Solenoid
A solenoid (/ˈsoʊlənɔɪd/[1]) is a type of electromagnet formed by a helical coil of wire whose length is substantially greater than its diameter,[2] which generates a controlled magnetic field. The coil can produce a uniform magnetic field in a volume of space when an electric current is passed through it.
André-Marie Ampère coined the term solenoid in 1823, having conceived of the device in 1820.[3]
The helical coil of a solenoid does not necessarily need to revolve around a
Solenoids provide magnetic focusing of electrons in vacuums, notably in television camera tubes such as vidicons and image orthicons. Electrons take helical paths within the magnetic field. These solenoids, focus coils, surround nearly the whole length of the tube.
Physics
Infinite continuous solenoid
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.
The magnetic field inside an infinitely long solenoid is homogeneous and its strength neither depends on the distance from the axis nor on the solenoid's cross-sectional area.
This is a derivation of the
Now consider the imaginary loop c that is located inside the solenoid. By
A similar argument can be applied to the loop a to conclude that the field outside the solenoid is radially uniform or constant. This last result, which holds strictly true only near the center of the solenoid where the field lines are parallel to its length, is important as it shows that the flux density outside is practically zero since the radii of the field outside the solenoid will tend to infinity. An intuitive argument can also be used to show that the flux density outside the solenoid is actually zero. Magnetic field lines only exist as loops, they cannot diverge from or converge to a point like electric field lines can (see Gauss's law for magnetism). The magnetic field lines follow the longitudinal path of the solenoid inside, so they must go in the opposite direction outside of the solenoid so that the lines can form loops. However, the volume outside the solenoid is much greater than the volume inside, so the density of magnetic field lines outside is greatly reduced. Now recall that the field outside is constant. In order for the total number of field lines to be conserved, the field outside must go to zero as the solenoid gets longer. Of course, if the solenoid is constructed as a wire spiral (as often done in practice), then it emanates an outside field the same way as a single wire, due to the current flowing overall down the length of the solenoid.
Applying Ampère's circuital law to the solenoid (see figure on the right) gives us
where is the magnetic flux density, is the length of the solenoid, is the
This equation is valid for a solenoid in free space, which means the permeability of the magnetic path is the same as permeability of free space, μ0.
If the solenoid is immersed in a material with relative permeability μr, then the field is increased by that amount:
In most solenoids, the solenoid is not immersed in a higher permeability material, but rather some portion of the space around the solenoid has the higher permeability material and some is just air (which behaves much like free space). In that scenario, the full effect of the high permeability material is not seen, but there will be an effective (or apparent) permeability μeff such that 1 ≤ μeff ≤ μr.
The inclusion of a
where μeff is the effective or apparent permeability of the core. The effective permeability is a function of the geometric properties of the core and its relative permeability. The terms relative permeability (a property of just the material) and effective permeability (a property of the whole structure) are often confused; they can differ by many orders of magnitude.
For an open magnetic structure, the relationship between the effective permeability and relative permeability is given as follows:
where k is the demagnetization factor of the core.[4]
Finite continuous solenoid
A finite solenoid is a solenoid with finite length. Continuous means that the solenoid is not formed by discrete coils but by a sheet of conductive material. We assume the current is uniformly distributed on the surface of the solenoid, with a surface
The magnetic field can be found using the vector potential, which for a finite solenoid with radius R and length l in cylindrical coordinates is[5][6]
Where:
- ,
- ,
- ,
- ,
- ,
- .
Here, , , and are complete elliptic integrals of the first, second, and third kind.
Using:
The magnetic flux density is obtained as[7][8][9]
On the symmetry axis, the radial component vanishes, and the axial field component is
Short solenoid estimate
For the case in which the radius is much larger than the length of the solenoid (), the magnetic flux density through the centre of the solenoid (in the z direction, parallel to the solenoid's length, where the coil is centered at z=0) can be estimated as the flux density of a single circular conductor loop:
Irregular solenoids
Within the category of finite solenoids, there are those that are sparsely wound with a single pitch, those that are sparsely wound with varying pitches (varied-pitch solenoid), and those with varying radii for different loops (non-cylindrical solenoids). They are called irregular solenoids. They have found applications in different areas, such as sparsely wound solenoids for wireless power transfer,[10][11] varied-pitch solenoids for magnetic resonance imaging (MRI),[12] and non-cylindrical solenoids for other medical devices.[13]
The calculation of the intrinsic inductance and capacitance cannot be done using those for the conventional solenoids, i.e. the tightly wound ones. New calculation methods were proposed for the calculation of intrinsic inductance[14](codes available at [15]) and capacitance.[16] (codes available at [17])
Inductance
As shown above, the magnetic flux density within the coil is practically constant and given by
where μ0 is the magnetic constant, the number of turns, the current and the length of the coil. Ignoring end effects, the total magnetic flux through the coil is obtained by multiplying the flux density by the cross-section area :
Combining this with the definition of inductance
the inductance of a solenoid follows as
A table of inductance for short solenoids of various diameter to length ratios has been calculated by Dellinger, Whittmore, and Ould.[18]
This, and the inductance of more complicated shapes, can be derived from Maxwell's equations. For rigid air-core coils, inductance is a function of coil geometry and number of turns, and is independent of current.
Similar analysis applies to a solenoid with a magnetic core, but only if the length of the coil is much greater than the product of the relative
See also
References
- ^ "solenoid: Meaning in the Cambridge English Dictionary". dictionary.cambridge.org. Archived from the original on 16 January 2017. Retrieved 16 January 2017.
- ^ or equivalently, the diameter of the coil is assumed to be infinitesimally small (Ampère 1823, p. 267: "des courants électriques formants de très-petits circuits autour de cette ligne, dans des plans infiniment rapprochés qui lui soient perpendiculaires").
- Académie des sciences of 22 December 1823, published in print in: Ampère, "Mémoire sur la théorie mathématique des phénomènes électro-dynamiques", Mémoires de l'Académie royale des sciences de l'Institut de France 6 (1827), Paris, F. Didot, pp. 267ff. (and figs. 29–33). "l'assemblage de tous les circuits qui l'entourent [viz. l'arc], assemblage auquel j'ai donné le nom de solénoïde électro-dynamique, du mot grec σωληνοειδὴς, dont la signification exprime précisement ce qui a la forme d'un canal, c'est-à-dire la surface de cette forme sur laquelle se trouvent tous les circuits." (p. 267).
- ^ Jiles, David. Introduction to magnetism and magnetic materials. CRC press, p. 48, 2015.
- ^ "Archived copy" (PDF). Archived (PDF) from the original on 10 April 2014. Retrieved 28 March 2013.
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: CS1 maint: archived copy as title (link) - ^ "Archived copy" (PDF). Archived (PDF) from the original on 19 July 2021. Retrieved 10 July 2021.
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: CS1 maint: archived copy as title (link) - S2CID 123686159.
- ^ Callaghan, Edmund E.; Maslen, Stephen H. (1 October 1960). "The magnetic field of a finite solenoid". NASA Technical Reports. NASA-TN-D-465 (E-900).
- S2CID 126037802.
- S2CID 17105396.
- ^ Zhou, Wenshen; Huang, Shao Ying (28 September 2017). "Novel coil design for wideband wireless power transfer". 2017 International Applied Computational Electromagnetics Society Symposium (ACES): 1–2.
- ^ Ren, Zhi Hua; Huang, Shao Ying (August 2018). "The design of a short solenoid with homogeneous B1 for a low-field portable MRI scanner using genetic algorithm". Proc. 26th ISMRM: 1720.[permanent dead link]
- S2CID 44197357.
- S2CID 182038533.
- ^ Zhou, Wenshen; Huang, Shao Ying (12 April 2021). "the code for accurate model for fast calculating the resonant frequency of an irregular solenoid".
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(help) - S2CID 229274298.
- ^ Zhou, Wenshen; Huang, Shao Ying (12 April 2021). "the code for accurate model for self-capacitance of irregular solenoids".
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External links
- Interactive Java Tutorial: Magnetic Field of a Solenoid, National High Magnetic Field Laboratory
- Discussion of Solenoids at Hyperphysics