Editing Solenoid (section)

=== 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 [[current density]] ''K''; in [[cylindrical coordinate]]s:
<math display="block">\vec{K} = \frac{I}{l} \hat{\phi} .</math>

The magnetic field can be found using the [[vector potential]], which for a finite solenoid with radius ''R'' and length ''l'' in cylindrical coordinates <math>(\rho, \phi, z)</math> is<ref>{{cite web |url=http://nukephysik101.files.wordpress.com/2011/07/finite-length-solenoid-potential-and-field.pdf |title=Archived copy |access-date=28 March 2013 |url-status=live |archive-url=https://web.archive.org/web/20140410113804/http://nukephysik101.files.wordpress.com/2011/07/finite-length-solenoid-potential-and-field.pdf |archive-date=10 April 2014  }}</ref><ref>{{cite web |url=http://nukephysik101.files.wordpress.com/2021/07/finite-length-solenoid-potential-and-field-1.pdf |title=Archived copy |access-date=10 July 2021 |url-status=live |archive-url=https://web.archive.org/web/20210719191335/https://nukephysik101.files.wordpress.com/2021/07/finite-length-solenoid-potential-and-field-1.pdf |archive-date=19 July 2021  }}</ref>
<math display="block">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_+},</math>

[[file:Finite Length Solenoid field radius 1 length 1.jpg|upright=1.5|thumb|[[Magnetic field]] lines and density created by a solenoid with surface [[current density]]]]

Where:

* <math>\zeta_{\pm}=z\pm \frac{l}{2}</math>,
* <math>n = \frac{4R\rho}{(R+\rho)^2}</math>,
* <math>m = \frac{4R\rho}{(R+\rho)^2+\zeta^2}</math>,
* <math>K(m)=\int_0^{\frac\pi 2}\frac{d\theta}{\sqrt{1-m \sin^2 \theta }}</math>,
* <math>E(m)=\int_0^{\frac\pi 2}{\sqrt{1-m \sin^2 \theta} } \,d\theta</math> ,
* <math>\Pi(n,m)=\int_0^{\frac\pi 2}\frac{d\theta}{(1-n \sin^2 \theta)\sqrt{1-m \sin^2 \theta }}</math> .

Here, <math>K(m)</math>, <math>E(m)</math>, and <math>\Pi(n,m)</math> are complete [[elliptic integral]]s of the first, second, and third kind.

Using:

<math display="block">\vec{B} = \nabla \times \vec{A},</math>

The magnetic flux density is obtained as<ref>{{cite journal |first1=Karl Friedrich |last1=Müller |title=Berechnung der Induktivität von Spulen |language=de |trans-title=Calculating the Inductance of Coils |journal=Archiv für Elektrotechnik |volume=17 |issue=3 |date=1 May 1926 |pages=336–353 |issn=1432-0487 |doi=10.1007/BF01655986 |s2cid=123686159 }}</ref><ref>{{cite journal |first1=Edmund E. |last1=Callaghan |first2=Stephen H. |last2=Maslen |title=The magnetic field of a finite solenoid |language=en |journal=NASA Technical Reports |volume=NASA-TN-D-465 |issue=E-900 |date=1 October 1960 |url=https://ntrs.nasa.gov/search.jsp?R=19980227402}}</ref><ref name="CaciagliBaars2018">{{cite journal |last1=Caciagli|first1=Alessio |last2=Baars|first2=Roel J. |last3=Philipse|first3=Albert P. |last4=Kuipers|first4=Bonny W.M. |title=Exact expression for the magnetic field of a finite cylinder with arbitrary uniform magnetization |journal=Journal of Magnetism and Magnetic Materials | volume=456 | year=2018|pages=423–432 | issn=0304-8853 | doi=10.1016/j.jmmm.2018.02.003 | bibcode=2018JMMM..456..423C |hdl=1874/363313 | s2cid=126037802|hdl-access=free }}</ref>
<math display="block">B_\rho = \frac{\mu_0 I}{4\pi} \frac{1}{l\,\rho} \left[\sqrt{(R+\rho)^2+\zeta^2} \biggl( (m-2)K(m) + 2 E(m)\biggr) \right]_{\zeta_-}^{\zeta_+},</math>
<math display="block">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_+}.</math>

On the symmetry axis, the radial component vanishes, and the axial field component is
<math display="block">B_z = \frac{\mu_0 NI}{2}\left( \frac{z+l/2}{l \sqrt{R^2+(z+l/2)^2}} - \frac{z-l/2}{l \sqrt{R^2+(z-l/2)^2}}\right).</math>
Inside the solenoid, far away from the ends (<math>l/2 - |z| \gg R</math>), this tends towards the constant value <math>B = \mu_0 N I/l</math>.