View source for Gauss's law for magnetism

{{Use American English|date = February 2019}}
{{Short description|Foundational law of classical magnetism}}
{{about|Gauss's law concerning the magnetic field|analogous laws concerning different fields|Gauss's law|and|Gauss's law for gravity|Gauss's theorem, a mathematical theorem relevant to all of these laws|Divergence theorem}}
{{Electromagnetism|cTopic=Magnetostatics}}

In [[physics]], '''Gauss's law for magnetism''' is one of the four [[Maxwell's equations]] that underlie [[classical electrodynamics]]. It states that the [[magnetic field]] {{math|'''B'''}} has [[divergence]] equal to zero,<ref name=Chow/> in other words, that it is a [[solenoidal vector field]]. It is equivalent to the statement that [[magnetic monopole]]s do not exist.<ref name=Jackson/> Rather than "magnetic charges", the basic entity for magnetism is the [[Magnetic dipole#Field from a magnetic dipole|magnetic dipole]]. (If monopoles were ever found, the law would have to be modified, as elaborated below.)

Gauss's law for magnetism can be written in two forms, a ''differential form'' and an ''integral form''. These forms are equivalent due to the [[divergence theorem]].

The name "Gauss's law for magnetism"<ref name=Chow>{{cite book|first=Tai L.|last=Chow|year=2006|title=Electromagnetic Theory: A modern perspective|url=https://books.google.com/books?id=dpnpMhw1zo8C&pg=PA153&dq=isbn:0763738271#PPA134,M1|page=134|publisher=[[Jones and Bartlett]]|isbn=0-7637-3827-1}}</ref> is not universally used. The law is also called "Absence of [[magnetic monopole|free magnetic poles]]";<ref name=Jackson>
{{cite book |first=John David |last=Jackson |year=1999 |title=Classical Electrodynamics |page=237 |edition=3rd |publisher=[[John Wiley & Sons|Wiley]] |isbn=0-471-30932-X}}</ref> one reference even explicitly says the law has "no name".<ref>{{cite book |first1=David J. |last1=Griffiths |year=1998 |title=Introduction to Electrodynamics |page=321 |edition=3rd |publisher=[[Prentice Hall]] |isbn=0-13-805326-X |url-access=registration |url=https://archive.org/details/introductiontoel00grif_0 }}</ref> It is also referred to as the "transversality requirement"<ref name=Joannopoulos>{{cite book |first1=John D. |last1=Joannopoulos |first2=Steve G. |last2=Johnson |first3=Joshua N. |last3=Winn |first4=Robert D. |last4=Meade |year=2008 |title=Photonic Crystals: Molding the Flow of Light |page=9 |edition=2nd |publisher=[[Princeton University Press]] |isbn=978-0-691-12456-8}}</ref> because for [[plane wave]]s it requires that the polarization be transverse to the direction of propagation.

==Differential form==

The differential form for Gauss's law for magnetism is:

{{Equation box 1
|indent =:
|equation = <math>\nabla\cdot\mathbf{B} = 0</math>
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F5FFFA}}

where {{math|∇ ·}} denotes [[divergence]], and {{math|'''B'''}} is the [[magnetic field]].

==Integral form==

[[File:SurfacesWithAndWithoutBoundary.svg|right|thumb|200px|Definition of a closed surface.<br>'''Left:''' Some examples of closed surfaces include the surface of a sphere, surface of a torus, and surface of a cube. The [[magnetic flux]] through any of these surfaces is zero.<br>'''Right:''' Some examples of non-closed surfaces include the [[disk (mathematics)|disk surface]], square surface, or hemisphere surface. They all have boundaries (red lines) and they do not fully enclose a 3D volume. The magnetic flux through these surfaces is ''not necessarily zero''.]]

The integral form of Gauss's law for magnetism states:

{{Equation box 1
|indent =:
|equation = {{oiint|intsubscpt=<math>\textstyle _S</math>|integrand=<math>\mathbf{B} \cdot \mathrm{d}\mathbf{A} = 0</math>}}
|cellpadding= 15
|border
|border colour = #0073CF
|background colour=#F5FFFA}}

where {{math|''S''}} is any [[closed surface]] (see image right), and {{math|d'''A'''}} is a [[Vector (geometric)|vector]], whose magnitude is the area of an [[infinitesimal]] piece of the surface {{math|''S''}}, and whose direction is the outward-pointing [[surface normal]] (see [[surface integral]] for more details).

The left-hand side of this equation is called the net [[flux]] of the magnetic field out of the surface, and Gauss's law for magnetism states that it is always zero.

The integral and differential forms of Gauss's law for magnetism are mathematically equivalent, due to the [[divergence theorem]]. That said, one or the other might be more convenient to use in a particular computation.

The law in this form states that for each volume element in space, there are exactly the same number of "magnetic field lines" entering and exiting the volume. No total "magnetic charge" can build up in any point in space. For example, the south pole of the magnet is exactly as strong as the north pole, and free-floating south poles without accompanying north poles (magnetic monopoles) are not allowed. In contrast, this is not true for other fields such as [[electric field]]s or [[gravitational field]]s, where total [[electric charge]] or [[mass]] can build up in a volume of space.

==Vector potential==
{{main|Magnetic vector potential}}

Due to the [[Helmholtz decomposition|Helmholtz decomposition theorem]], Gauss's law for magnetism is equivalent to the following statement:<ref>{{cite book |first1=W. H. A.|last1=Schilders |title=Handbook of Numerical Analysis |url=https://books.google.com/books?id=F_E9SAe6ny0C&pg=PA13 |page=13 |isbn=978-0-444-51375-5 |date=2005 |display-authors=etal}}</ref><ref>{{cite book | first=John David |last=Jackson | year=1999 | title=Classical Electrodynamics | publisher=[[John Wiley & Sons|Wiley]] | page=180 | edition=3rd | isbn=0-471-30932-X}}</ref>

{{block indent|1=''There exists a vector field ''{{math|'''A'''}}'' such that''
{{block indent|1=<math>\mathbf{B} = \nabla\times\mathbf{A}</math>.}}}}

The vector field {{math|'''A'''}} is called the [[magnetic vector potential]].

Note that there is more than one possible {{math|'''A'''}} which satisfies this equation for a given {{math|'''B'''}} field. In fact, there are infinitely many: any field of the form {{math|∇''ϕ''}} can be added onto {{math|'''A'''}} to get an alternative choice for {{math|'''A'''}}, by the identity (see [[Vector calculus identities]]):

{{block indent|1=<math>\nabla\times \mathbf{A} = \nabla\times(\mathbf{A} + \nabla \phi)</math>}}

since the curl of a gradient is the [[Null vector|zero]] [[vector field]]:

{{block indent|1=<math>\nabla\times \nabla \phi=\boldsymbol{0}</math>}}

This arbitrariness in {{math|'''A'''}} is called [[gauge freedom]].

==Field lines==
{{main|Field line}}

The magnetic field {{math|'''B'''}}, like any vector field, can be depicted via [[field line]]s (also called ''flux lines'') – that is, a set of curves whose direction corresponds to the direction of {{math|'''B'''}}, and whose areal density is proportional to the magnitude of {{math|'''B'''}}. Gauss's law for magnetism is equivalent to the statement that the field lines have neither a beginning nor an end: Each one either forms a closed loop, winds around forever without ever quite joining back up to itself exactly, or extends to infinity.

==Modification if magnetic monopoles exist==
{{main|Magnetic monopole}}

If [[magnetic monopoles]] were discovered, then Gauss's law for magnetism would state the divergence of {{math|'''B'''}} would be proportional to the ''[[magnetic charge]] density'' {{math|''ρ''<sub>m</sub>}}, analogous to Gauss's law for electric field. For zero net magnetic charge density ({{math|1=''ρ''<sub>m</sub> = 0}}), the original form of Gauss's magnetism law is the result.

The modified formula in [[SI units]] is not standard; in one variation, magnetic charge has units of [[Weber (unit)|webers]], in another it has units of [[ampere]]-[[meter]]s.

{| class="wikitable"
|-
! Units
! Equation
|-
| [[Gaussian units|cgs]] units<ref>{{cite journal|first=F.|last=Moulin|year=2001|title=Magnetic monopoles and Lorentz force|journal=[[Il Nuovo Cimento B]]|volume=116 |issue=8 |pages=869–877|arxiv=math-ph/0203043|bibcode = 2001NCimB.116..869M}}</ref>
|| <math>\nabla\cdot\mathbf{B} = 4\pi\rho_{\mathrm m}</math>
|-
| [[SI units]] ([[Weber (unit)|weber]] convention)<ref>{{cite book |first=John David |last=Jackson |year=1999 |title=Classical Electrodynamics |edition=3rd |page=273, eq. 6.150 |publisher=[[John Wiley & Sons|Wiley]]}}</ref>
|| <math>\nabla\cdot\mathbf{B} = \rho_{\mathrm m}</math>
|-
| SI units ([[ampere]]-[[meter]] convention)<ref>See for example equation 4 in {{cite journal |first1=M. |last1=Nowakowski |first2=N. G. |last2=Kelkar |year=2005 |title=Faraday's law in the presence of magnetic monopoles |journal=[[Europhysics Letters]] |volume=71 |issue=3 |pages=346 |arxiv=physics/0508099 |doi=10.1209/epl/i2004-10545-2|bibcode = 2005EL.....71..346N }}</ref>
|| <math>\nabla\cdot\mathbf{B} = \mu_0\rho_{\mathrm m}</math>
|-
|}

where {{math|''μ''<sub>0</sub>}} is the [[vacuum permeability]].

So far, no magnetic monopoles have been found, despite extensive search.<ref>[http://pdg.lbl.gov/2016/reviews/rpp2016-rev-mag-monopole-searches.pdf Magnetic Monopoles], report from [[Particle data group]], updated August 2015 by D. Milstead and E.J. Weinberg. "To date there have been no confirmed observations of exotic particles possessing magnetic charge."</ref>

==History==
This idea of the nonexistence of magnetic monopoles originated in 1269 by [[Petrus Peregrinus de Maricourt]]. His work heavily influenced [[William Gilbert (astronomer)|William Gilbert]], whose 1600 work [[De Magnete]] spread the idea further. In the early 1800s [[Michael Faraday]] reintroduced this law, and it subsequently made its way into [[James Clerk Maxwell]]'s electromagnetic field equations.

==Numerical computation==
In numerical computation, the numerical solution may not satisfy Gauss's law for magnetism due to the discretization errors of the numerical methods. However, in many cases, e.g., for [[magnetohydrodynamics]], it is important to preserve Gauss's law for magnetism precisely (up to the machine precision). Violation of Gauss's law for magnetism on the discrete level will introduce a strong non-physical force. In view of energy conservation, violation of this condition leads to a non-conservative energy integral, and the error is proportional to the divergence of magnetic field<ref>{{cite journal |last1=Brackbill |first1=J.U |last2=Barnes |first2=D.C |title=The Effect of Nonzero ∇ · B on the numerical solution of the magnetohydrodynamic equations |journal=Journal of Computational Physics |date=May 1980 |volume=35 |issue=3 |pages=426–430 |doi=10.1016/0021-9991(80)90079-0}}</ref>.

There are various ways to preserve Gauss's law for magnetism in numerical methods, including the divergence-cleaning techniques<ref>{{cite journal |last1=Tóth |first1=Gábor |title=The ∇·B=0 Constraint in Shock-Capturing Magnetohydrodynamics Codes |journal=Journal of Computational Physics |date=1 July 2000 |volume=161 |issue=2 |pages=605–652 |doi=10.1006/jcph.2000.6519 |url=https://doi.org/10.1006/jcph.2000.6519 |issn=0021-9991}}</ref>, the constrained transport method<ref>{{cite journal |last1=Hernquist |first1=Lars |last2=Vogelsberger |first2=Mark |last3=Mocz |first3=Philip |title=A constrained transport scheme for MHD on unstructured static and moving meshes |journal=Monthly Notices of the Royal Astronomical Society |date=21 July 2014 |volume=442 |issue=1 |pages=43–55 |doi=10.1093/mnras/stu865 |url=https://doi.org/10.1093/mnras/stu865 |language=en |issn=0035-8711}}</ref>, potential-based formulations<ref>{{cite book |last1=Jardin |first1=Stephen |title=Computational Methods in Plasma Physics |date=2010 |publisher=CRC Press |location=Boca Raton |isbn=9780429075537 |edition=1st}}</ref> and de Rham complex based finite element methods<ref>{{cite journal |last1=Hu |first1=Kaibo |last2=Ma |first2=Yicong |last3=Xu |first3=Jinchao |title=Stable finite element methods preserving ∇·B=0 exactly for MHD models |journal=Numerische Mathematik |date=1 February 2017 |volume=135 |issue=2 |pages=371–396 |doi=10.1007/s00211-016-0803-4 |url=https://doi.org/10.1007/s00211-016-0803-4 |language=en |issn=0945-3245}}</ref><ref>{{cite journal |last1=Ma |first1=Yicong |last2=Hu |first2=Kaibo |last3=Hu |first3=Xiaozhe |last4=Xu |first4=Jinchao |title=Robust preconditioners for incompressible MHD models |journal=Journal of Computational Physics |date=July 2016 |volume=316 |pages=721–746 |doi=10.1016/j.jcp.2016.04.019}}</ref> where stable and structure-preserving algorithms are constructed on unstructured meshes with finite element differential forms.

==See also==

{{Wikipedia books|Maxwell's equations}}
*[[Magnetic moment]]
*[[Vector calculus]]
*[[Integral]]
*[[Flux]]
*[[Gaussian surface]]
*[[Faraday's law of induction]]
*[[Ampère's circuital law]]
*[[Lorenz gauge condition]]

==References==
{{reflist|30em}}

{{DEFAULTSORT:Gauss's Law For Magnetism}}
[[Category:Magnetism]]
[[Category:Magnetic monopoles]]
[[Category:Maxwell's equations]]
[[Category:Carl Friedrich Gauss|Magnetism]]