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An infinite solenoid has infinite length but finite diameter. "Continuous" means that the solenoid is not formed by discrete finite-width coils but by many infinitely thin coils with no space between them; in this abstraction, the solenoid is often viewed as a cylindrical sheet of conductive material.
Consider an infinite solenoid (ideal solenoid) with n turns per length unit, through which a current () flows. The magnetic field inside the solenoid is, = (1) while the field outside the solenoid is null. From the second and third Maxwell's equations,
A solenoid is a one-dimensional homogeneous indecomposable continuum that has the structure of an abelian compact topological group. Solenoids were first introduced by Vietoris for the n i = 2 {\displaystyle n_{i}=2} case, [ 2 ] and by van Dantzig the n i = n {\displaystyle n_{i}=n} case, where n ≥ 2 {\displaystyle n\geq 2} is fixed. [ 3 ]
The solenoid can be useful for positioning, stopping mid-stroke, or for low velocity actuation; especially in a closed loop control system. A uni-directional solenoid would actuate against an opposing force or a dual solenoid system would be self cycling. The proportional concept is more fully described in SAE publication 860759 (1986).
The strength of the magnetic field decreases with distance from the wire. (For an infinite length wire the strength is inversely proportional to the distance.) A Solenoid with electric current running through it behaves like a magnet. Bending a current-carrying wire into a loop concentrates the magnetic field inside the loop while weakening it ...
The length of is the largest length of any of its chains. If no such largest length exists, we say that has infinite length. Clearly, if the length of a chain equals the length of the module, one has = and =.
An example of a solenoidal vector field, (,) = (,) In vector calculus a solenoidal vector field (also known as an incompressible vector field, a divergence-free vector field, or a transverse vector field) is a vector field v with divergence zero at all points in the field: =
The method divides the domain concerned into sections of infinite length. In contrast with a finite element which is approximated by polynomial expressions on a finite support, the unbounded length of the infinite element is fitted with functions allowing the evaluation of the field at the asymptote.