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[[Image:Hhcoil.jpg|thumb|A Helmholtz coil]]
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.
[[Image:Helmholtz coils.png|thumb|255px|Helmholtz coil schematic drawing]]
A '''Helmholtz coil''' is a device for producing a region of nearly uniform [[magnetic field]]. It is named in honor of the German physicist [[Hermann von Helmholtz]].


== Description ==
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]
A Helmholtz pair consists of two identical circular [[coil|magnetic
* Only registered users will be able to execute this rendering mode.
coils]] that are placed symmetrically one on each side of the
* Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.
experimental area along a common axis, and separated by a distance
<math>h</math> equal to the radius <math>R</math> of the coil. Each coil carries an equal
[[electrical current]] flowing in the same direction.


Setting <math>h=R</math>, which is what defines a Helmholtz pair, minimizes the nonuniformity of the field at the center of the coils, in the sense of setting <math>\partial^{2}B/\partial x^{2} = 0</math><ref>[http://www.purcellsolutions.com/2011/06/purcell-physics-problem-6-13-solution.html Helmholtz Coil in CGS units]</ref> (meaning that the first nonzero derivative is <math>\partial^{4}B/\partial x^{4}</math> as explained below), but leaves about 7% variation in field strength between the center and the planes of the coils.
Registered users will be able to choose between the following three rendering modes:  
A slightly larger value of <math>h</math> reduces the difference in field between the center and the planes of the coils, at the expense of worsening the field’s uniformity in the region near the center, as measured by <math>\partial^{2}B/\partial x^{2}</math>.<ref>[http://www.lightandmatter.com/html_books/0sn/ch11/ch11.html Electromagnetism<!-- Bot generated title -->]</ref>


In some applications, a Helmholtz coil is used to cancel out the [[Earth's magnetic field]], producing a region with a magnetic field intensity much closer to zero.<ref>
'''MathML'''
[http://www.circuitcellar.com/library/print/0606/Wotiz191/5.htm "Earth Field Magnetometer: Helmholtz coil"] by Richard Wotiz 2004
:<math forcemathmode="mathml">E=mc^2</math>
</ref>


== Mathematics ==
<!--'''PNG''' (currently default in production)
[[Image:VFPt helmholtz coil thumb.svg|thumb|255px|Magnetic field lines in a plane
:<math forcemathmode="png">E=mc^2</math>
bisecting the current loops. Note the field is approximately uniform in between the coil pair. (In this picture the coils are placed one beside the other: the axis is horizontal)]]
[[Image:Helmholtz zfield.png|thumb|255px|Magnetic field induction along the axis crossing the center of coils; ''z''&nbsp;=&nbsp;0 is the point in the middle of distance between coils.]]
[[Image: B mag.helmholtz.contour.png|thumb|255px|Contours showing the magnitude of the magnetic field
near the coil pair. Inside the central 'octopus' the field is within
1% of its central value ''B''<sub>0</sub>. The five contours are for
field magnitudes of <math>0.5 B_0</math>, <math>0.8 B_0</math>, <math>0.9 B_0</math>, <math>0.95 B_0</math>, and <math>0.99B_0</math> .]]
The calculation of the exact magnetic field at any point in space is mathematically complex and involves the study of [[Bessel function]]s. Things are simpler along the axis of the coil-pair, and it is convenient to think about the [[Taylor series]] expansion of the field strength as a function of
<math>x</math>, the distance from the central point of the coil-pair along the axis.
By symmetry the odd order terms in the expansion are zero. By separating the coils so that charge <math>x=0</math> is an [[inflection point]] for each coil separately we can guarantee that
the order <math>x^2</math> term is also zero, and hence the leading non-uniform term is of order <math>x^4</math>. One can easily show that the inflection point for a simple coil is <math>R/2</math>
from the coil center along the axis; hence the location of each coil at <math>x=\pm R/2</math>


A simple calculation gives the correct value of the field at the center point. If the radius is ''R'', the number of turns in each coil is ''n'' and the current flowing through the coils is ''I'', then the magnetic flux density, B at the midpoint between the coils will be given by
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


:<math> B = {\left ( \frac{4}{5} \right )}^{3/2} \frac{\mu_0 n I}{R}</math>
<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>\mu_0</math> is the [[permeability of free space]] (<math>1.26 \times 10^{-6} \text{ T}\cdot\text{m/A}</math>).
==Demos==


=== Derivation ===
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


Start with the formula for the on-axis field due to a single wire loop [http://hyperphysics.phy-astr.gsu.edu/HBASE/magnetic/curloo.html#c3] (which is itself derived from the [[Biot-Savart law]]):


:<math> B = \frac{\mu_0 I R^2}{2(R^2+x^2)^{3/2}}</math>
* 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)]
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** 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.


::Where:
==Test pages ==


:<math>\mu_0\;</math> = the [[permeability constant]] = <math> 4\pi \times 10^{-7} \text{ T}\cdot\text{m/A} = 1.257 \times 10^{-6} \text{ T}\cdot\text{m/A}</math>
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]


:<math>I\;</math> = coil current, in [[ampere]]s
*[[Inputtypes|Inputtypes (private Wikis only)]]
:<math>R\;</math> = coil radius, in meters
*[[Url2Image|Url2Image (private Wikis only)]]
:<math>x\;</math> = coil distance, on axis, to point, in meters
==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 .
However the coil consists of a number of wire loops, the total current in the coil is given by
 
:<math>nI\;</math> = total current
 
::Where:
 
:<math>n\;</math> = number of wire loops in one coil
 
Adding this to the formula:
 
:<math> B = \frac{\mu_0 n I R^2}{2(R^2+x^2)^{3/2}}</math>
 
In a Helmholtz coil, a point halfway between the two loops has an x value equal to R/2, so let's perform that substitution:
 
:<math> B = \frac{\mu_0 n I R^2}{2(R^2+(R/2)^2)^{3/2}}</math>
 
There are also two coils instead of one, so let's multiply the formula by 2, then simplify the formula:
 
:<math> B = \frac{2\mu_0 n I R^2}{2(R^2+(R/2)^2)^{3/2}}</math>
 
:<math> B = {\left ( \frac{4}{5} \right )}^{3/2} \frac{\mu_0 n I}{R}</math>
 
==Maxwell coils==
To improve the uniformity of the field in the space inside the coils, additional coils can be added around the outside.  [[James Clerk Maxwell]] showed in 1873 that a third larger-diameter coil located midway between the two Helmholtz coils can reduce the variance of the field on the axis to zero up to the sixth derivative of position.  This is sometimes called a [[Maxwell coil]].
 
== See also ==
 
* [[Maxwell coil]]
* [[Solenoid]]
* [[Halbach array]]
 
== References ==
 
<references/>
 
== External links ==
{{Commons category|Helmholtz coils}}
* [http://www.netdenizen.com/emagnet/helmholtz/idealhelmholtz.htm On-Axis Field of an Ideal Helmholtz Coil]
* [http://www.netdenizen.com/emagnet/helmholtz/realhelmholtz.htm Axial field of a real Helmholtz coil pair]
* ''[http://demonstrations.wolfram.com/HelmholtzCoilFields/ Helmholtz-Coil Fields]'' by Franz Kraft, [[The Wolfram Demonstrations Project]].
* Kevin Kuns (2007) [http://plasmalab.pbwiki.com/f/bfield.pdf Calculation of Magnetic Field inside Plasma Chamber], uses [[elliptic integral]]s and their [[derivative]]s to compute off-axis fields, from [[PBworks]].
 
[[Category:Electromagnetic coils]]
[[Category:Magnetic devices]]
 
[[de:Helmholtz-Spule]]
[[fr:Bobines d'Helmholtz]]
[[he:סליל הלמהולץ]]
[[pl:Cewka Helmholtza]]
[[ro:Bobină Helmholtz]]
[[ru:Кольца Гельмгольца]]
[[uk:Котушка Гельмгольца]]
[[vi:Cuộn Helmholtz]]
[[zh:亥姆霍茲線圈]]

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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