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[[Image:Taylor cone photo.jpg|thumb|right|250 px|Photograph of a meniscus of polyvinyl alcohol in aqueous solution showing a fibre drawn from a Taylor cone by the process of electrospinning.]]
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.
A '''Taylor cone''' refers to the cone observed in [[electrospinning]], [[electrospray]]ing and hydrodynamic spray processes from which a jet of charged particles emanates above a threshold voltage. Aside from [[electrospray ionization]] in [[mass spectrometry]] the Taylor cone is important in [[FEEP]] and [[colloid thruster]]s used in fine control and high efficiency (low power) thrust of spacecraft.


== History ==
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This cone was described by Sir [[Geoffrey Ingram Taylor]] in 1964 before electrospray was "discovered".<ref>{{cite journal | author = Sir Geoffrey Taylor | year = 1964 | title = Disintegration of Water Droplets in an Electric Field | journal = [[Proceedings_of_the_Royal_Society_A#Proceedings_of_the_Royal_Society_A|Proc. Roy. Soc. London. Ser. A]] | volume = 280 | pages = 383  | issue = 1382 | doi = 10.1098/rspa.1964.0151 | jstor=2415876|bibcode = 1964RSPSA.280..383T }}</ref> This work followed on the work of Zeleny<ref>{{cite journal| author=Zeleny, J. |title = The Electrical Discharge from Liquid Points, and a Hydrostatic Method of Measuring the Electric Intensity at Their Surfaces. | journal = [[Physical Review]] | year = 1914 | volume = 3| issue=2 | pages = 69 | doi = 10.1103/PhysRev.3.69|bibcode = 1914PhRv....3...69Z }}</ref> who, in 1917, photographed a cone-jet of glycerine under high electric field and the work of several others:  Wilson and Taylor (1925),<ref> {{cite journal|title=The bursting of soap bubbles in a uniform electric field|journal=Proc. Cambridge Philos. Soc.|year=1925|first=C. T.|last=Wilson|coauthors= G. I Taylor|volume=22|issue=05|pages=728|id= |url=|format=|accessdate=|doi=10.1017/S0305004100009609 |bibcode = 1925PCPS...22..728W }}</ref> Nolan (1926)<ref> {{cite journal|title= |journal=Proc. R. Ir. Acad. Sect. A|year=1926|first=J. J.|last=Nolan|coauthors=|volume=37|issue=|pages=28|id= |url=|format=|accessdate= }}</ref> and Macky (1931).<ref name='rspa.1931.0168'> {{cite journal|title=Some Investigations on the Deformation and Breaking of Water Drops in Strong Electric Fields|journal=Proceedings of the Royal Society of London. Series A|date=October 1, 1931|first=W. A.|last=Macky|coauthors=|volume=133|issue=822|pages=565–587|doi= 10.1098/rspa.1931.0168|bibcode = 1931RSPSA.133..565M }}|url=http://www.journals.royalsoc.ac.uk/content/c6188343042555vw/fulltext.pdf|format=PDF|accessdate=2007-10-25</ref> Taylor was primarily interested in the behavior of water droplets in strong electric fields, such as in thunderstorms.
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== Taylor cone formation ==
'''MathML'''
[[Image:Taylor cone.jpg|thumb|right|300 px|Electrospray diagram depicting the Taylor cone, jet and plume]]
:<math forcemathmode="mathml">E=mc^2</math>
When a small volume of electrically conductive liquid is exposed to an electric field, the shape of liquid starts to deform from the shape caused by surface tension alone. As the voltage is increased the effect of the electric field becomes more prominent and as it approaches exerting a similar amount of force on the droplet as the surface tension does a cone shape begins to form with convex sides and a rounded tip. This approaches the shape of a [[Cone (geometry)|cone]] with a whole angle (width) of 98.6°<ref>{{cite journal | author = Sir Geoffrey Taylor | year = 1964 | title = Disintegration of Water Droplets in an Electric Field | journal = [[Proceedings_of_the_Royal_Society_A#Proceedings_of_the_Royal_Society_A|Proc. Roy. Soc. London. Ser. A]] | volume = 280 | pages = 392  | issue = 1382 | doi = 10.1098/rspa.1964.0151 | jstor=2415876|bibcode = 1964RSPSA.280..383T }}</ref>. When a certain threshold voltage has been reached the slightly rounded tip inverts and emits a jet of liquid. This is called a cone-jet and is the beginning of the [[electrospray]]ing process in which ions may be transferred to the gas phase. It is generally found that in order to achieve a stable cone-jet a slightly higher than threshold voltage must be used. As the voltage is increased even more other modes of droplet disintegration are found. The term Taylor cone can specifically refer to the theoretical limit of a perfect cone of exactly  the predicted angle or generally refer to the approximately conical portion of a cone-jet after the electrospraying process has begun.


== Theory ==
<!--'''PNG'''  (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


Sir [[Geoffrey Ingram Taylor]] in 1964 described this phenomenon, theoretically derived based on general assumptions that the requirements to form a perfect cone under such conditions required a semi-vertical angle of 49.3° (a whole angle of 98.6°) and demonstrated that the shape of such a cone approached the theoretical shape just before jet formation.  This angle is known as the '''Taylor angle'''.  This angle is more precisely <math>\pi-\theta _0\,</math> where <math>\theta _0\,</math> is the first zero of <math>P _{1/2} (\cos\theta _0)\,</math> (the [[Legendre polynomial]] of order 1/2).
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


Taylor's derivation is based on two assumptions: (1) that the surface of the cone is an equipotential surface and (2) that the cone exists in a steady state equilibrium. To meet both of these criteria the electric field must have [[azimuth]]al symmetry and have <math>\sqrt{R}\,</math> dependence to counter the surface tension to produce the cone. The solution to this problem is:
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].


:<math>V=V_0+AR^{1/2}P _{1/2} (\cos\theta _0)\,</math>
==Demos==


where <math>V=V_0\,</math> (equipotential surface) exists at a value of <math>\theta _0</math> (regardless of R) producing an equipotential cone. The angle necessary for <math>V=V_0\,</math> for all R is a zero of <math>P _{1/2} (\cos\theta _0)\,</math> between 0 and <math>\pi\,</math> which there is only one at 130.7099°.  The complement of this angle is the Taylor angle.
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


== References ==
{{reflist}}


==External links==
* accessibility:
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


[[Category:Mass spectrometry]]
==Test pages ==


[[de:Taylor-Kegel]]
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
[[it:Cono di Taylor]]
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==Bug reporting==
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .

Latest revision as of 23:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

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