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Consider an infinite solenoid (ideal solenoid) with n turns per length unit, through which a current () flows. The magnetic field inside the solenoid is, The magnetic field inside the solenoid is, B = μ n I ( t ) {\displaystyle \mathbf {B} =\mu nI(t)} (1)
Figure 1: An infinite solenoid with three arbitrary Ampèrian loops labelled a, b, and c. Integrating over path c demonstrates that the magnetic field inside the solenoid must be radially uniform. An infinite solenoid has infinite length but finite diameter.
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 ]
Of course, the ideal case of infinite length is not realizable, and in practice the finite length of the cylinders produces end effects, which introduce non-uniformities in the field. [ 12 ] [ 13 ] The difficulty of manufacturing a cylinder with a continuously varying magnetization also usually leads to the design being broken into segments.
A simple example is a semi-infinite domain bounded by a rigid wall and filled with viscous fluid. [12] At time t = 0 {\displaystyle t=0} the wall is made to move with constant speed U {\displaystyle U} in a fixed direction (for definiteness, say the x {\displaystyle x} direction and consider only the x − y {\displaystyle x-y} plane), one can ...
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The length of is the largest length of any of its chains. If no such largest length exists, we say that M {\displaystyle M} has infinite length . Clearly, if the length of a chain equals the length of the module, one has M 0 = 0 {\displaystyle M_{0}=0} and M n = M . {\displaystyle M_{n}=M.}
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