Solenoidal vector field

As source- free or free of sources, a vector field is called in physics and potential theory, the field lines in the area under consideration do not have a starting point. Source is free, for example, the outer compartment of a motor or gravity field if it contains no mass points or charges.

In nature, this ideal situation is indeed hardly given because it almost anywhere - are residual gas molecules, dust particles and free electrons - even in interplanetary space. For the scientific practice and in astronomy source freedom is instead given wherever the matter and gas density is below some particles per cm3. For the laboratory physics, the best technically producible high vacuum may be the reference value, which lies far with a residual gas pressure of about 10-11 mbar above.

Field lines and divergence

In electrodynamics and fluid dynamics swelling free spaces are characterized by the fact that in the considered space section enter as many field lines as escape. This behavior of the field lines can be mathematically described by the divergence of the vector field.

The mathematics is called the term " source -free" also divergence-free, because the lack of sources is coupled with the disappearance of divergence: true In a source-free vector field

Here stands for the divergence operator ( see also nabla operator ). Conversely, the presence of sources is characterized by a nonzero value.

Interpretation and examples

Physically can be the divergence of a vector field as a measure of the source strength interpret, because after the Gauss integral of the divergence over a volume is equal to the flux through the surface of the volume. Accordingly, we describe a vector field whose divergence is zero, as a source -free, because here is for arbitrary closed surfaces of the flow is equal to zero, that is, net no longer flows out than in. There is thus in the inside of the volume enclosed by the surface of neither sources nor sinks.

Important examples for source- free fields in physics are the magnetic field and the velocity field of incompressible flows, which are source- free due to the continuity equation.

For twice continuously differentiable vector fields is considered according to the set of black

Thus, the rotation of such a vector field is always free source.