Sources and sinks
In the physical sciences, engineering and mathematics, sources and sinks is an analogy used to describe properties of vector fields. It generalizes the idea of fluid sources and sinks across different scientific disciplines. These terms describe points, regions, or entities where a vector field originates or terminates. This analogy is usually invoked when discussing the continuity equation, the divergence of the field and the divergence theorem. The analogy sometimes includes swirls and saddles for points that are neither of the two.
In the case of electric fields the idea of flow is replaced by field lines and the sources and sinks are electric charges.
Description and fluid dynamics analogy
In physics, a vector field is a function that returns a vector and is defined for each point in a region of space. The idea of sources and sinks applies to if it follows a continuity equation of the formwhere is time, is some quantity density associated to, and is the source-sink term. The points in space where are called a sources and when are called sinks. The integral version of the continuity equation is given by the divergence theorem.
These concepts originate from sources and sinks in fluid dynamics, where the flow is conserved per the continuity equation related to conservation of mass, given by
where is the mass density of the fluid, is the flow velocity vector, and is the source-sink flow. This equation implies that any emerging or disappearing amount of flow in a given volume must have a source or a sink, respectively. Sources represent locations where fluid is added to the region of space, and sinks represent points of removal of fluid. The term is positive for a source and negative for a sink. Note that for incompressible flow or time-independent systems, is directly related to the divergence as
For this kind of flow, solenoidal vector fields have no source or sinks. When at a given point but the curl , the point is sometimes called a swirl. And when both divergence and curl are zero, the point is sometimes called a saddle.
Other examples in physics
Electromagnetism
In electrodynamics, the current density behaves similar to hydrodynamics as it also follows a continuity equation due to the charge conservation:where this time is the charge density, is the current density vector, and is the current source-sink term. The current source and current sinks are where the current density emerges or vanishes, respectively.
The concept is also used for the electromagnetic fields, where fluid flow is replaced by field lines. For an electric field, a source is a point where electric field lines emanate, such as a positive charge, while a sink is where field lines converge, such as a negative charge. This happens because electric fields follow Gauss's law given by
where is the vacuum permittivity. In this sense, for a magnetic field there are no sources or sinks because there are no magnetic monopoles as described by Gauss's law for magnetism which states that
Electric and magnetic fields also carry energy as described by Poynting's theorem, given by
where is the electromagnetic energy density, is the Poynting vector and can be considered as an energy source-sink term.
Newtonian gravity
Similar to electric and magnetic fields, one can discuss the case of a Newtonian gravitational field described by Gauss's law for gravity,where is the gravitational constant. As gravity is only attractive, there are only gravitational sinks but no sources. Sinks are represented by point masses.