Maxwell's equations

Maxwell's equations are the set of four equations, attributed to James Clerk Maxwell (written by Oliver Heaviside), that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter.

Summary of the equations

All variables that are in bold represent vector quantities.

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General case

where:

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: ho is the free electric charge density (SI unit: coulomb per cubic metre), not including dipole charges bound in a material

Related Topics:
Electric charge - SI - Coulomb - Cubic metre

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: mathbf{B} is the magnetic flux density (SI unit: tesla, volt × second per square metre), also called the magnetic induction.

Related Topics:
Magnetic flux density - Tesla - Volt - Second - Square metre

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: mathbf{D} is the electric displacement field (SI unit: coulomb per square metre).

Related Topics:
Electric displacement field - Square metre

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: mathbf{A} is the area of the Gaussian surface

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: mathbf{E} is the electric field (SI unit: volt per metre),

Related Topics:
Electric field - Volt - Metre

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: mathbf{H} is the magnetic field strength (SI unit: ampere per metre)

Related Topics:
Magnetic field strength - Ampere

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: mathbf{J} is the current density (SI unit: ampere per square metre)

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: abla cdot is the divergence operator (SI unit: 1 per metre),

Related Topics:
Divergence - Operator

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: abla imes is the curl operator (SI unit: 1 per metre).

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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.

Related Topics:
SI - Planck units - Cgs

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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:

Related Topics:
Magnetic monopole - Electric field - Magnetic field - Lorentz force

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: mathbf{F} = q (mathbf{E} + mathbf{v} imes mathbf{B}),

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where q is the charge on the particle and mathbf{v} is the particle velocity. This is slightly different when expressed in the cgs system of units below.

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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).

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In linear materials

In linear materials, the D and H fields are related to E and B by:

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:mathbf{D} = arepsilon mathbf{E}

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:mathbf{B} = mu mathbf{H}

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where:

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ε is the electrical permittivity

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μ is the magnetic permeability

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(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.)

Related Topics:
Kerr - Pockels effect

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In non-dispersive, isotropic media, ε and μ are time-independent scalars, and Maxwell's equations reduce to

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: abla cdot arepsilon mathbf{E} = ho

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: abla cdot mathbf{B} = 0

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: abla imes mathbf{E} = -rac{partial mathbf{B}} {partial t}

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: abla imes mathbf{rac{B}{mu}} = mathbf{J} + arepsilon rac{partial mathbf{E}} {partial t}

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In a uniform (homogeneous) medium, ε and μ are constants independent of position, and can thus be furthermore interchanged with the spatial derivatives.

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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).

Related Topics:
Tensor - Matrices - Birefringent - Material dispersion - Frequency - Kramers-Kronig relations

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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 ε0 and μ0 (neglecting very slight nonlinearities due to quantum effects). If there is no current or electric charge present in the vacuum, we obtain the Maxwell's equations in free space:

Related Topics:
ε₀ - μ₀

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: abla cdot mathbf{E} = 0

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: abla cdot mathbf{B} = 0

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: abla imes mathbf{E} = -rac{partialmathbf{B}} {partial t}

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: abla imes mathbf{B} = mu_0 arepsilon_0 rac{partial mathbf{E}} {partial t}

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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

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:c = rac{1}{sqrt{mu_0 arepsilon_0}}

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Maxwell discovered that this quantity c is simply the speed of light in vacuum, and thus that light is a form of electromagnetic radiation.

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