Encyclopedia Magnetica™

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====== Maxwell's equations ======

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| //[[user/Stan Zurek]], Maxwell's equations, Encyclopedia Magnetica//, \\ @PAGEL@  |


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**Maxwell's equations** or **Maxwell-Heaviside equations** - a set of mathematical equations which describe the behaviour of electromagnetic field in space and time.[(Maxwell>[[https://doi.org/10.1098/rstl.1865.0008|James Clerk Maxwell, 1865 VIII. A dynamical theory of the electromagnetic field, Philosophical Transactions of the Royal Society of London, 155: 459–512. https://doi.org/10.1098/rstl.1865.0008]])][(Griffiths>[[http://books.google.com/books?isbn=0321856562|David J. Griffiths, Introduction to electrodynamics, 4th ed., Pearson, Boston, 2013, ISBN 0321856562]])][(Purcell>[[https://isbnsearch.org/isbn/9781107014022|E.M. Purcell, D.J. Morin, Electricity and magnetism, 3rd edition, Cambridge University Press, 2013, ISBN 9781107014022]])][(Fleisch_Maxwell>[[https://isbnsearch.org/isbn/9780521877619|Daniel Fleisch, A Student’s Guide to Maxwell’s Equations, Cambridge University Press, Cambridge, 2008, ISBN 9780521877619]])][(Feynman>[[https://www.feynmanlectures.caltech.edu/II_36.html|Richard Feynman, Robert Leighton, Matthew Sands, Ferromagnetism, The Feynman Lectures on Physics, Vol. II, Basic Books, ISBN: 9780465079988]])][(Band>[[https://isbnsearch.org/isbn/9780471899310|Yehuda B. Band, Light and Matter: Electromagnetism, Optics, Spectroscopy and Lasers, John Wiley & Sons, 2006, ISBN 9780471899310]])][(Fiorillo>[[https://isbnsearch.org/isbn/9780122572517|Fausto Fiorillo, Measurement and Characterization of Magnetic Materials, Academic Press, 2005, ISBN 9780122572517]])]

In his original publication in 1865,[(Maxwell)] **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.[(Heaviside>[[https://macsphere.mcmaster.ca/bitstream/11375/14746/1/fulltext.pdf|Oliver Heaviside, Electromagnetic theory, Vol. I, 1893, Ernest Benn Limited]])] There are four basic electromagnetic equations, supplemented by additional [[constitutive relations]], as well as expressions of energy conservation, and [[electromagnetic force]].

To fully quantify an electromagnetic vector field it is necessary to determine its [[divergence]] as well as [[curl]].[(Griffiths)] For other fields, such as [[thermal field|thermal]], it is also useful to calculate [[gradient]], for example by employing [[vector calculus]].

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===== Electromagnetic quantities =====

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

^  Electromagnetic quantities expressed as vector fields [(Griffiths)][(Band)]  ^^^^^
|  |  name  |  symbol  |  SI unit  |  CGS unit  |
|  magnetic  |  [[Magnetic flux density]]  |  $$ \vec{B}  $$  |  (T) ≡ \\ (V·s / m<sup>2</sup>)  |  (G)  |
| ::: |  [[Magnetic field strength]]  |  $$ \vec{H}  $$  |  (A/m)  |  (Oe)  |
| ::: |  [[Magnetisation]]  |  $$ \vec{M}  $$  |  (A/m)  |  (Oe)  |
| ::: |  [[Magnetic polarisation]]  |  $$ \vec{P}  $$  |  (T)  |  (G)  |
|  electric  |  [[Current density]]  |  $$ \vec{J} $$  |  (A/m<sup>2</sup>)  |  (A/cm<sup>2</sup>)  |
| ::: |  [[Electric field]]  |  $$ \vec{E}  $$  |  (V/m)  |  (statV/cm)  |
| ::: |  [[Electric displacement field]]  |  $$ \vec{D}  $$  |  (C/m<sup>2</sup>)  |  (statV/cm)  |
| ::: |  [[Electric polarisation]]  |  $$ \vec{P}  $$  |  (C/m<sup>2</sup>)  |  (statV/cm)  |
===== Maxwell's equations in general =====

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 dipole moment|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.[(Griffiths)] 

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^  Maxwell's equations, valid in general[(Griffiths)]  ^^^
| |  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<sup>3</sup>), //ε<sub>0</sub>// - [[electric permittivity of vacuum]] (F/m), //q// - [[electric charge]] (C), //l// - increment of path for integral (m), //a// - increment of surface for integral (m<sup>2</sup>), //μ<sub>0</sub>// - [[magnetic permeability of vacuum]] (H/m), //J// - [[electric current density]] (A/m<sup>2</sup>), //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 dipole moment|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).[(Griffiths)] 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//.[(Feynman)]

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^  Maxwell's equations, valid in matter[(Griffiths)]  ^^^
| |  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<sup>3</sup>), //a// - increment of surface for integral (m<sup>2</sup>), //Q// - [[electric charge]] (C), //t// - time (s), //l// - increment of path for integral (m), //J// - [[electric current density]] (A/m<sup>2</sup>), //I// - [[electric current]] (A), //S// - closed surface (region of integral), //C// - closed curve (path of integral)    |||

==== Constitutive relationships ====




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===== See also =====
  *[[Electromagnetism]]

===== References =====
~~REFNOTES~~

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