A flux tube is a generally tube-like (cylindrical) region of space containing a magnetic field, B, such that the cylindrical sides of the tube are everywhere parallel to the magnetic field lines. It is a graphical visual aid for visualizing a magnetic field. Since no magnetic flux passes through the sides of the tube, the flux through any cross section of the tube is equal, and the flux entering the tube at one end is equal to the flux leaving the tube at the other. Both the cross-sectional area of the tube and the magnetic field strength may vary along the length of the tube, but the magnetic flux inside is always constant.
As used in astrophysics, a flux tube generally means an area of space through which a strong magnetic field passes, in which the behavior of matter (usually ionized gas or plasma) is strongly influenced by the field. They are commonly found around stars, including the Sun, which has many flux tubes from tens to hundreds of kilometers in diameter. Sunspots are also associated with larger flux tubes of 2500 km diameter. Some planets also have flux tubes. A well-known example is the flux tube between Jupiter and its moon Io.
Definition
The flux of a vector field passing through any closed orientable surface is the surface integral of the field over the surface. For example, for a vector field consisting of the velocity of a volume of liquid in motion, and an imaginary surface within the liquid, the flux is the volume of liquid passing through the surface per unit time.
A flux tube can be defined passing through any closed, orientable surface in a vector field , as the set of all points on the field lines passing through the boundary of . This set forms a hollow tube. The tube follows the field lines, possibly turning, twisting, and changing its cross sectional size and shape as the field lines converge or diverge. Since no field lines pass through the tube walls there is no flux through the walls of the tube, so all the field lines enter and leave through the end surfaces. Thus a flux tube divides all the field lines into two sets; those passing through the inside of the tube, and those outside. Consider the volume bounded by the tube and any two surfaces and intersecting it. If the field has sources or sinks within the tube the flux out of this volume will be nonzero. However, if the field is divergenceless (solenoidal, ) then from the divergence theorem the sum of the flux leaving the volume through these two surfaces will be zero, so the flux leaving through will be equal to the flux entering through . In other words, the flux within the tube through any surface intersecting the tube is equal, the tube encloses a constant quantity of flux along its length. The strength (magnitude) of the vector field, and the cross sectional area of the tube varies along its length, but the surface integral of the field over any surface spanning the tube is equal.
Since from Maxwell's equations (specifically Gauss's law for magnetism) magnetic fields are divergenceless, magnetic flux tubes have this property, so flux tubes are mainly used as an aid in visualizing magnetic fields. However flux tubes can also be useful for visualizing other vector fields in regions of zero divergence, such as electric fields in regions where there are no charges and gravitational fields in regions where there is no mass.
In particle physics, the hadron particles that make up all matter, such as neutrons and protons, are composed of more basic particles called quarks, which are bound together by thin flux tubes of strong nuclear force field. The flux tube model is important in explaining the so-called color confinement mechanism, why quarks are never seen separately in particle experiments.
Types
Flux rope: Twisted magnetic flux tube. Fibril field: Magnetic flux tube that does not have a magnetic field outside the tube.
History
In 1861, James Clerk Maxwell gave rise to the concept of a flux tube inspired by Michael Faraday's work in electrical and magnetic behavior in his paper titled "On Physical Lines of Force". Maxwell described flux tubes as: If upon any surface which cuts the lines of fluid motion we draw a closed curve, and if from every point of this curve we draw lines of motion, these lines of motion will generate a tubular surface which we may call a tube of fluid motion.
Flux tube strength
The flux tube's strength, , is defined to be the magnetic flux through a surface intersecting the tube, equal to the surface integral of the magnetic field over