Field line

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The figure at left shows the electric field lines of two equal positive charges. The figure at right shows the electric field lines of a dipole.

Iron filings arrange themselves so as to approximately depict some magnetic field lines. The magnetic field is created a permanent magnet.

A field line is a locus that is defined by a vector field and a starting location within the field. Field lines are useful for visualizing vector fields, which are otherwise hard to depict. Note that, like longitude and latitude lines on a globe, or topographic lines on a topographic map, these lines are not physical lines that are actually present at certain locations; they are merely visualization tools.

[edit] Precise definition

A vector field defines a direction at all points in space; a field line for that vector field may be constructed by tracing a topographic path in the direction of the vector field. More precisely, the tangent line to the path at each point is required to be parallel to the vector field at that point.

A complete description of the geometry of all the field lines of a vector field is sufficient to completely specify the direction of the vector field everywhere. In order to also depict the magnitude, a selection of field lines is drawn such that the density of field lines (number of field lines per unit area) at any location is proportional to the magnitude of the vector field at that point. This is almost always the case, for example, when field lines are used to depict electric and magnetic fields.

[edit] Examples

If the vector field describes a velocity field, then the field lines follow stream lines in the flow. Perhaps the most familiar example of a vector field described by field lines is the magnetic field, which is often depicted using field lines emanating from a magnet.

[edit] Divergence and curl

Field lines can be used to trace familiar quantities from vector calculus:

• Divergence may be easily seen through field lines, assuming the lines are drawn such that the density of field lines is proportional to the magnitude of the field (see above). In this case, the divergence may be seen as the beginning and ending of field lines. In a solenoidal vector field (i.e. a vector field where the divergence is zero everywhere), the field lines neither begin nor end; they either form closed loops, or go off to infinity in both directions. If a vector field has positive divergence in some area, there will be field lines starting from points in that area. If a vector field has negative divergence in some area, there will be field lines ending at points in that area.

• Curl may be seen as a helical shape of field lines.

[edit] Physical significance

Gravitational field line plot of the effective potential of a two-body system (the Sun and Earth here) due to gravity and the centrifugal force. Objects revolving with the same orbital period as the Earth will begin to move according to the contour lines showing equipotential surfaces. The arrows indicate the gradients of the potential around the five Lagrange points — downhill toward or away from them, but at the points themselves these forces are balanced.

While field lines are a "mere" mathematical construction, in some circumstance they take on physical significance. In the context of plasma physics, electrons or ions that happen to be on the same field line interact strongly, while particles on different field lines in general do not interact. This is the same behavior that the particles of iron filings exhibit in a magnetif field.

The iron filings in the photo appear to be aligning themselves with discrete field lines, but in actuality, they are creating the field lines by concentrating the magnetic field along a random topographic path, and not along any line that actually exists in the field. In other words, the lines formed by the iron filings would not exist without the iron filings, and so the magnetic field lines you see are a not a demonstration of a lines in the magnetic field. Magnetic fields are continuous, and do not have discrete lines.

[edit] References

• Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.). Prentice Hall. pp. 65–67 and 232. ISBN 0-13-805326-X. 
• "Visualization of Fields and the Divergence and Curl" course notes from a course at the Massachusetts Institute of Technology.

[edit] See also

• Force field (physics)
• External ray — field lines of Douady-Hubbard potential of Mandelbrot set or filled-in Julia sets
• Line of force