Editing Maxwell's equations (section)

== Conceptual descriptions ==
=== Gauss's law ===
{{Main|Gauss's law}}
[[File:VFPt charges plus minus thumb.svg|thumb|upright=0.5|Electric field from positive to negative charges]]
[[Gauss's law]] describes the relationship between an [[electric field]] and [[electric charge]]s: an electric field points away from positive charges and towards negative charges, and the net [[electric flux|outflow]] of the electric field through a [[Gaussian surface|closed surface]] is proportional to the enclosed charge, including bound charge due to polarization of material. The coefficient of the proportion is the [[vacuum permittivity|permittivity of free space]].

=== Gauss's law for magnetism ===
{{Main|Gauss's law for magnetism}}
[[Image:VFPt dipole magnetic1.svg|right|thumb|[[Gauss's law for magnetism]]: magnetic field lines never begin nor end but form loops or extend to infinity as shown here with the magnetic field due to a ring of current.]]

[[Gauss's law for magnetism]] states that electric charges have no magnetic analogues, called [[magnetic monopole]]s; no north or south magnetic poles exist in isolation.<ref name=VideoGlossary>{{cite web | url =http://videoglossary.lbl.gov/#n45 | title =Maxwell's equations | last =Jackson | first =John | website =Science Video Glossary | publisher =Berkeley Lab | access-date =2016-06-04 | archive-date =2019-01-29 | archive-url =https://web.archive.org/web/20190129113142/https://videoglossary.lbl.gov/#n45 | url-status =dead }}</ref> Instead, the magnetic field of a material is attributed to a [[dipole]], and the net outflow of the magnetic field through a closed surface is zero. Magnetic dipoles may be represented as loops of current or inseparable pairs of equal and opposite "magnetic charges". Precisely, the total [[magnetic flux]] through a Gaussian surface is zero, and the magnetic field is a [[solenoidal vector field]].<ref group="note">The absence of sinks/sources of the field does not imply that the field lines must be closed or escape to infinity. They can also wrap around indefinitely, without self-intersections. Moreover, around points where the field is zero (that cannot be intersected by field lines, because their direction would not be defined), there can be the simultaneous begin of some lines and end of other lines. This happens, for instance, in the middle between two identical cylindrical magnets, whose north poles face each other. In the middle between those magnets, the field is zero and the axial field lines coming from the magnets end. At the same time, an infinite number of divergent lines emanate radially from this point. The simultaneous presence of lines which end and begin around the point preserves the divergence-free character of the field. For a detailed discussion of non-closed field lines, see L.&nbsp;Zilberti [https://zenodo.org/record/4518772#.YCJU_WhKjIU "The Misconception of Closed Magnetic Flux Lines"], IEEE Magnetics Letters, vol.&nbsp;8, art. 1306005, 2017.</ref>

=== Faraday's law ===
{{Main|Faraday's law of induction}}
[[File:Magnetosphere rendition.jpg|thumb|upright=1.45|left|In a [[geomagnetic storm]], solar wind plasma impacts [[Earth's magnetic field]] causing a time-dependent change in the field, thus inducing electric fields in Earth's atmosphere and conductive [[lithosphere]] which can destabilize [[power grid]]s. (Not to scale.)]]

The [[Faraday's law of induction#Maxwell–Faraday equation|Maxwell–Faraday]] version of [[Faraday's law of induction]] describes how a time-varying [[magnetic field]] corresponds to [[Curl (mathematics)|curl]] of an [[electric field]].<ref name="VideoGlossary" /> In integral form, it states that the work per unit charge required to move a charge around a closed loop equals the rate of change of the magnetic flux through the enclosed surface.

The [[electromagnetic induction]] is the operating principle behind many [[electric generator]]s: for example, a rotating [[bar magnet]] creates a changing magnetic field and generates an electric field in a nearby wire.

=== Ampère–Maxwell law ===
{{Main|Ampère's circuital law}}
[[Image:Magnetic core.jpg|right|thumb|[[Magnetic-core memory]] (1954) is an application of [[Ampère's circuital law]]. Each [[magnetic core|core]] stores one [[bit]] of data.]]

The original law of Ampère states that magnetic fields relate to [[electric current]]. [[Ampère–Maxwell law|Maxwell's addition]] states that magnetic fields also relate to changing electric fields, which Maxwell called [[displacement current]]. The integral form states that electric and displacement currents are associated with a proportional magnetic field along any enclosing curve.

Maxwell's modification of Ampère's circuital law is important because the laws of Ampère and Gauss must otherwise be adjusted for static fields.<ref>J. D. Jackson, ''Classical Electrodynamics'', section 6.3</ref>{{clarify|date=May 2022}} As a consequence, it predicts that a rotating magnetic field occurs with a changing electric field.<ref name="VideoGlossary" /><ref>[https://books.google.com/books?id=1DZz341Pp50C&pg=PA809 ''Principles of physics: a calculus-based text''], by R. A. Serway, J. W. Jewett, page 809.</ref> A further consequence is the existence of self-sustaining [[electromagnetic waves]] which [[electromagnetic wave equation|travel through empty space]].

The speed calculated for electromagnetic waves, which could be predicted from experiments on charges and currents,<ref group="note">The quantity we would now call {{math|(''ε''{{sub|0}}''μ''{{sub|0}})<sup>−1/2</sup>}}, with units of velocity, was directly measured before Maxwell's equations, in an 1855 experiment by [[Wilhelm Eduard Weber]] and [[Rudolf Kohlrausch]]. They charged a [[leyden jar]] (a kind of [[capacitor]]), and measured the [[Coulomb's law|electrostatic force]] associated with the potential; then, they discharged it while measuring the [[Ampère's force law|magnetic force]] from the current in the discharge wire. Their result was {{val|3.107|e=8|ul=m/s}}, remarkably close to the speed of light. See Joseph F. Keithley, [https://books.google.com/books?id=uwgNAtqSHuQC&pg=PA115 ''The story of electrical and magnetic measurements: from 500&nbsp;B.C. to the 1940s'', p.&nbsp;115].</ref> matches the [[speed of light]]; indeed, [[light]] ''is'' one form of [[electromagnetic radiation]] (as are [[X-ray]]s, [[radio wave]]s, and others). Maxwell understood the connection between electromagnetic waves and light in 1861, thereby unifying the theories of [[electromagnetism]] and [[optics]].