# Solenoid

An illustration of a solenoid
Magnetic field created by a seven-loop solenoid (cross-sectional view) described using field lines

A solenoid (/ˈslənɔɪd/[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. The term solenoid was coined in 1823 by André-Marie Ampère.[3]

The helical coil of a solenoid does not necessarily need to revolve around a straight-line axis; for example, William Sturgeon's electromagnet of 1824 consisted of a solenoid bent into a horseshoe shape (similarly to an arc spring).

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.

In engineering, the term "solenoid" refers not only to the electromagnet but to a complete apparatus providing an actuator that converts electrical energy to mechanical energy.

## Physics

### Infinite continuous solenoid

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. "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 magnetic flux density around a solenoid that is long enough so that fringe effects can be ignored. In Figure 1, we immediately know that the flux density vector points in the positive z direction inside the solenoid, and in the negative z direction outside the solenoid. We confirm this by applying the right hand grip rule for the field around a wire. If we wrap our right hand around a wire with the thumb pointing in the direction of the current, the curl of the fingers shows how the field behaves. Since we are dealing with a long solenoid, all of the components of the magnetic field not pointing upwards cancel out by symmetry. Outside, a similar cancellation occurs, and the field is only pointing downwards.

Now consider the imaginary loop c that is located inside the solenoid. By Ampère's law, we know that the line integral of B (the magnetic flux density vector) around this loop is zero, since it encloses no electrical currents (it can be also assumed that the circuital electric field passing through the loop is constant under such conditions: a constant or constantly changing current through the solenoid). We have shown above that the field is pointing upwards inside the solenoid, so the horizontal portions of loop c do not contribute anything to the integral. Thus the integral of the up side 1 is equal to the integral of the down side 2. Since we can arbitrarily change the dimensions of the loop and get the same result, the only physical explanation is that the integrands are actually equal, that is, the magnetic field inside the solenoid is radially uniform. Note, though, that nothing prohibits it from varying longitudinally, which in fact, it does.

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.

The picture shows how Ampère's law can be applied to the solenoid

Applying Ampère's circuital law to the solenoid (see figure on the right) gives us

${\displaystyle Bl=\mu _{0}NI,}$

where ${\displaystyle B}$ is the magnetic flux density, ${\displaystyle l}$ is the length of the solenoid, ${\displaystyle \mu _{0}}$ is the magnetic constant, ${\displaystyle N}$ the number of turns, and ${\displaystyle I}$ the current. From this we get

${\displaystyle B=\mu _{0}{\frac {NI}{l}}.}$

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:

${\displaystyle B=\mu _{0}\mu _{\mathrm {r} }{\frac {NI}{l}}.}$

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 ferromagnetic core, such as iron, increases the magnitude of the magnetic flux density in the solenoid and raises the effective permeability of the magnetic path. This is expressed by the formula

${\displaystyle B=\mu _{0}\mu _{\mathrm {eff} }{\frac {NI}{l}}=\mu {\frac {NI}{l}},}$

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:

${\displaystyle \mu _{\mathrm {eff} }={\frac {\mu _{r}}{1+k(\mu _{r}-1)}},}$

where k is the demagnetization factor of the core.[4]

### Finite continuous solenoid

Magnetic field line and density created by a solenoid with surface current density

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 current density K; in cylindrical coordinates:

${\displaystyle {\vec {K}}={\frac {I}{l}}{\hat {\phi }}.}$

The magnetic field can be found using the vector potential, which for a finite solenoid with radius R and length l in cylindrical coordinates ${\displaystyle (\rho ,\phi ,z)}$ is[5][6]

${\displaystyle A_{\phi }={\frac {\mu _{0}I}{\pi }}{\frac {R}{l}}\left[{\frac {\zeta }{\sqrt {(R+\rho )^{2}+\zeta ^{2}}}}\left({\frac {m+n-mn}{mn}}K(m)-{\frac {1}{m}}E(m)+{\frac {n-1}{n}}\Pi (n,m)\right)\right]_{\zeta _{-}}^{\zeta _{+}},}$

Where:

• ${\displaystyle \zeta _{\pm }=z\pm {\frac {l}{2}}}$,
• ${\displaystyle n={\frac {4R\rho }{(R+\rho )^{2}}}}$,
• ${\displaystyle m={\frac {4R\rho }{(R+\rho )^{2}+\zeta ^{2}}}}$,
• ${\displaystyle K(m)=\int _{0}^{\frac {\pi }{2}}{\frac {d\theta }{\sqrt {1-m\sin ^{2}\theta }}}}$,
• ${\displaystyle E(m)=\int _{0}^{\frac {\pi }{2}}{\sqrt {1-m\sin ^{2}\theta }}\,d\theta }$ ,
• ${\displaystyle \Pi (n,m)=\int _{0}^{\frac {\pi }{2}}{\frac {d\theta }{(1-n\sin ^{2}\theta ){\sqrt {1-m\sin ^{2}\theta }}}}}$ .

Here, ${\displaystyle K(m)}$, ${\displaystyle E(m)}$, and ${\displaystyle \Pi (n,m)}$ are complete elliptic integrals of the first, second, and third kind.

Using:

${\displaystyle {\vec {B}}=\nabla \times {\vec {A}},}$

The magnetic flux density is obtained as[7][8][9]

${\displaystyle B_{\rho }={\frac {\mu _{0}I}{4\pi }}{\frac {1}{l\,\rho }}\left[{\sqrt {(R+\rho )^{2}+\zeta ^{2}}}{\biggl (}(m-2)K(m)+2E(m){\biggr )}\right]_{\zeta _{-}}^{\zeta _{+}},}$
${\displaystyle B_{z}={\frac {\mu _{0}I}{2\pi }}{\frac {1}{l}}\left[{\frac {\zeta }{\sqrt {(R+\rho )^{2}+\zeta ^{2}}}}\left(K(m)+{\frac {R-\rho }{R+\rho }}\Pi (n,m)\right)\right]_{\zeta _{-}}^{\zeta _{+}}.}$

On the symmetry axis, the radial component vanishes, and the axial field component is

${\displaystyle B_{z}={\frac {\mu _{0}NI}{2}}\left({\frac {l/2-z}{l{\sqrt {R^{2}+(l/2-z)^{2}}}}}+{\frac {l/2+z}{l{\sqrt {R^{2}+(l/2+z)^{2}}}}}\right).}$
Inside the solenoid, far away from the ends (${\displaystyle l/2-|z|\gg R}$), this tends towards the constant value ${\displaystyle B=\mu _{0}NI/l}$.

### Short solenoid estimate

For the case in which the radius is much larger than the length of the solenoid (${\displaystyle R\gg l}$), 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:

${\displaystyle B_{z}\approx {\frac {\mu _{0}INR^{2}}{2{\sqrt {R^{2}+z^{2}}}^{3}}}}$

### Irregular solenoids

Examples of irregular solenoids (a) sparse solenoid, (b) varied pitch solenoid, (c) non-cylindrical solenoid

Within the category of finite solenoids, there are those that are sparsely wound with a single pitch, sparsely wound with varying pitches (varied-pitch solenoid), or those with a varying radius 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 traditional 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 ${\displaystyle B}$ within the coil is practically constant and given by

${\displaystyle B=\mu _{0}{\frac {NI}{l}},}$

where μ0 is the magnetic constant, ${\displaystyle N}$ the number of turns, ${\displaystyle I}$ the current and ${\displaystyle l}$ the length of the coil. Ignoring end effects, the total magnetic flux through the coil is obtained by multiplying the flux density ${\displaystyle B}$ by the cross-section area ${\displaystyle A}$:

${\displaystyle \Phi =\mu _{0}{\frac {NIA}{l}}.}$

Combining this with the definition of inductance

${\displaystyle L={\frac {N\Phi }{I}},}$

the inductance of a solenoid follows as

${\displaystyle L=\mu _{0}{\frac {N^{2}A}{l}}.}$

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 permeability of the magnetic core and the diameter. That limits the simple analysis to low-permeability cores, or extremely long thin solenoids. The presence of a core can be taken into account in the above equations by replacing the magnetic constant μ0 with μ or μ0μr, where μ represents permeability and μr relative permeability. Note that since the permeability of ferromagnetic materials changes with applied magnetic flux, the inductance of a coil with a ferromagnetic core will generally vary with current.

## References

1. ^ "solenoid: Meaning in the Cambridge English Dictionary". dictionary.cambridge.org. Archived from the original on 16 January 2017. Retrieved 16 January 2017.
2. ^ 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").
3. ^ Session of the 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).
4. ^ Jiles, David. Introduction to magnetism and magnetic materials. CRC press, p. 48, 2015.
5. ^ "Archived copy" (PDF). Archived (PDF) from the original on 10 April 2014. Retrieved 28 March 2013.{{cite web}}: CS1 maint: archived copy as title (link)
6. ^ "Archived copy" (PDF). Archived (PDF) from the original on 19 July 2021. Retrieved 10 July 2021.{{cite web}}: CS1 maint: archived copy as title (link)
7. ^ Müller, Karl Friedrich (1 May 1926). "Berechnung der Induktivität von Spulen" [Calculating the Inductance of Coils]. Archiv für Elektrotechnik (in German). 17 (3): 336–353. doi:10.1007/BF01655986. ISSN 1432-0487. S2CID 123686159.
8. ^ 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).
9. ^ Caciagli, Alessio; Baars, Roel J.; Philipse, Albert P.; Kuipers, Bonny W.M. (2018). "Exact expression for the magnetic field of a finite cylinder with arbitrary uniform magnetization". Journal of Magnetism and Magnetic Materials. 456: 423–432. Bibcode:2018JMMM..456..423C. doi:10.1016/j.jmmm.2018.02.003. ISSN 0304-8853. S2CID 126037802.
10. ^ Kurs, André; Karalis, Aristeidis; Moffatt, Robert; Joannopoulos, J. D.; Fisher, Peter; Soljačić, Marin (6 July 2007). "Wireless Power Transfer via Strongly Coupled Magnetic Resonances". Science. 317 (5834): 83–86. Bibcode:2007Sci...317...83K. doi:10.1126/science.1143254. PMID 17556549. S2CID 17105396.
11. ^ 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.
12. ^ 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.
13. ^ Jian, L.; Shi, Y.; Liang, J.; Liu, C.; Xu, G. (June 2013). "A Novel Targeted Magnetic Fluid Hyperthermia System Using HTS Coil Array for Tumor Treatment". IEEE Transactions on Applied Superconductivity. 23 (3): 4400104. Bibcode:2013ITAS...23Q0104J. doi:10.1109/TASC.2012.2230051. S2CID 44197357.
14. ^ Zhou, Wenshen; Huang, Shao Ying (July 2019). "An Accurate Model for Fast Calculating the Resonant Frequency of an Irregular Solenoid". IEEE Transactions on Microwave Theory and Techniques. 67 (7): 2663–2673. Bibcode:2019ITMTT..67.2663Z. doi:10.1109/TMTT.2019.2915514. S2CID 182038533.
15. ^ Zhou, Wenshen; Huang, Shao Ying (12 April 2021). "the code for accurate model for fast calculating the resonant frequency of an irregular solenoid". {{cite journal}}: Cite journal requires |journal= (help)
16. ^ Zhou, Wenshen; Huang, Shao Ying (October 2020). "Modeling the Self-Capacitance of an Irregular Solenoid". IEEE Transactions on Electromagnetic Compatibility. 63 (3): 783–791. doi:10.1109/TEMC.2020.3031075. ISSN 0018-9375. S2CID 229274298.
17. ^ Zhou, Wenshen; Huang, Shao Ying (12 April 2021). "the code for accurate model for self-capacitance of irregular solenoids". {{cite journal}}: Cite journal requires |journal= (help)
18. ^ D. Howard Dellinger; L. E. Whittmore & R. S. Ould (1924). Radio Instruments and Measurements. NBS Circular. Vol. C74. ISBN 9780849302527. Retrieved 7 September 2009.