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Difference between revisions of "Solenoidal field"

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''tubular field''
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A [[Vector field|vector field]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086050/s0860501.png" /> having neither sources nor sinks, i.e. its [[Divergence|divergence]] vanishes at all its points. The flow of a solenoidal field through any closed piecewise-smooth oriented boundary of any domain is equal to zero. Solenoidal fields are characterized by their so-called vector potential, that is, a vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086050/s0860502.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086050/s0860503.png" />. Examples of solenoidal fields are field of velocities of an incompressible liquid and the magnetic field within an infinite solenoid.
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''tubular field''
  
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A [[Vector field|vector field]] in  $  \mathbf R  ^ {3} $
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having neither sources nor sinks, i.e. its [[Divergence|divergence]] vanishes at all its points. The flow of a solenoidal field through any closed piecewise-smooth oriented boundary of any domain is equal to zero. Solenoidal fields are characterized by their so-called vector potential, that is, a vector field  $  A $
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such that  $  \mathbf a =  \mathop{\rm curl}  A $.
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Examples of solenoidal fields are field of velocities of an incompressible liquid and the magnetic field within an infinite solenoid.
  
 
====Comments====
 
====Comments====
A solenoid is a long spiral coil of wire, usually cylindrical, through which a current can be passed to produce a magnetic field. More abstractly, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086050/s0860504.png" /> be a vector field (on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086050/s0860505.png" />) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086050/s0860506.png" />. Consider a surface consisting of a cylinder along the vector lines together with surfaces normal to the lines at both ends. Such a tube is called a solenoid.
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A solenoid is a long spiral coil of wire, usually cylindrical, through which a current can be passed to produce a magnetic field. More abstractly, let $  \mathbf a $
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be a vector field (on $  \mathbf R  ^ {3} $)  
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with $  \mathop{\rm div} ( \mathbf a ) = 0 $.  
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Consider a surface consisting of a cylinder along the vector lines together with surfaces normal to the lines at both ends. Such a tube is called a solenoid.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.A. Hylleras,  "Mathematical and theoretical physics" , '''1''' , Wiley (Interscience)  (1970)  pp. 70ff</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B.G. Levich,  "Theoretical physics" , '''1. Theory of the electromagnetic field''' , North-Holland  (1970)  pp. 6; 364; 366</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G.K. Batchelor,  "An introduction to fluid dynamics" , Cambridge Univ. Press  (1974)  pp. 75; 167</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  K. Rektorys (ed.) , ''Applicable mathematics'' , Iliffe  (1969)  pp. 272</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.A. Hylleras,  "Mathematical and theoretical physics" , '''1''' , Wiley (Interscience)  (1970)  pp. 70ff</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B.G. Levich,  "Theoretical physics" , '''1. Theory of the electromagnetic field''' , North-Holland  (1970)  pp. 6; 364; 366</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G.K. Batchelor,  "An introduction to fluid dynamics" , Cambridge Univ. Press  (1974)  pp. 75; 167</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  K. Rektorys (ed.) , ''Applicable mathematics'' , Iliffe  (1969)  pp. 272</TD></TR></table>

Latest revision as of 08:14, 6 June 2020


tubular field

A vector field in $ \mathbf R ^ {3} $ having neither sources nor sinks, i.e. its divergence vanishes at all its points. The flow of a solenoidal field through any closed piecewise-smooth oriented boundary of any domain is equal to zero. Solenoidal fields are characterized by their so-called vector potential, that is, a vector field $ A $ such that $ \mathbf a = \mathop{\rm curl} A $. Examples of solenoidal fields are field of velocities of an incompressible liquid and the magnetic field within an infinite solenoid.

Comments

A solenoid is a long spiral coil of wire, usually cylindrical, through which a current can be passed to produce a magnetic field. More abstractly, let $ \mathbf a $ be a vector field (on $ \mathbf R ^ {3} $) with $ \mathop{\rm div} ( \mathbf a ) = 0 $. Consider a surface consisting of a cylinder along the vector lines together with surfaces normal to the lines at both ends. Such a tube is called a solenoid.

References

[a1] E.A. Hylleras, "Mathematical and theoretical physics" , 1 , Wiley (Interscience) (1970) pp. 70ff
[a2] B.G. Levich, "Theoretical physics" , 1. Theory of the electromagnetic field , North-Holland (1970) pp. 6; 364; 366
[a3] G.K. Batchelor, "An introduction to fluid dynamics" , Cambridge Univ. Press (1974) pp. 75; 167
[a4] K. Rektorys (ed.) , Applicable mathematics , Iliffe (1969) pp. 272
How to Cite This Entry:
Solenoidal field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Solenoidal_field&oldid=19139
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article