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the simultaneous begin and end of field lines takes place. This situation happens, for instance, in the middle between two identical positive electric point charges. There, the field vanishes and the lines coming axially from the charges end. At the same time, in the transverse plane passing through the middle point, an infinite number of field lines diverge radially. The concomitant presence of the lines that end and begin preserves the divergence-free character of the field in the point.
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confusing most field line diagrams are of this type. Since at each point where it is nonzero and finite the vector field has a unique direction, field lines can never intersect, so there is exactly one field line passing through each point at which the vector field is nonzero and finite. Points where the field is zero or infinite have no field line through them, since direction cannot be defined there, but can be the
28:
1013:(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.
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vector field may be depicted by different sets of field lines. A field line diagram is necessarily an incomplete description of a vector field, since it gives no information about the field between the drawn field lines, and the choice of how many and which lines to show determines how much useful information the diagram gives.
148:. Then the density of field lines (number of field lines per unit perpendicular area) at any location is proportional to the magnitude of the vector field at that point. Areas in which neighboring field lines are converging (getting closer together) indicates that the field is getting stronger in that direction.
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By repeating this and connecting the points, the field line can be extended as far as desired. This is only an approximation to the actual field line, since each straight segment isn't actually tangent to the field along its length, just at its starting point. But by using a small enough value for
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form closed loops, extend to infinity in both directions, or continue indefinitely without ever crossing itself. However, as stated above, a special situation may occur around points where the field is zero (that cannot be intersected by field lines, because their direction would not be defined) and
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properties of the filings they damp the field to either side, creating the apparent spaces between the lines that we see. Of course the two stages described here happen concurrently until an equilibrium is achieved. Because the intrinsic magnetism of the filings modifies the field, the lines shown
1098:
The iron filings in the photo appear to be aligning themselves with discrete field lines, but the situation is more complex. It is easy to visualize as a two-stage-process: first, the filings are spread evenly over the magnetic field but all aligned in the direction of the field. Then, based on the
128:
Since there are an infinite number of points in any region, an infinite number of field lines can be drawn; but only a limited number can be shown on a field line diagram. Therefore which field lines are shown is a choice made by the person or computer program which draws the diagram, and a single
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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. If the vector field is the resultant of radial
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to the field vector at each point. A field line is usually shown as a directed line segment, with an arrowhead indicating the direction of the vector field. For two-dimensional fields the field lines are plane curves; since a plane drawing of a 3-dimensional set of field lines can be visually
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198:, sinks at negative charges, and neither elsewhere, so electric field lines start at positive charges and end at negative charges. A gravitational field has no sources, it has sinks at masses, and it has neither elsewhere, gravitational field lines come from infinity and end at masses. A
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Note that for this kind of drawing, where the field-line density is intended to be proportional to the field magnitude, it is important to represent all three dimensions. For example, consider the electric field arising from a single, isolated
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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 magnetic field.
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at each point along its length. A diagram showing a representative set of neighboring field lines is a common way of depicting a vector field in scientific and mathematical literature; this is called a
19:
This article is about the modern use of "field lines" as a way to depict electromagnetic and other vector fields. For the role of these lines in the early history and philosophy of electromagnetism, see
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219:. The electric field lines in this case are straight lines that emanate from the charge uniformly in all directions in three-dimensional space. This means that their density is proportional to
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The figure at left shows the electric field lines of two isolated equal positive charges. The figure at right shows the electric field lines of two isolated equal charges of opposite sign.
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In physics, drawings of field lines are mainly useful in cases where the sources and sinks, if any, have a physical meaning, as opposed to e.g. the case of a force field of a
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for that vector field and may be constructed by starting at a point and tracing a line through space that follows the direction of the vector field, by making the field line
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inverse-square law fields with respect to one or more sources then this corresponds to the fact that the divergence of such a field is zero outside the sources. In a
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for this case. However, if the electric field lines for this setup were just drawn on a two-dimensional plane, their two-dimensional density would be proportional to
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899:, taking a greater number of shorter steps, the field line can be approximated as closely as desired. The field line can be extended in the opposite direction from
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by the filings are only an approximation of the field lines of the original magnetic field. Magnetic fields are continuous, and do not have discrete lines.
868:{\displaystyle \mathbf {x} _{\text{i+1}}=\mathbf {x} _{\text{i}}+{\mathbf {F} (\mathbf {x} _{\text{i}}) \over |\mathbf {F} (\mathbf {x} _{\text{i}})|}ds}
619:{\displaystyle \mathbf {x} _{\text{1}}=\mathbf {x} _{\text{0}}+{\mathbf {F} (\mathbf {x} _{\text{0}}) \over |\mathbf {F} (\mathbf {x} _{\text{0}})|}ds}
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Field lines depicting the electric field created by a positive charge (left), negative charge (center), and uncharged object (right).
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While field lines are a "mere" mathematical construction, in some circumstances they take on physical significance. In
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of the field, field line diagrams are often drawn so that each line represents the same quantity of
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in the flow. Perhaps the most familiar example of a vector field described by field lines is the
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479:{\displaystyle \mathbf {F} (\mathbf {x} _{\text{0}})/|\mathbf {F} (\mathbf {x} _{\text{0}})|}
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a field line can be constructed iteratively by finding the field vector at that point
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Interactive Java applet showing the electric field lines of selected pairs of charges
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defines a direction and magnitude at each point in space. A field line is an
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course notes from a course at the
Massachusetts Institute of Technology.
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by taking each step in the opposite direction by using a negative step
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1239:. Hypermedia Teaching Facility, Massachusetts Institute of Technology
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1233:"Section 2.7: Visualization of Fields and the Divergence and Curl"
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and end on negative charges. In fields which are divergenceless (
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along the field direction a new point on the line can be found
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Field lines can be used to trace familiar quantities from
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722:{\displaystyle \mathbf {F} (\mathbf {x} _{\text{2}})}
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397:{\displaystyle \mathbf {F} (\mathbf {x} _{\text{0}})}
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206:), so its field lines have no start or end: they can
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1354:Am. J. Phys., Vol. 64, No. 6. (1996), pp. 714–724
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159:) and end on points of negative divergence (
1231:Haus, Herman A.; Mechior, James R. (1998).
729:of the field line is found. At each point
328:{\displaystyle \mathbf {F} (\mathbf {x} )}
286:, an incorrect result for this situation.
1376:Introduction to Electrodynamics (3rd ed.)
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921:{\displaystyle \mathbf {x} _{\text{0}}}
751:{\displaystyle \mathbf {x} _{\text{i}}}
666:is found and moving a further distance
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151:In vector fields which have nonzero
1175:. John Wiley and Sons. p. 64.
132:An individual field line shows the
1258:Lieberherr, Martin (6 July 2010).
1204:Vectors in Physics and Engineering
14:
1237:Electromagnetic fields and energy
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1302:Zilberti, Luca (25 April 2017).
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140:. In order to also depict the
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626:Then the field at that point
1372:Griffiths, David J. (1998).
1078:, the velocity field lines (
298:Construction of a field line
76:among many other types. In
1264:American Journal of Physics
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1441:Numerical function drawing
1380:. Prentice Hall. pp.
96:Definition and description
80:, field lines showing the
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1320:10.1109/LMAG.2017.2698038
1147:Line integral convolution
1119:Field lines of Julia sets
1026:conservative vector field
204:Gauss's law for magnetism
202:has no sources or sinks (
64:. They are used to show
194:has sources at positive
167:lines begin on positive
1011:solenoidal vector field
247:{\displaystyle 1/r^{2}}
1308:IEEE Magnetics Letters
1201:Durrant, Alan (1996).
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279:{\displaystyle 1/r}
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188:Gauss's law
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1430:Categories
1243:9 November
1153:References
1099:scale and
1006:Divergence
173:solenoidal
153:divergence
86:fluid flow
41:visual aid
37:field line
1088:electrons
1028:, e.g. a
1024:(i.e., a
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142:magnitude
138:magnitude
134:direction
123:endpoints
51:which is
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1338:39584751
1108:See also
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958:Examples
82:velocity
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1272:Bibcode
1038:helical
196:charges
157:sources
118:tangent
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1386:ISBN
1245:2019
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1177:ISBN
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1016:The
208:only
146:flux
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1316:doi
1280:doi
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