64:
295:). 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.
1254:
67:
66:
299:
71:
70:
65:
72:
50:
199:
69:
2838:
42:
68:
3388:, 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). English translation: "the assembly of all the circuits that surround it , assembly to which I gave the name
253:
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
757:
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
290:
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
291:
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
621:
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
282:
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
2584:
226:
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
2857:
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(codes available at ) and capacitance. (codes available at )
2845:
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
1772:
184:
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.
867:
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
2143:
1249:
3252:
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
1548:
3356:
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
2361:
2368:
277:
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
1638:
745:
2827:
850:
1455:
616:
934:
1931:
1645:
1378:
1314:
1938:
3119:
3233:
3565:
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".
2944:
532:
991:
3392:, from the Greek word σωληνοειδὴς, whose meaning precisely expresses what has the shape of a channel, that is to say the surface of this shape on which all the circuits are located".
3169:
2147:
3464:
986:
3418:
2681:
360:
2634:
1869:
434:
2715:
1462:
654:, 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
1834:
1805:
241:
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
3062:
3042:
3018:
2998:
2978:
2890:
478:
458:
407:
383:
1555:
234:
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.
3711:
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".
660:
2731:
2579:{\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).}
283:
magnetic field inside the solenoid is radially uniform. Note, though, that nothing prohibits it from varying longitudinally, which in fact, it does.
3248:
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
273:(the magnetic flux density vector) around this loop is zero, since it encloses no electrical currents (it can be also assumed that the circuital
767:
3969:
882:
3335:
1883:
3754:
Zhou, Wenshen; Huang, Shao Ying (July 2019). "An
Accurate Model for Fast Calculating the Resonant Frequency of an Irregular Solenoid".
3489:
3443:
3238:
A table of inductance for short solenoids of various diameter to length ratios has been calculated by
Dellinger, Whittmore, and Ould.
3457:
3293:
materials changes with applied magnetic flux, the inductance of a coil with a ferromagnetic core will generally vary with current.
1385:
761:
For an open magnetic structure, the relationship between the effective permeability and relative permeability is given as follows:
558:
3411:
3503:
Müller, Karl
Friedrich (1 May 1926). "Berechnung der Induktivität von Spulen" [Calculating the Inductance of Coils].
3372:
of 22 December 1823, published in print in: Ampère, "Mémoire sur la théorie mathématique des phénomènes électro-dynamiques",
1767:{\displaystyle \Pi (n,m)=\int _{0}^{\frac {\pi }{2}}{\frac {d\theta }{(1-n\sin ^{2}\theta ){\sqrt {1-m\sin ^{2}\theta }}}}}
3245:. For rigid air-core coils, inductance is a function of coil geometry and number of turns, and is independent of current.
2854:, varied-pitch solenoids for magnetic resonance imaging (MRI), and non-cylindrical solenoids for other medical devices.
1321:
1273:
3616:
Kurs, André; Karalis, Aristeidis; Moffatt, Robert; Joannopoulos, J. D.; Fisher, Peter; Soljačić, Marin (6 July 2007).
3923:
3070:
2867:
3180:
3694:"The design of a short solenoid with homogeneous B1 for a low-field portable MRI scanner using genetic algorithm"
3286:
3249:
2957:
538:
2898:
486:
3992:
158:
3133:
2841:
Examples of irregular solenoids (a) sparse solenoid, (b) varied-pitch solenoid, (c) non-cylindrical solenoid
30:
This article is about the electromagnet. For the device that converts electricity to mechanical energy, see
292:
3833:
Zhou, Wenshen; Huang, Shao Ying (October 2020). "Modeling the Self-Capacitance of an
Irregular Solenoid".
311:
303:
2138:{\displaystyle B_{\rho }={\frac {\mu _{0}I}{4\pi }}{\frac {1}{l\,\rho }}\left_{\zeta _{-}}^{\zeta _{+}},}
3693:
953:
35:
17:
3975:
3952:
3898:
3819:
2639:
320:
3799:"the code for accurate model for fast calculating the resonant frequency of an irregular solenoid"
3368:
2589:
2851:
1253:
1244:{\displaystyle A_{\phi }={\frac {\mu _{0}I}{\pi }}{\frac {R}{l}}\left_{\zeta _{-}}^{\zeta _{+}},}
152:
in 1823, having conceived of the device in 1820. The French term originally created by Ampère is
145:
3384:). "l'assemblage de tous les circuits qui l'entourent , assemblage auquel j'ai donné le nom de
3327:
3242:
1839:
1543:{\displaystyle K(m)=\int _{0}^{\frac {\pi }{2}}{\frac {d\theta }{\sqrt {1-m\sin ^{2}\theta }}}}
876:
31:
3913:
2356:{\displaystyle B_{z}={\frac {\mu _{0}I}{2\pi }}{\frac {1}{l}}\left_{\zeta _{-}}^{\zeta _{+}}.}
412:
177:'s electromagnet of 1824 consisted of a solenoid bent into a horseshoe shape (similarly to an
3885:
3806:
238:
2694:
254:
symmetry. Outside, a similar cancellation occurs, and the field is only pointing downwards.
3763:
3720:
3629:
3574:
250:
1810:
1781:
8:
3767:
3724:
3633:
3578:
2850:. They have found applications in different areas, such as sparsely wound solenoids for
3997:
3940:
3858:
3779:
3736:
3655:
3598:
3528:
3401:
Jiles, David. Introduction to magnetism and magnetic materials. CRC press, p. 48, 2015.
3047:
3027:
3003:
2983:
2963:
2875:
1633:{\displaystyle E(m)=\int _{0}^{\frac {\pi }{2}}{\sqrt {1-m\sin ^{2}\theta }}\,d\theta }
463:
443:
392:
368:
2365:
On the symmetry axis, the radial component vanishes, and the axial field component is
262:
203:
134:
whose length is substantially greater than its diameter, which generates a controlled
3919:
3862:
3850:
3783:
3647:
3602:
3590:
3532:
3520:
3483:
3437:
3381:
3377:
1872:
437:
3740:
3659:
3842:
3771:
3728:
3637:
3582:
3512:
939:
740:{\displaystyle B=\mu _{0}\mu _{\mathrm {eff} }{\frac {NI}{l}}=\mu {\frac {NI}{l}},}
174:
139:
90:
3679:
2017 International
Applied Computational Electromagnetics Society Symposium (ACES)
2822:{\displaystyle B_{z}\approx {\frac {\mu _{0}INR^{2}}{2{\sqrt {R^{2}+z^{2}}}^{3}}}}
222:
demonstrates that the magnetic field inside the solenoid must be radially uniform.
3547:
2691:
For the case in which the radius is much larger than the length of the solenoid (
1261:
869:
622:
material is not seen, but there will be an effective (or apparent) permeability
3674:
3586:
3302:
1257:
386:
274:
231:
135:
53:
3846:
3775:
3732:
2725:=0) can be estimated as the flux density of a single circular conductor loop:
298:
3986:
3877:
3854:
3594:
3524:
3290:
3021:
647:
266:
170:
169:
The helical coil of a solenoid does not necessarily need to revolve around a
138:. The coil can produce a uniform magnetic field in a volume of space when an
123:
3911:
3798:
3642:
3617:
2721:
direction, parallel to the solenoid's length, where the coil is centered at
3651:
77:
845:{\displaystyle \mu _{\mathrm {eff} }={\frac {\mu _{r}}{1+k(\mu _{r}-1)}},}
3878:"the code for accurate model for self-capacitance of irregular solenoids"
3241:
This, and the inductance of more complicated shapes, can be derived from
49:
2717:), the magnetic flux density through the centre of the solenoid (in the
198:
56:
created by a seven-loop solenoid (cross-sectional view) described using
3516:
3125:
178:
57:
2837:
548:
If the solenoid is immersed in a material with relative permeability μ
537:
This equation is valid for a solenoid in free space, which means the
76:
Magnetic field demonstration with solenoid-shaped insulated wire and
3307:
3618:"Wireless Power Transfer via Strongly Coupled Magnetic Resonances"
3374:
Mémoires de l'Académie royale des sciences de l'Institut de France
541:
of the magnetic path is the same as permeability of free space, μ
249:
direction outside the solenoid. We confirm this by applying the
3912:
D. Howard
Dellinger; L. E. Whittmore & R. S. Ould (1924).
3615:
111:
3024:
through the coil is obtained by multiplying the flux density
1450:{\displaystyle m={\frac {4R\rho }{(R+\rho )^{2}+\zeta ^{2}}}}
758:
often confused; they can differ by many orders of magnitude.
611:{\displaystyle B=\mu _{0}\mu _{\mathrm {r} }{\frac {NI}{l}}.}
127:
41:
3564:
3546:
Callaghan, Edmund E.; Maslen, Stephen H. (1 October 1960).
651:
131:
105:
99:
3675:"Novel coil design for wideband wireless power transfer"
3020:
the length of the coil. Ignoring end effects, the total
929:{\displaystyle {\vec {K}}={\frac {I}{l}}{\hat {\phi }}.}
3970:
Interactive Java
Tutorial: Magnetic Field of a Solenoid
3328:"solenoid: Meaning in the Cambridge English Dictionary"
156:, which is a French transliteration of the Greek word
3673:
Zhou, Wenshen; Huang, Shao Ying (28 September 2017).
3183:
3136:
3073:
3050:
3030:
3006:
2986:
2966:
2901:
2892:
within the coil is practically constant and given by
2878:
2734:
2697:
2642:
2592:
2371:
2150:
1941:
1886:
1842:
1813:
1784:
1648:
1558:
1465:
1388:
1324:
1276:
1260:
lines and density created by a solenoid with surface
994:
956:
885:
770:
663:
561:
489:
466:
446:
415:
395:
371:
323:
108:
96:
3756:
IEEE Transactions on
Microwave Theory and Techniques
1926:{\displaystyle {\vec {B}}=\nabla \times {\vec {A}},}
202:
Figure 1: An infinite solenoid with three arbitrary
114:
102:
314:to the solenoid (see figure on the right) gives us
245:direction inside the solenoid, and in the negative
93:
3835:IEEE Transactions on Electromagnetic Compatibility
3227:
3163:
3113:
3056:
3036:
3012:
2992:
2972:
2938:
2884:
2821:
2709:
2675:
2628:
2578:
2355:
2137:
1925:
1863:
1828:
1799:
1766:
1632:
1542:
1449:
1373:{\displaystyle n={\frac {4R\rho }{(R+\rho )^{2}}}}
1372:
1308:
1243:
980:
928:
844:
739:
610:
526:
472:
452:
428:
401:
377:
354:
3876:Zhou, Wenshen; Huang, Shao Ying (12 April 2021).
3797:Zhou, Wenshen; Huang, Shao Ying (12 April 2021).
2097:
2045:
1309:{\displaystyle \zeta _{\pm }=z\pm {\frac {l}{2}}}
3984:
3558:
3710:
3545:
3114:{\displaystyle \Phi =\mu _{0}{\frac {NIA}{l}}.}
193:
3713:IEEE Transactions on Applied Superconductivity
3692:Ren, Zhi Hua; Huang, Shao Ying (August 2018).
3228:{\displaystyle L=\mu _{0}{\frac {N^{2}A}{l}}.}
552:, then the field is increased by that amount:
286:A similar argument can be applied to the loop
27:Type of electromagnet formed by a coil of wire
2586:Inside the solenoid, far away from the ends (
862:
3567:Journal of Magnetism and Magnetic Materials
859:is the demagnetization factor of the core.
2939:{\displaystyle B=\mu _{0}{\frac {NI}{l}},}
2872:As shown above, the magnetic flux density
2686:
942:, which for a finite solenoid with radius
938:The magnetic field can be found using the
527:{\displaystyle B=\mu _{0}{\frac {NI}{l}}.}
3972:, National High Magnetic Field Laboratory
3875:
3832:
3796:
3753:
3672:
3641:
3548:"The magnetic field of a finite solenoid"
2636:), this tends towards the constant value
1991:
1935:The magnetic flux density is obtained as
1623:
3691:
3174:the inductance of a solenoid follows as
2836:
1252:
297:
261:that is located inside the solenoid. By
197:
62:
48:
40:
3976:Discussion of Solenoids at Hyperphysics
14:
3985:
3502:
3488:: CS1 maint: archived copy as title (
3442:: CS1 maint: archived copy as title (
3289:. Note that since the permeability of
3164:{\displaystyle L={\frac {N\Phi }{I}},}
3124:Combining this with the definition of
2832:
1875:of the first, second, and third kind.
3338:from the original on 16 January 2017
24:
3915:Radio Instruments and Measurements
3424:from the original on 10 April 2014
3149:
3074:
2297:
1902:
1843:
1649:
1185:
783:
780:
777:
692:
689:
686:
584:
25:
4009:
3963:
3470:from the original on 19 July 2021
2868:Inductance with physical symmetry
257:Now consider the imaginary loop
89:
3905:
3869:
3826:
3790:
3747:
3704:
3685:
3666:
3609:
981:{\displaystyle (\rho ,\phi ,z)}
409:is the length of the solenoid,
3539:
3496:
3450:
3404:
3395:
3360:
3350:
3320:
2676:{\displaystyle B=\mu _{0}NI/l}
2616:
2608:
2554:
2533:
2476:
2455:
2312:
2300:
2265:
2259:
2226:
2213:
2092:
2086:
2074:
2068:
2062:
2050:
2019:
2006:
1914:
1893:
1858:
1846:
1823:
1817:
1794:
1788:
1729:
1701:
1664:
1652:
1568:
1562:
1475:
1469:
1422:
1409:
1358:
1345:
1200:
1188:
1161:
1155:
1136:
1130:
1065:
1052:
975:
957:
917:
892:
833:
814:
480:the current. From this we get
355:{\displaystyle Bl=\mu _{0}NI,}
306:can be applied to the solenoid
13:
1:
3313:
2861:
45:An illustration of a solenoid
3376:6 (1827), Paris, F. Didot,
3279:represents permeability and
2629:{\displaystyle l/2-|z|\gg R}
237:This is a derivation of the
194:Infinite continuous solenoid
7:
3386:solénoïde électro-dynamique
3296:
950:in cylindrical coordinates
10:
4014:
3587:10.1016/j.jmmm.2018.02.003
3044:by the cross-section area
2865:
863:Finite continuous solenoid
188:
29:
3847:10.1109/TEMC.2020.3031075
3776:10.1109/TMTT.2019.2915514
3733:10.1109/TASC.2012.2230051
3505:Archiv für Elektrotechnik
1864:{\displaystyle \Pi (n,m)}
460:the number of turns, and
293:Gauss's law for magnetism
36:Solenoid (disambiguation)
3554:. NASA-TN-D-465 (E-900).
3390:electro-dynamic solenoid
3332:dictionary.cambridge.org
629:such that 1 ≤
429:{\displaystyle \mu _{0}}
218:. Integrating over path
3643:10.1126/science.1143254
2852:wireless power transfer
2687:Short solenoid estimate
877:cylindrical coordinates
3893:Cite journal requires
3814:Cite journal requires
3552:NASA Technical Reports
3229:
3165:
3115:
3058:
3038:
3014:
2994:
2974:
2940:
2886:
2842:
2823:
2711:
2710:{\displaystyle R\gg l}
2677:
2630:
2580:
2357:
2139:
1927:
1865:
1830:
1801:
1768:
1634:
1544:
1451:
1374:
1310:
1264:
1245:
982:
930:
846:
741:
612:
528:
474:
454:
430:
403:
379:
356:
312:Ampère's circuital law
307:
223:
142:is passed through it.
80:
60:
46:
34:. For other uses, see
32:solenoid (engineering)
3993:Electromagnetic coils
3369:Académie des sciences
3287:relative permeability
3230:
3166:
3116:
3059:
3039:
3015:
2995:
2980:the number of turns,
2975:
2941:
2887:
2840:
2824:
2712:
2678:
2631:
2581:
2358:
2140:
1928:
1866:
1831:
1802:
1769:
1635:
1545:
1452:
1375:
1311:
1256:
1246:
983:
931:
847:
742:
613:
529:
475:
455:
431:
404:
387:magnetic flux density
380:
357:
301:
239:magnetic flux density
201:
75:
52:
44:
3181:
3134:
3071:
3048:
3028:
3004:
2984:
2964:
2899:
2876:
2732:
2695:
2640:
2590:
2369:
2148:
1939:
1884:
1840:
1829:{\displaystyle E(m)}
1811:
1800:{\displaystyle K(m)}
1782:
1646:
1556:
1463:
1386:
1322:
1274:
992:
954:
883:
768:
661:
559:
487:
464:
444:
413:
393:
369:
321:
251:right hand grip rule
3768:2019ITMTT..67.2663Z
3725:2013ITAS...23Q0104J
3634:2007Sci...317...83K
3579:2018JMMM..456..423C
3357:perpendiculaires").
3243:Maxwell's equations
2848:irregular solenoids
2833:Irregular solenoids
2349:
2131:
1689:
1593:
1500:
1237:
646:The inclusion of a
265:, we know that the
173:axis; for example,
3517:10.1007/BF01655986
3225:
3161:
3111:
3054:
3034:
3010:
2990:
2970:
2936:
2882:
2843:
2819:
2707:
2673:
2626:
2576:
2353:
2201:
2135:
1998:
1923:
1873:elliptic integrals
1861:
1826:
1797:
1764:
1670:
1630:
1574:
1540:
1481:
1447:
1370:
1306:
1265:
1241:
1040:
978:
926:
842:
737:
608:
524:
470:
450:
426:
399:
375:
352:
308:
224:
146:André-Marie Ampère
81:
61:
47:
3918:. Vol. C74.
3220:
3156:
3106:
3057:{\displaystyle A}
3037:{\displaystyle B}
3013:{\displaystyle l}
2993:{\displaystyle I}
2973:{\displaystyle N}
2958:magnetic constant
2931:
2885:{\displaystyle B}
2817:
2808:
2566:
2563:
2488:
2485:
2408:
2295:
2249:
2248:
2199:
2189:
2041:
1996:
1980:
1917:
1896:
1762:
1759:
1687:
1621:
1591:
1538:
1537:
1498:
1445:
1368:
1304:
1183:
1150:
1125:
1088:
1087:
1038:
1028:
920:
909:
895:
837:
732:
711:
603:
519:
473:{\displaystyle I}
453:{\displaystyle N}
438:magnetic constant
402:{\displaystyle l}
378:{\displaystyle B}
73:
16:(Redirected from
4005:
3957:
3956:
3950:
3946:
3944:
3936:
3934:
3932:
3909:
3903:
3902:
3896:
3891:
3889:
3881:
3873:
3867:
3866:
3830:
3824:
3823:
3817:
3812:
3810:
3802:
3794:
3788:
3787:
3762:(7): 2663–2673.
3751:
3745:
3744:
3708:
3702:
3701:
3698:Proc. 26th ISMRM
3689:
3683:
3682:
3670:
3664:
3663:
3645:
3613:
3607:
3606:
3562:
3556:
3555:
3543:
3537:
3536:
3500:
3494:
3493:
3487:
3479:
3477:
3475:
3469:
3462:
3454:
3448:
3447:
3441:
3433:
3431:
3429:
3423:
3416:
3408:
3402:
3399:
3393:
3364:
3358:
3354:
3348:
3347:
3345:
3343:
3324:
3234:
3232:
3231:
3226:
3221:
3216:
3212:
3211:
3201:
3199:
3198:
3170:
3168:
3167:
3162:
3157:
3152:
3144:
3120:
3118:
3117:
3112:
3107:
3102:
3091:
3089:
3088:
3063:
3061:
3060:
3055:
3043:
3041:
3040:
3035:
3019:
3017:
3016:
3011:
3000:the current and
2999:
2997:
2996:
2991:
2979:
2977:
2976:
2971:
2945:
2943:
2942:
2937:
2932:
2927:
2919:
2917:
2916:
2891:
2889:
2888:
2883:
2828:
2826:
2825:
2820:
2818:
2816:
2815:
2814:
2809:
2807:
2806:
2794:
2793:
2784:
2777:
2776:
2775:
2760:
2759:
2749:
2744:
2743:
2716:
2714:
2713:
2708:
2682:
2680:
2679:
2674:
2669:
2658:
2657:
2635:
2633:
2632:
2627:
2619:
2611:
2600:
2585:
2583:
2582:
2577:
2572:
2568:
2567:
2565:
2564:
2562:
2561:
2543:
2529:
2528:
2519:
2513:
2503:
2494:
2489:
2487:
2486:
2484:
2483:
2465:
2451:
2450:
2441:
2435:
2425:
2416:
2409:
2404:
2397:
2396:
2386:
2381:
2380:
2362:
2360:
2359:
2354:
2348:
2347:
2346:
2336:
2335:
2334:
2324:
2320:
2319:
2315:
2296:
2294:
2283:
2272:
2250:
2247:
2246:
2234:
2233:
2212:
2208:
2200:
2192:
2190:
2188:
2180:
2176:
2175:
2165:
2160:
2159:
2144:
2142:
2141:
2136:
2130:
2129:
2128:
2118:
2117:
2116:
2106:
2102:
2101:
2100:
2049:
2048:
2042:
2040:
2039:
2027:
2026:
2005:
1997:
1995:
1983:
1981:
1979:
1971:
1967:
1966:
1956:
1951:
1950:
1932:
1930:
1929:
1924:
1919:
1918:
1910:
1898:
1897:
1889:
1870:
1868:
1867:
1862:
1835:
1833:
1832:
1827:
1806:
1804:
1803:
1798:
1773:
1771:
1770:
1765:
1763:
1761:
1760:
1752:
1751:
1733:
1722:
1721:
1699:
1691:
1688:
1680:
1678:
1639:
1637:
1636:
1631:
1622:
1614:
1613:
1595:
1592:
1584:
1582:
1549:
1547:
1546:
1541:
1539:
1530:
1529:
1511:
1510:
1502:
1499:
1491:
1489:
1456:
1454:
1453:
1448:
1446:
1444:
1443:
1442:
1430:
1429:
1407:
1396:
1379:
1377:
1376:
1371:
1369:
1367:
1366:
1365:
1343:
1332:
1315:
1313:
1312:
1307:
1305:
1297:
1286:
1285:
1250:
1248:
1247:
1242:
1236:
1235:
1234:
1224:
1223:
1222:
1212:
1208:
1207:
1203:
1184:
1179:
1168:
1151:
1143:
1126:
1124:
1116:
1096:
1089:
1086:
1085:
1073:
1072:
1051:
1047:
1039:
1031:
1029:
1024:
1020:
1019:
1009:
1004:
1003:
987:
985:
984:
979:
940:vector potential
935:
933:
932:
927:
922:
921:
913:
910:
902:
897:
896:
888:
851:
849:
848:
843:
838:
836:
826:
825:
803:
802:
793:
788:
787:
786:
746:
744:
743:
738:
733:
728:
720:
712:
707:
699:
697:
696:
695:
679:
678:
617:
615:
614:
609:
604:
599:
591:
589:
588:
587:
577:
576:
533:
531:
530:
525:
520:
515:
507:
505:
504:
479:
477:
476:
471:
459:
457:
456:
451:
435:
433:
432:
427:
425:
424:
408:
406:
405:
400:
384:
382:
381:
376:
361:
359:
358:
353:
342:
341:
175:William Sturgeon
148:coined the term
140:electric current
121:
120:
117:
116:
113:
110:
107:
104:
101:
98:
95:
74:
21:
4013:
4012:
4008:
4007:
4006:
4004:
4003:
4002:
3983:
3982:
3966:
3961:
3960:
3948:
3947:
3938:
3937:
3930:
3928:
3926:
3910:
3906:
3894:
3892:
3883:
3882:
3874:
3870:
3831:
3827:
3815:
3813:
3804:
3803:
3795:
3791:
3752:
3748:
3709:
3705:
3690:
3686:
3671:
3667:
3628:(5834): 83–86.
3614:
3610:
3563:
3559:
3544:
3540:
3501:
3497:
3481:
3480:
3473:
3471:
3467:
3460:
3458:"Archived copy"
3456:
3455:
3451:
3435:
3434:
3427:
3425:
3421:
3414:
3412:"Archived copy"
3410:
3409:
3405:
3400:
3396:
3366:Session of the
3365:
3361:
3355:
3351:
3341:
3339:
3326:
3325:
3321:
3316:
3299:
3284:
3273:
3269:
3258:
3207:
3203:
3202:
3200:
3194:
3190:
3182:
3179:
3178:
3145:
3143:
3135:
3132:
3131:
3092:
3090:
3084:
3080:
3072:
3069:
3068:
3049:
3046:
3045:
3029:
3026:
3025:
3005:
3002:
3001:
2985:
2982:
2981:
2965:
2962:
2961:
2955:
2920:
2918:
2912:
2908:
2900:
2897:
2896:
2877:
2874:
2873:
2870:
2864:
2835:
2810:
2802:
2798:
2789:
2785:
2783:
2782:
2778:
2771:
2767:
2755:
2751:
2750:
2748:
2739:
2735:
2733:
2730:
2729:
2696:
2693:
2692:
2689:
2665:
2653:
2649:
2641:
2638:
2637:
2615:
2607:
2596:
2591:
2588:
2587:
2557:
2553:
2539:
2524:
2520:
2518:
2514:
2499:
2495:
2493:
2479:
2475:
2461:
2446:
2442:
2440:
2436:
2421:
2417:
2415:
2414:
2410:
2392:
2388:
2387:
2385:
2376:
2372:
2370:
2367:
2366:
2342:
2338:
2337:
2330:
2326:
2325:
2284:
2273:
2271:
2255:
2251:
2242:
2238:
2229:
2225:
2207:
2206:
2202:
2191:
2181:
2171:
2167:
2166:
2164:
2155:
2151:
2149:
2146:
2145:
2124:
2120:
2119:
2112:
2108:
2107:
2096:
2095:
2044:
2043:
2035:
2031:
2022:
2018:
2004:
2003:
1999:
1987:
1982:
1972:
1962:
1958:
1957:
1955:
1946:
1942:
1940:
1937:
1936:
1909:
1908:
1888:
1887:
1885:
1882:
1881:
1841:
1838:
1837:
1812:
1809:
1808:
1783:
1780:
1779:
1747:
1743:
1732:
1717:
1713:
1700:
1692:
1690:
1679:
1674:
1647:
1644:
1643:
1609:
1605:
1594:
1583:
1578:
1557:
1554:
1553:
1525:
1521:
1503:
1501:
1490:
1485:
1464:
1461:
1460:
1438:
1434:
1425:
1421:
1408:
1397:
1395:
1387:
1384:
1383:
1361:
1357:
1344:
1333:
1331:
1323:
1320:
1319:
1296:
1281:
1277:
1275:
1272:
1271:
1262:current density
1230:
1226:
1225:
1218:
1214:
1213:
1169:
1167:
1142:
1117:
1097:
1095:
1094:
1090:
1081:
1077:
1068:
1064:
1046:
1045:
1041:
1030:
1015:
1011:
1010:
1008:
999:
995:
993:
990:
989:
955:
952:
951:
912:
911:
901:
887:
886:
884:
881:
880:
870:current density
865:
821:
817:
804:
798:
794:
792:
776:
775:
771:
769:
766:
765:
756:
721:
719:
700:
698:
685:
684:
680:
674:
670:
662:
659:
658:
642:
635:
628:
592:
590:
583:
582:
578:
572:
568:
560:
557:
556:
551:
544:
508:
506:
500:
496:
488:
485:
484:
465:
462:
461:
445:
442:
441:
420:
416:
414:
411:
410:
394:
391:
390:
370:
367:
366:
337:
333:
322:
319:
318:
196:
191:
122:) is a type of
92:
88:
63:
39:
28:
23:
22:
15:
12:
11:
5:
4011:
4001:
4000:
3995:
3979:
3978:
3973:
3965:
3964:External links
3962:
3959:
3958:
3949:|journal=
3924:
3904:
3895:|journal=
3868:
3841:(3): 783–791.
3825:
3816:|journal=
3789:
3746:
3719:(3): 4400104.
3703:
3684:
3665:
3608:
3557:
3538:
3511:(3): 336–353.
3495:
3449:
3403:
3394:
3359:
3349:
3318:
3317:
3315:
3312:
3311:
3310:
3305:
3303:Helmholtz coil
3298:
3295:
3282:
3271:
3267:
3256:
3236:
3235:
3224:
3219:
3215:
3210:
3206:
3197:
3193:
3189:
3186:
3172:
3171:
3160:
3155:
3151:
3148:
3142:
3139:
3122:
3121:
3110:
3105:
3101:
3098:
3095:
3087:
3083:
3079:
3076:
3053:
3033:
3009:
2989:
2969:
2953:
2947:
2946:
2935:
2930:
2926:
2923:
2915:
2911:
2907:
2904:
2881:
2863:
2860:
2834:
2831:
2830:
2829:
2813:
2805:
2801:
2797:
2792:
2788:
2781:
2774:
2770:
2766:
2763:
2758:
2754:
2747:
2742:
2738:
2706:
2703:
2700:
2688:
2685:
2672:
2668:
2664:
2661:
2656:
2652:
2648:
2645:
2625:
2622:
2618:
2614:
2610:
2606:
2603:
2599:
2595:
2575:
2571:
2560:
2556:
2552:
2549:
2546:
2542:
2538:
2535:
2532:
2527:
2523:
2517:
2512:
2509:
2506:
2502:
2498:
2492:
2482:
2478:
2474:
2471:
2468:
2464:
2460:
2457:
2454:
2449:
2445:
2439:
2434:
2431:
2428:
2424:
2420:
2413:
2407:
2403:
2400:
2395:
2391:
2384:
2379:
2375:
2352:
2345:
2341:
2333:
2329:
2323:
2318:
2314:
2311:
2308:
2305:
2302:
2299:
2293:
2290:
2287:
2282:
2279:
2276:
2270:
2267:
2264:
2261:
2258:
2254:
2245:
2241:
2237:
2232:
2228:
2224:
2221:
2218:
2215:
2211:
2205:
2198:
2195:
2187:
2184:
2179:
2174:
2170:
2163:
2158:
2154:
2134:
2127:
2123:
2115:
2111:
2105:
2099:
2094:
2091:
2088:
2085:
2082:
2079:
2076:
2073:
2070:
2067:
2064:
2061:
2058:
2055:
2052:
2047:
2038:
2034:
2030:
2025:
2021:
2017:
2014:
2011:
2008:
2002:
1994:
1990:
1986:
1978:
1975:
1970:
1965:
1961:
1954:
1949:
1945:
1922:
1916:
1913:
1907:
1904:
1901:
1895:
1892:
1860:
1857:
1854:
1851:
1848:
1845:
1825:
1822:
1819:
1816:
1796:
1793:
1790:
1787:
1776:
1775:
1758:
1755:
1750:
1746:
1742:
1739:
1736:
1731:
1728:
1725:
1720:
1716:
1712:
1709:
1706:
1703:
1698:
1695:
1686:
1683:
1677:
1673:
1669:
1666:
1663:
1660:
1657:
1654:
1651:
1641:
1629:
1626:
1620:
1617:
1612:
1608:
1604:
1601:
1598:
1590:
1587:
1581:
1577:
1573:
1570:
1567:
1564:
1561:
1551:
1536:
1533:
1528:
1524:
1520:
1517:
1514:
1509:
1506:
1497:
1494:
1488:
1484:
1480:
1477:
1474:
1471:
1468:
1458:
1441:
1437:
1433:
1428:
1424:
1420:
1417:
1414:
1411:
1406:
1403:
1400:
1394:
1391:
1381:
1364:
1360:
1356:
1353:
1350:
1347:
1342:
1339:
1336:
1330:
1327:
1317:
1303:
1300:
1295:
1292:
1289:
1284:
1280:
1258:Magnetic field
1240:
1233:
1229:
1221:
1217:
1211:
1206:
1202:
1199:
1196:
1193:
1190:
1187:
1182:
1178:
1175:
1172:
1166:
1163:
1160:
1157:
1154:
1149:
1146:
1141:
1138:
1135:
1132:
1129:
1123:
1120:
1115:
1112:
1109:
1106:
1103:
1100:
1093:
1084:
1080:
1076:
1071:
1067:
1063:
1060:
1057:
1054:
1050:
1044:
1037:
1034:
1027:
1023:
1018:
1014:
1007:
1002:
998:
977:
974:
971:
968:
965:
962:
959:
925:
919:
916:
908:
905:
900:
894:
891:
864:
861:
853:
852:
841:
835:
832:
829:
824:
820:
816:
813:
810:
807:
801:
797:
791:
785:
782:
779:
774:
754:
748:
747:
736:
731:
727:
724:
718:
715:
710:
706:
703:
694:
691:
688:
683:
677:
673:
669:
666:
650:core, such as
640:
633:
626:
619:
618:
607:
602:
598:
595:
586:
581:
575:
571:
567:
564:
549:
542:
535:
534:
523:
518:
514:
511:
503:
499:
495:
492:
469:
449:
423:
419:
398:
374:
363:
362:
351:
348:
345:
340:
336:
332:
329:
326:
275:electric field
232:magnetic field
204:Ampèrian loops
195:
192:
190:
187:
136:magnetic field
54:Magnetic field
26:
9:
6:
4:
3:
2:
4010:
3999:
3996:
3994:
3991:
3990:
3988:
3981:
3977:
3974:
3971:
3968:
3967:
3954:
3942:
3927:
3925:9780849302527
3921:
3917:
3916:
3908:
3900:
3887:
3879:
3872:
3864:
3860:
3856:
3852:
3848:
3844:
3840:
3836:
3829:
3821:
3808:
3800:
3793:
3785:
3781:
3777:
3773:
3769:
3765:
3761:
3757:
3750:
3742:
3738:
3734:
3730:
3726:
3722:
3718:
3714:
3707:
3699:
3695:
3688:
3680:
3676:
3669:
3661:
3657:
3653:
3649:
3644:
3639:
3635:
3631:
3627:
3623:
3619:
3612:
3604:
3600:
3596:
3592:
3588:
3584:
3580:
3576:
3572:
3568:
3561:
3553:
3549:
3542:
3534:
3530:
3526:
3522:
3518:
3514:
3510:
3507:(in German).
3506:
3499:
3491:
3485:
3466:
3459:
3453:
3445:
3439:
3420:
3413:
3407:
3398:
3391:
3387:
3383:
3379:
3375:
3371:
3370:
3363:
3353:
3337:
3333:
3329:
3323:
3319:
3309:
3306:
3304:
3301:
3300:
3294:
3292:
3291:ferromagnetic
3288:
3285:
3278:
3274:
3263:
3259:
3251:
3246:
3244:
3239:
3222:
3217:
3213:
3208:
3204:
3195:
3191:
3187:
3184:
3177:
3176:
3175:
3158:
3153:
3146:
3140:
3137:
3130:
3129:
3128:
3127:
3108:
3103:
3099:
3096:
3093:
3085:
3081:
3077:
3067:
3066:
3065:
3051:
3031:
3023:
3022:magnetic flux
3007:
2987:
2967:
2959:
2952:
2933:
2928:
2924:
2921:
2913:
2909:
2905:
2902:
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2879:
2869:
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2853:
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2839:
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2803:
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2768:
2764:
2761:
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2547:
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2500:
2496:
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2422:
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2373:
2363:
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2309:
2306:
2303:
2291:
2288:
2285:
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2274:
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2262:
2256:
2252:
2243:
2239:
2235:
2230:
2222:
2219:
2216:
2209:
2203:
2196:
2193:
2185:
2182:
2177:
2172:
2168:
2161:
2156:
2152:
2132:
2125:
2121:
2113:
2109:
2103:
2089:
2083:
2080:
2077:
2071:
2065:
2059:
2056:
2053:
2036:
2032:
2028:
2023:
2015:
2012:
2009:
2000:
1992:
1988:
1984:
1976:
1973:
1968:
1963:
1959:
1952:
1947:
1943:
1933:
1920:
1911:
1905:
1899:
1890:
1879:
1876:
1874:
1871:are complete
1855:
1852:
1849:
1820:
1814:
1791:
1785:
1756:
1753:
1748:
1744:
1740:
1737:
1734:
1726:
1723:
1718:
1714:
1710:
1707:
1704:
1696:
1693:
1684:
1681:
1675:
1671:
1667:
1661:
1658:
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1642:
1627:
1624:
1618:
1615:
1610:
1606:
1602:
1599:
1596:
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1585:
1579:
1575:
1571:
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1552:
1534:
1531:
1526:
1522:
1518:
1515:
1512:
1507:
1504:
1495:
1492:
1486:
1482:
1478:
1472:
1466:
1459:
1439:
1435:
1431:
1426:
1418:
1415:
1412:
1404:
1401:
1398:
1392:
1389:
1382:
1362:
1354:
1351:
1348:
1340:
1337:
1334:
1328:
1325:
1318:
1301:
1298:
1293:
1290:
1287:
1282:
1278:
1270:
1269:
1268:
1263:
1259:
1255:
1251:
1238:
1231:
1227:
1219:
1215:
1209:
1204:
1197:
1194:
1191:
1180:
1176:
1173:
1170:
1164:
1158:
1152:
1147:
1144:
1139:
1133:
1127:
1121:
1118:
1113:
1110:
1107:
1104:
1101:
1098:
1091:
1082:
1078:
1074:
1069:
1061:
1058:
1055:
1048:
1042:
1035:
1032:
1025:
1021:
1016:
1012:
1005:
1000:
996:
972:
969:
966:
963:
960:
949:
945:
941:
936:
923:
914:
906:
903:
898:
889:
878:
874:
871:
860:
858:
839:
830:
827:
822:
818:
811:
808:
805:
799:
795:
789:
772:
764:
763:
762:
759:
753:
734:
729:
725:
722:
716:
713:
708:
704:
701:
681:
675:
671:
667:
664:
657:
656:
655:
653:
649:
648:ferromagnetic
644:
639:
636: ≤
632:
625:
605:
600:
596:
593:
579:
573:
569:
565:
562:
555:
554:
553:
546:
540:
521:
516:
512:
509:
501:
497:
493:
490:
483:
482:
481:
467:
447:
439:
421:
417:
396:
388:
372:
349:
346:
343:
338:
334:
330:
327:
324:
317:
316:
315:
313:
305:
300:
296:
294:
289:
284:
281:
276:
272:
268:
267:line integral
264:
260:
255:
252:
248:
244:
240:
235:
233:
228:
221:
217:
213:
209:
205:
200:
186:
182:
180:
176:
172:
171:straight-line
167:
165:
161:
160:
155:
151:
147:
143:
141:
137:
133:
129:
125:
124:electromagnet
119:
86:
79:
59:
55:
51:
43:
37:
33:
19:
3980:
3929:. Retrieved
3914:
3907:
3886:cite journal
3871:
3838:
3834:
3828:
3807:cite journal
3792:
3759:
3755:
3749:
3716:
3712:
3706:
3697:
3687:
3678:
3668:
3625:
3621:
3611:
3570:
3566:
3560:
3551:
3541:
3508:
3504:
3498:
3472:. Retrieved
3452:
3426:. Retrieved
3406:
3397:
3389:
3385:
3373:
3367:
3362:
3352:
3340:. Retrieved
3331:
3322:
3280:
3276:
3265:
3261:
3254:
3250:permeability
3247:
3240:
3237:
3173:
3123:
2950:
2948:
2871:
2856:
2847:
2844:
2722:
2718:
2690:
2364:
1934:
1880:
1877:
1777:
1266:
947:
943:
937:
872:
866:
856:
854:
760:
751:
749:
645:
637:
630:
623:
620:
547:
539:permeability
536:
364:
309:
304:Ampère's law
287:
285:
279:
270:
263:Ampère's law
258:
256:
246:
242:
236:
229:
225:
219:
215:
211:
207:
183:
168:
163:
162:which means
157:
153:
149:
144:
126:formed by a
84:
82:
78:iron filings
3931:7 September
3573:: 423–432.
3382:figs. 29–33
946:and length
159:σωληνοειδὴς
58:field lines
3987:Categories
3378:pp. 267ff.
3342:16 January
3314:References
3126:inductance
2866:See also:
2862:Inductance
227:material.
179:arc spring
3998:Actuators
3951:ignored (
3941:cite book
3863:229274298
3855:0018-9375
3784:182038533
3603:126037802
3595:0304-8853
3533:123686159
3525:1432-0487
3192:μ
3150:Φ
3082:μ
3075:Φ
2910:μ
2753:μ
2746:≈
2702:≫
2651:μ
2621:≫
2605:−
2470:−
2430:−
2390:μ
2340:ζ
2332:−
2328:ζ
2298:Π
2292:ρ
2281:ρ
2278:−
2240:ζ
2223:ρ
2210:ζ
2186:π
2169:μ
2122:ζ
2114:−
2110:ζ
2057:−
2033:ζ
2016:ρ
1993:ρ
1977:π
1960:μ
1948:ρ
1915:→
1906:×
1903:∇
1894:→
1844:Π
1757:θ
1754:
1738:−
1727:θ
1724:
1708:−
1697:θ
1682:π
1672:∫
1650:Π
1628:θ
1619:θ
1616:
1600:−
1586:π
1576:∫
1535:θ
1532:
1516:−
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1483:∫
1436:ζ
1419:ρ
1405:ρ
1355:ρ
1341:ρ
1294:±
1283:±
1279:ζ
1228:ζ
1220:−
1216:ζ
1186:Π
1174:−
1140:−
1108:−
1079:ζ
1062:ρ
1049:ζ
1026:π
1013:μ
1001:ϕ
967:ϕ
961:ρ
918:^
915:ϕ
893:→
828:−
819:μ
796:μ
773:μ
717:μ
682:μ
672:μ
580:μ
570:μ
498:μ
418:μ
335:μ
310:Applying
206:labelled
154:solénoïde
18:Solenoids
3741:44197357
3660:17105396
3652:17556549
3484:cite web
3465:Archived
3438:cite web
3428:28 March
3419:Archived
3336:Archived
3308:Inductor
3297:See also
3275:, where
150:solenoid
130:coil of
85:solenoid
3764:Bibcode
3721:Bibcode
3700:: 1720.
3630:Bibcode
3622:Science
3575:Bibcode
3474:10 July
2956:is the
1878:Using:
1267:Where:
436:is the
385:is the
189:Physics
164:tubular
128:helical
3922:
3861:
3853:
3782:
3739:
3681:: 1–2.
3658:
3650:
3601:
3593:
3531:
3523:
2949:where
1836:, and
1778:Here,
855:where
750:where
365:where
214:, and
3859:S2CID
3780:S2CID
3737:S2CID
3656:S2CID
3599:S2CID
3529:S2CID
3468:(PDF)
3461:(PDF)
3422:(PDF)
3415:(PDF)
3380:(and
3260:with
875:; in
3953:help
3933:2009
3920:ISBN
3899:help
3851:ISSN
3820:help
3648:PMID
3591:ISSN
3521:ISSN
3490:link
3476:2021
3444:link
3430:2013
3344:2017
652:iron
302:How
230:The
132:wire
3843:doi
3772:doi
3729:doi
3638:doi
3626:317
3583:doi
3571:456
3513:doi
3264:or
1745:sin
1715:sin
1607:sin
1523:sin
988:is
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634:eff
627:eff
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3432:.
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3141:=
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