Solenoidal vector field explained

In vector calculus a solenoidal vector field (also known as an incompressible vector field, a divergence-free vector field, or a transverse vector field) is a vector field v with divergence zero at all points in the field:$$\backslash nabla\; \backslash cdot\; \backslash mathbf\; =\; 0.$$A common way of expressing this property is to say that the field has no sources or sinks.^[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

d S 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:$$\backslash mathbf\; =\; \backslash nabla\; \backslash times\; \backslash mathbf$$automatically results in the identity (as can be shown, for example, using Cartesian coordinates):$$\backslash nabla\; \backslash cdot\; \backslash mathbf\; =\; \backslash nabla\; \backslash cdot\; (\backslash nabla\; \backslash times\; \backslash mathbf)\; =\; 0.$$The converse also holds: for any solenoidal v there exists a vector potential A such that

v = \nabla x A . (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.

Examples

• The magnetic field B (see Gauss's law for magnetism)
• The velocity field of an incompressible fluid flow
• The vorticity field
• The electric field E in neutral regions (

\rho _{e} = 0 );

• The current density J where the charge density is unvarying, $\backslash frac\; =\; 0$.
• The magnetic vector potential A in Coulomb gauge

See also

• Longitudinal and transverse vector fields
• Stream function
• Conservative vector field

Notes and References

1. 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.