# Difference between revisions of "Solenoidal field"

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+ | $#A+1 = 6 n = 0 | ||

+ | $#C+1 = 6 : ~/encyclopedia/old_files/data/S086/S.0806050 Solenoidal field, | ||

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+ | ''tubular field'' | ||

+ | A [[Vector field|vector field]] in $ \mathbf R ^ {3} $ | ||

+ | 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 $ | ||

+ | 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==== | ====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 | + | 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==== | ====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