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--- maxwell_equations [2023/08/28 11:14] – [Maxwell's equations in matter] stan_zurek
+++ maxwell_equations [2024/06/07 15:00] (current) – [Maxwell's equations] stan_zurek
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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~~
+{{tag> Maxwell's_equations Maxwell's-Heaviside_equations Counter}}