# Difference between revisions of "Solenoidal field"

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.

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.