### Table of Contents

# Maxwell's equations

Stan Zurek, Maxwell's equations, Encyclopedia Magnetica, http://www.e-magnetica.pl/doku.php/maxwell_equations |

* This page is being edited and may be incomplete or incorrect.

**Maxwell's equations** or **Maxwell-Heaviside equations** - a set of mathematical equations which describe the behaviour of electromagnetic field in space and time.^{1)}^{2)}^{3)}^{4)}^{5)}^{6)}^{7)}

In his original publication in 1865,^{8)} **James Clerk Maxwell** listed 20 equations, which were split for each orthogonal coordinates (hence the large number of equations). These equations were later rationalised by **Oliver Heaviside**, who expressed them in a vector form which is know today.^{9)} There are four basic electromagnetic equations, supplemented by additional constitutive relations, as well as expressions of energy conservation, and force.

To fully quantify an electromagnetic vector field it is necessary to determine its divergence as well as curl.^{10)} For other fields, such as thermal, it is also useful to calculate gradient.

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## Electromagnetic quantities

The **Maxwell-Heaviside equations** make use of several basic electromagnetic quantities, which mathematically represent vector fields.

Electromagnetic quantities expressed as vector fields ^{11)}^{12)} |
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name | symbol | SI unit | CGS unit | |

magnetic | Magnetic flux density | $$ \vec{B} $$ | (T) ≡ (V·s / m^{2}) | (G) |

Magnetic field strength | $$ \vec{H} $$ | (A/m) | (Oe) | |

Magnetisation | $$ \vec{M} $$ | (A/m) | (Oe) | |

Magnetic polarisation | $$ \vec{P} $$ | (T) | (G) | |

electric | Electric field | $$ \vec{E} $$ | (V/m) | (statV/cm) |

Electric displacement field | $$ \vec{D} $$ | (C/m^{2}) | (statV/cm) | |

Electric polarisation | $$ \vec{P} $$ | (C/m^{2}) | (statV/cm) |

## General Maxwell's equations

From the viewpoint of theoretical physics, the equations can be expressed in a form which is always valid, in vacuum or in material. However, this requires a full knowledge of microscopic magnetic moments (at the level of subatomic particles) which makes their direct application very difficult for matter, which comprises of a very larger number of atoms.^{13)}

Maxwell's equations, valid in general^{14)} |
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differential form | integral form | |

Gauss's law for electric field | $$ ∇·\vec{E} = \frac{ρ}{ε_0} $$ | $$ \oint_S \vec{E} · d\vec{a} = \frac{q}{ε_0} $$ |

Faraday's law of induction | $$ ∇×\vec{E} = -\frac{∂\vec{B}}{∂t} $$ | $$ \oint_C \vec{E}·d\vec{l} = - \frac{d}{dt}· \int_S \vec{B}·d\vec{a} $$ |

Gauss's law for magnetic field | $$ ∇·\vec{B} = 0 $$ | $$ \oint_S \vec{B}·d\vec{a} = 0 $$ |

Ampère-Maxwell's circuital law | $$ ∇×\vec{B} = μ_0 · \vec{J} + μ_0 · ε_0 · \frac{∂\vec{E}}{∂t} $$ | $$ \oint_C \vec{B}·d\vec{l} = μ_0 · I + μ_0 · ε_0 · \frac{d}{dt}· \int_S \vec{E}·d\vec{a} $$ |

where: ρ - electric charge density (C/m^{3}), ε - electric permittivity of vacuum (F/m), _{0}q - electric charge (C), l - increment of path for integral (m), a - increment of surface for integral (m^{2}), μ - magnetic permeability of vacuum (H/m), _{0}J - electric current density (A/m^{2}), I - electric current (A), S - closed surface (region of integral), C - closed curve (path of integral) |

## Maxwell's equations in matter

In matter, there are localised magnetic moments which respond to the magnetic field penetrating the matter. It is possible to express the response of the matter as a vector field which is averaged (smoothed out) over the whole volume of the material, so that the vector field is expressed in effect as a macroscopic quantity, rather than microscopic variation (which can very wildly).^{15)} These averaged out quantities are measurable experimentally on a macroscopic scale, hence useful for a direct experimental verification and technical purposes.

However, this approach requires further information about the relationship between the excitation and response of the matter, which can be quantified for example in the form of the magnetic permeability *μ* or electric permittivity *ε*.^{16)}

Maxwell's equations, valid in matter^{17)} |
||
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differential form | integral form | |

Gauss's law for electric field | $$ ∇·\vec{D} = ρ $$ | $$ \oint_S \vec{D} · d\vec{a} = Q $$ |

Faraday's law of induction | $$ ∇×\vec{E} = -\frac{∂\vec{B}}{∂t} $$ | $$ \oint_C \vec{E}·d\vec{l} = - \frac{d}{dt}· \int_S \vec{B}·d\vec{a} $$ |

Gauss's law for magnetic field | $$ ∇·\vec{B} = 0 $$ | $$ \oint_S \vec{B}·d\vec{a} = 0 $$ |

Ampère-Maxwell's circuital law | $$ ∇×\vec{H} = \vec{J} + \frac{∂\vec{D}}{∂t} $$ | $$ \oint_C \vec{H}·d\vec{l} = I + \frac{d}{dt}· \int_S \vec{D}·d\vec{a} $$ |

where: ρ - electric charge density (C/m^{3}), a - increment of surface for integral (m^{2}), Q - electric charge (C), t - time (s), l - increment of path for integral (m), J - electric current density (A/m^{2}), I - electric current (A), S - closed surface (region of integral), C - closed curve (path of integral) |

### Constitutive relationships

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## See also

## References

^{2), 10), 11), 13), 14), 15), 17)}David J. Griffiths, Introduction to electrodynamics, 4th ed., Pearson, Boston, 2013, ISBN 0321856562