A common way of expressing this property is to say that the field has no sources or sinks.[note 1]
Properties
The
divergence theorem gives an equivalent integral definition of a solenoidal field; namely that for any closed surface, the net total flux through the surface must be zero:
where is the outward normal to each surface element.
The
fundamental theorem of vector calculus states that any vector field can be expressed as the sum of an
irrotational and a solenoidal field. The condition of zero divergence is satisfied whenever a vector field v has only a
vector potential component, because the definition of the vector potential A as:
automatically results in the
identity (as can be shown, for example, using Cartesian coordinates):
The
converse also holds: for any solenoidal v there exists a vector potential A such that (Strictly speaking, this holds subject to certain technical conditions on v, see
Helmholtz decomposition.)
Etymology
Solenoidal has its origin in the Greek word for
solenoid, which is σωληνοειδές (sōlēnoeidēs) meaning pipe-shaped, from σωλην (sōlēn) or pipe.
^This statement does not mean that the field lines of a solenoidal field must be closed, neither that they cannot begin or end. For a detailed discussion of the subject, see J. Slepian: "Lines of Force in Electric and Magnetic Fields", American Journal of Physics, vol. 19, pp. 87-90, 1951, and L. Zilberti: "The Misconception of Closed Magnetic Flux Lines", IEEE Magnetics Letters, vol. 8, art. 1306005, 2017.