbf{A}} is the area of a differential square on the closed surface A with an outward facing surface normal defining its direction, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q_\mbox{enclosed}} is the free charge enclosed by the surface.
In a linear material, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{D}} is directly related to the electric field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{E}} via a material-dependent constant called the permittivity, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon} :
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{D} = \varepsilon \mathbf{E}} .
Any material can be treated as linear, as long as the electric field is not extremely strong. The permittivity of free space is referred to as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon_0} , and appears in:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \cdot \mathbf{E} = \frac{\rho_t}{\varepsilon_0}}
where, again, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{E}} is the electric field (in units of V/m), Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho_t} is the total charge density (including bound charges), and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon_0} (approximately 8.854 pF/m) is the permittivity of free space. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon} can also be written as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon_0 \cdot \varepsilon_r} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon_r} is the material's relative permittivity or its dielectric constant.
Compare Poisson's equation.
The structure of the magnetic field
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \cdot \mathbf{B} = 0}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{B}} is the magnetic flux density (in units of teslas, T), also called the magnetic induction.
Equivalent integral form:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \oint_A \mathbf{B} \cdot d\mathbf{A} = 0}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d\mathbf{A}} is the area of a differential square on the surface Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 the mathematical formulation of the assumption that there are no magnetic monopoles.
A changing magnetic flux and the electric field
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \times \mathbf{E} = -\frac {\partial \mathbf{B}}{\partial t}}
Equivalent integral Form:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \oint_{s} \mathbf{E} \cdot d\mathbf{s} = - \frac {d\Phi_{\mathbf{B}}} {dt}} where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_{\mathbf{B}} = \int_{A} \mathbf{B} \cdot d\mathbf{A}}
where
Φ_{B} is the magnetic flux through the area A described by the second equation
E is the electric field generated by the magnetic flux
s is a closed path in which current is induced, such as a wire.
The electromotive force (sometimes denoted Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{E}} , not to be confused with the permittivity above) is equal to the value of this integral.
This law corresponds to the Faraday's law of electromagnetic induction.
Some textbooks show the right hand sign of the Integral form with an N (representing the number of coils of wire that are around the edge of A) in front of the flux derivative. The N can be taken care of in calculating A (multiple wire coils means multiple surfaces for the flux to go through), and it is an engineering detail so it has been omitted here.
The negative sign is necessary to maintain conservation of energy. It is so important that it even has its own name, Lenz's law.
This equation relates the electric and magnetic fields, but it also has a lot of practical applications, too. This equation describes how electric motors and electric generators work. Specifically, it demonstrates that a voltage can be generated by varying the magnetic flux passing through a given area over time, such as by uniformly rotating a loop of wire through a fixed magnetic field. In a motor or generator, the fixed excitation is provided by the field circuit and the varying voltage is measured across the armature circuit. In some types of motors/generators, the field circuit is mounted on the rotor and the armature circuit is mounted on the stator, but other types of motors/generators employ the reverse configuration.
Maxwell's equations apply to a right-handed coordinate system. To apply them unmodified to a left handed system would mean a reversal of polarity of magnetic fields (not inconsistent, but confusingly against convention).
The source of the magnetic field
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \times \mathbf{H} = \mathbf{J} + \frac {\partial \mathbf{D}} {\partial t}}
where H is the magnetic field strength (in units of A/m), related to the magnetic flux B by a constant called the permeability, μ (B = μH), and J is the current density, defined by: J = ∫ρ_{q}vdV where v is a vector field called the drift velocity that describes the velocities of that charge carriers which have a density described by the scalar function ρ_{q}.
In free space, the permeability μ is the permeability of free space, μ_{0}, which is defined to be exactly 4π×10^{-7} W/A·m. Also, the permittivity becomes the permittivity of free space ε_{0}. Thus, in free space, the equation becomes:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0\varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}}
Equivalent integral form:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \oint_s \mathbf{B} \cdot d\mathbf{s} = \mu_0 I_\mbox{encircled} + \mu_0\varepsilon_0 \int_A \frac{\partial \mathbf{E}}{\partial t} \cdot d \mathbf{A}}
s is the edge of the open surface A (any surface with the curve s as its edge will do), and I_{encircled} is the current encircled by the curve s (the current through any surface is defined by the equation: I_{through A} = ∫_{A}J·dA).
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.
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:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \cdot \mathbf{E} = 4\pi\rho}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \cdot \mathbf{B} = 0}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \times \mathbf{E} = -\frac{1}{c} \frac{\partial \mathbf{B}} {\partial t}}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \times \mathbf{B} = \frac{1}{c} \frac{ \partial \mathbf{E}} {\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:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \cdot \mathbf{E} = 0}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \cdot \mathbf{B} = 0}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \times \mathbf{E} = -\frac{1}{c} \frac{\partial \mathbf{B}} {\partial t}}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \times \mathbf{B} = \frac{1}{c} \frac{\partial \mathbf{E}}{\partial t} }
The force exerted upon a charged particle by the electric field and magnetic field is given by the Lorentz force equation:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{F} = q (\mathbf{E} + \frac{\mathbf{v}}{c} \times \mathbf{B}),}
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q \ } is the charge on the particle and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{v} \ } is the particle velocity. This is slightly different from the SI-unit expression above. For example, here the magnetic field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{B} \ } has the same units as the electric field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{E} \ } .
Formulation of Maxwell's equations in special relativity
Mathematical note: In this section the abstract index notation will be used.
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, the vacuum Maxwell's equations are written in terms of four-vectors and tensors in the "manifestly covariant" form:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J^ b = \partial_a F^{ab} \,\!} ,
and
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 = \partial_c F_{ab} + \partial_b F_{ca} + \partial_a F_{bc}}
the last of which is equivalent to:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 = \epsilon_{dabc}\partial^a F^{bc} \,\! }
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \, J^a} is the 4-current, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \, F^{ab}} is the field strength tensor (written as a 4 × 4 matrix), Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \, \epsilon_{abcd}} is the Levi-Civita symbol, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \partial_a = (\partial/\partial ct, \nabla)} is the 4-gradient (so that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \partial_a \partial^a} is the d'Alembertian operator). (The Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} in the first equation is implicitly summed over, according to Einstein notation.) The first tensor equation expresses the two inhomogeneous Maxwell's equations: Gauss' law and Ampere's law with Maxwell's correction. The second equation expresses the other two, homogenous equations: Faraday's law of induction and the absence of magnetic monopoles.
More explicitly, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J^a = \, (c \rho, \vec J) } (as a contravariant vector), in terms of the charge density ρ and the current density Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec J} . The 4-current satisfies the continuity equation
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J^a{}_{,a} \, = 0}
In terms of the 4-potential (as a contravariant vector) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^{a} = \left(\phi, \vec A c \right)} , where φ is the electric potential and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec A} is the magnetic vector potential in the Lorenz gauge Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left ( \partial_a A^a = 0 \right )} , F can be expressed as:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F^{ab} = \partial^b A^a - \partial^a A^b \,\!}
which leads to the 4 × 4 matrix rank-2 tensor:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F^{ab} = \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) .}
The fact that both electric and magnetic fields are combined into a single tensor expresses the fact that, according to relativity, both of these 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.
Using the tensor form of Maxwell's equations, the first equation implies
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Box F^{ab} = 0} (See Electromagnetic four-potential for the relationship between the d'Alembertian of the four-potential and the four-current, expressed in terms of the older vector operator notation).
Different authors sometimes employ different sign conventions for the above tensors and 4-vectors (which does not affect the physical interpretation).
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \, F^{ab}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \, F_{ab}} are not the same: they are related by the Minkowski metric tensor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \eta} : Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_{ab} =\, \eta_{ac} \eta_{bd} F^{cd}} . This introduces sign changes in the mixed spacetime components of F'; more complex metric dualities are encountered in general relativity.
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
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d\bold{F}=0}
where d denotes the exterior derivative - a differential operator acting on forms - and the source equation
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d{*\bold{F}}=*\bold{J}}
where * is the Hodge star (dual) operator. Here, the fields are represented in natural units where ε_{0} is 1. Here, J is a 1-form called the "electric current" or "current form" satisfying the continuity equation
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d{*\bold{J}}=0}
In a linear, macroscopic theory, the influence of matter on the electromagnetic field is described through a linear transformation in the space of 2-forms, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Lambda^2} . We call this the constitutive transformation
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C:\Lambda^2\ni\bold{F}\mapsto \bold{G}=C\bold{F}\in\Lambda^2}
The rôle of this transformation is comparable to the Hodge duality transformation and we write the Maxwell equations in the presence of matter as:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d*\bold{G} = *\bold{J}}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d\bold{F} = 0}
When the fields are expressed as linear combinations (of exterior products) of basis forms Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bold{\theta}^p} ,
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bold{F} = F_{pq}\bold{\theta}^p\wedge\bold{\theta}^q}
the constitutive relation takes the form
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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.
This shows that the expression of Maxwell's equations in terms of differential forms leads to a further notational simplification. Whereas Maxwell's equations were once eight scalar equations, they could be written as two tensor equations, from which the propagation of electromagnetic disturbances and the continuity equation could be derived with a little effort. Using the differential forms notation however, leads to an even simpler derivation of these results. The price to pay for this simplification is that one needs knowledge of more technical mathematics.
Classical electrodynamics as a line bundle
An elegant and intuitive way to formulate Maxwell's equations is to use line bundles or principal bundles with fibre U(1). The connection on the line bundle is d+A with A the four-vector comprised of the electric potential and the magnetic vector potential. The curvature of the connection F=dA is the field strength. Some feel that this formulation allows a more natural description of the Aharonov-Bohm effect, namely in terms of the holonomy of a curve on a line bundle. (See Micheal 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 Bulliten, 28 (1985) )no. 2 pp 129-164.)
See also
- Proca's equations
- gauge theory for more details
- vector calculus.
- natural units
- Lorentz-Heaviside units.
- Michael Faraday
- Heinrich Hertz
- Electromagnetic wave equation
References
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
- Larmor, J. (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).
- Lorentz, H. A. (1899) "Simplified theory of electrical and optical phnomena in moving systems", Proc. Acad. Science Amsterdam, I, 427-43.
- Lorentz, H. A. (1904) "Electromagnetic phenomena in a system moving with any velocity less than that of light", Proc. Acad. Science Amsterdam, IV, 669-78.
- Poincare, H. (1900) "La theorie de Lorentz et la Principe de Reaction", Archives Neerlandaies, V, 253-78.
- Poincaré, H (1901) Science and Hypothesis
- Poincare, H. (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
- Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.), Prentice Hall. ISBN 013805326X.
- Tipler, Paul (2004). Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics (5th ed.), W. H. Freeman. ISBN 0716708108.
- Purcell, Edward M. (1985). Electricity and Magnetism, McGraw-Hill. ISBN 0070049084..
- Banesh Hoffman, Relativity and Its Roots (Freeman, New York, 1983).
- Charles F. Stevens, The Six Core Theories of Modern Physics, (MIT Press, 1995) ISBN 0-262-69188-4.
Graduate
- Jackson, John D. (1998). Classical Electrodynamics (3rd ed.), Wiley. ISBN 047130932X.
- Landau, L. D., The Classical Theory of Fields (Course of Theoretical Physics: Volume 2), (Butterworth-Heinemann: Oxford, 1987).
- Maxwell, James C. (1954). A Treatise on Electricity and Magnetism, Dover. ISBN 0486606376.
- Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, Gravitation, (1970) W.H. Freeman, New York; ISBN 0-7167-0344-0. (Provides a treatment of Maxwell's equations in terms of differential forms.)
External links
- Fitzpatrick, Richard, "Lecture series: Relativity and electromagnetism". Advanced Classical Electromagnetism, PHY387K. University of Texas at Austin, Fall 1996.
- MISN-0-513 Maxwell's Equations: Electricity and Magnetism (PDF file) by R. Young for Project PHYSNET.
- Feynman's derivation of Maxwell equations and extra dimensions.
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