Tarski's exponential function problem: Difference between revisions

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{{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}}
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{{Electromagnetism|cTopic=Magnetostatics}}
 
In [[physics]], '''Gauss's law for magnetism''' is one of [[Maxwell's equations]]—the four equations that underlie [[classical electrodynamics]]. It states that the [[magnetic field]] '''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. Rather than "magnetic charges", the basic entity for magnetism is the [[Magnetic_dipole#Field_from_a_magnetic_dipole|magnetic dipole]]. (Of course, 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
|author=Tai L. Chow
|year=2006
|title=Electromagnetic Theory: A modern perspective
|url=http://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
|author=John David Jackson
|year=1999
|title=Classical Electrodynamics
|page=237 |edition=3rd
|publisher=[[John Wiley & Sons|Wiley]]
|isbn=0-471-30932-X
}}</ref> (or some variant); one reference even explicitly says the law has "no name".<ref>
{{cite book
|author=David J. Griffiths
|year=1998
|title=Introduction to Electrodynamics
|page=321 |edition=3rd
|publisher=[[Prentice Hall]]
|isbn=0-13-805326-X
}}</ref> It is also referred to as the "transversality requirement"<ref name=Joannopoulos>
{{cite book
|author=John D. Joannopoulos, Steve G. Johnson, Joshua N. Winn, Robert D. 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 ∇• denotes [[divergence]], and '''B''' is the [[magnetic field]].
 
==Integral form==
 
[[File:SurfacesWithAndWithoutBoundary.svg|right|thumb|200px|Definition of a closed surface. 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. 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
| preintegral =
| intsubscpt = <math>{\scriptstyle S}</math>
| integrand = <math>\mathbf{B} \cdot \mathrm{d}\mathbf{A} = 0</math>}}
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F5FFFA}}
 
where ''S'' is any [[closed surface]] (see image right), and d'''A''' is a [[Vector (geometric)|vector]], whose magnitude is the area of an [[infinitesimal]] piece of the surface ''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.
 
==In terms of 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
|author=W.H.A. Schilders ''et al.''
  |year=
|title=Handbook of Numerical Analysis
|url=http://books.google.com/books?id=F_E9SAe6ny0C&pg=PA13
  |page=13
|publisher=
|isbn=978-0-444-51375-5
|date=2005-05-23
}}</ref><ref>
{{cite book
| author=John David Jackson
| year=1999
| title=Classical Electrodynamics
  | publisher=[[John Wiley & Sons|Wiley]]
| page=180 | edition=3rd
| isbn=0-471-30932-X
}}</ref>
 
:''There exists a vector field '''A''' such that''
 
::<math>\mathbf{B} = \nabla\times\mathbf{A}</math>.
 
The vector field '''A''' is called the [[magnetic vector potential]].
 
Note that there is more than one possible '''A''' which satisfies this equation for a given '''B''' field. In fact, there are infinitely many: any field of the form ∇φ can be added onto '''A''' to get an alternative choice for '''A''', by the identity (see [[Vector calculus identities]]):
 
:<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]]:
 
:<math>\nabla\times \nabla \phi=\boldsymbol{0}</math>
 
This arbitrariness in '''A''' is called [[gauge freedom]].
 
==In terms of field lines==
{{main|Field line}}
 
The magnetic field '''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 '''B''', and whose areal density is proportional to the magnitude of '''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 '''B''' would be proportional to the ''[[magnetic charge]] density'' ρ<sub>m</sub>, analogous to Gauss's law for electric field. For zero net magnetic charge density (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
|author=F. Moulin
|year=2001
|title=Magnetic monopoles and Lorentz force
|journal=[[Il Nuovo Cimento B]]
|volume=116 |issue=8 |pages=869&ndash;877
|arxiv=math-ph/0203043
|bibcode = 2001NCimB.116..869M
}}</ref>
|| <math>\nabla\cdot\mathbf{B} = 4\pi\rho_m</math>
|-
| [[SI units]] ([[Weber (unit)|weber]] convention)<ref>
{{cite book
|author=John David Jackson
|year=1999
|title=Classical Electrodynamics
|edition=3rd
|page=273, eq. (6.150)
|publisher=[[John Wiley & Sons|Wiley]]
|isbn=
}}</ref>
|| <math>\nabla\cdot\mathbf{B} = \rho_m</math>
|-
| SI units ([[ampere]]-[[meter]] convention)<ref>See for example equation (4) in {{cite journal
|author=M. Nowakowski, N. G. 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_m</math>
|-
|}
 
where ''μ''<sub>0</sub> is the [[vacuum permeability]].
 
So far no magnetic monopoles have been found, despite extensive search.
 
==History==
 
The equation <math>\mathbf{B} = \nabla\times\mathbf{A}</math> was one of [[A dynamical theory of the electromagnetic field|Maxwell's original eight equations]]. However, the interpretation was somewhat different: Maxwell's '''A''' field directly corresponded to an important physical quantity which he believed corresponded to Faraday's ''electrotonic state'',<ref name=Huray>
{{cite book
|author=Paul G. Hurray
|year=2010
|title=Maxwell's Equations
|url=http://books.google.com/books?id=0QsDgdd0MhMC&pg=PA22
|page=22
|publisher=
|isbn=978-0-470-54276-7
}}</ref> while the modern interpretation emphasizes [[gauge freedom]], the idea that there are many possible '''A''' fields, all equally valid.<ref name=Huray/>
 
==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}}
 
{{DEFAULTSORT:Gauss's Law For Magnetism}}
[[Category:Magnetism]]
[[Category:Maxwell's equations]]

Latest revision as of 11:19, 8 August 2014

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