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| [[Image:Hhcoil.jpg|thumb|A Helmholtz coil]]
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| [[Image:Helmholtz coils modified.jpg|thumb|255px|Helmholtz coil schematic drawing]]
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| A '''Helmholtz coil''' is a device for producing a region of nearly uniform [[magnetic field]], named after the German physicist [[Hermann von Helmholtz]]. It consists of two [[solenoid]] [[electromagnet]]s on the same axis. Besides creating magnetic fields, Helmholtz coils are also used in scientific apparatus to cancel external magnetic fields, such as the Earth's magnetic field.
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| == Description ==
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| A Helmholtz pair consists of two identical circular [[coil|magnetic coils]] ([[solenoid]]s) that are placed symmetrically along a common axis, one on each side of the experimental area, 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.
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| 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.
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| 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>
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| 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>
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| [http://www.circuitcellar.com/library/print/0606/Wotiz191/5.htm "Earth Field Magnetometer: Helmholtz coil"] by Richard Wotiz 2004
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| </ref>
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| == Mathematics ==
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| [[Image:VFPt helmholtz coil thumb.svg|thumb|255px|Magnetic field lines in a plane
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| 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)]]
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| [[Image:Helmholtz zfield.png|thumb|255px|Magnetic field induction along the axis crossing the center of coils; ''z'' = 0 is the point in the middle of distance between coils.]]
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| [[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
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| 1% of its central value ''B''<sub>0</sub>. The eight contours are for field magnitudes of 0.5 ''B''<sub>0</sub>, 0.8 ''B''<sub>0</sub>, 0.9 ''B''<sub>0</sub>, 0.95 ''B''<sub>0</sub>, 0.99 ''B''<sub>0</sub>, 1.01 ''B''<sub>0</sub>, 1.05 ''B''<sub>0</sub>, and 1.1 ''B''<sub>0</sub>.]]
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| 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
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| <math>x</math>, the distance from the central point of the coil-pair along the axis.
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| 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
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| 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>
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| from the coil center along the axis; hence the location of each coil at <math>x=\pm R/2</math>
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| 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
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| :<math> B = {\left ( \frac{4}{5} \right )}^{3/2} \frac{\mu_0 n I}{R}</math>
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| <math>\mu_0</math> is the [[permeability of free space]] (<math>4\pi \times 10^{-7} \text{ T}\cdot\text{m/A}</math>).
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| === Derivation ===
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| Start with the formula for the on-axis field due to a single wire loop (which is itself derived from the [[Biot-Savart law]]):<ref>http://hyperphysics.phy-astr.gsu.edu/HBASE/magnetic/curloo.html#c3</ref>
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| :<math> B_1(x) = \frac{\mu_0 I R^2}{2(R^2+x^2)^{3/2}}</math>
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| ::Where:
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| :<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>
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| :<math>I\;</math> = coil current, in [[ampere]]s
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| :<math>R\;</math> = coil radius, in meters
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| :<math>x\;</math> = coil distance, on axis, to point, in meters
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| The Helmholtz coils consists of ''n'' turns of wire, so the equivalent current in a one-turn coil is ''n'' times the current ''I'' in the ''n''-turn coil. Substituting ''nI'' for ''I'' in the above formula gives the field for an ''n''-turn coil.
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| :<math> B_1(x) = \frac{\mu_0 n I R^2}{2(R^2+x^2)^{3/2}}</math>
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| In a Helmholtz coil, a point halfway between the two loops has an ''x'' value equal to R/2, so calculate the field strength at that point:
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| :<math> B_1\left(\frac{R}{2}\right) = \frac{\mu_0 n I R^2}{2(R^2+(R/2)^2)^{3/2}}</math>
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| There are also two coils instead of one (the coil above is at ''x''=0; there is a second coil at ''x''=''R''). From symmetry, the field strength at the midpoint will be twice the single coil value:
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| <!-- It would be better to use coils at -R/2 and +R/2 and give field expression for arb x -->
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| :<math>\begin{align}
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| B\left(\frac{R}{2}\right) &= 2 B_1(R/2) \\
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| &= \frac{2\mu_0 n I R^2}{2(R^2+(R/2)^2)^{3/2}}
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| = \frac{\mu_0 n I R^2}{(R^2+(R/2)^2)^{3/2}} \\
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| &= \frac{\mu_0 n I R^2}{(R^2+\frac{1}{4}R^2)^{3/2}}
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| = \frac{\mu_0 n I R^2}{(\frac{5}{4}R^2)^{3/2}} \\
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| &= {\left ( \frac{4}{5} \right )}^{3/2} \frac{\mu_0 n I}{R} \\
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| &= {\left ( \frac{8}{5\sqrt{5}} \right )} \frac{\mu_0 n I}{R} \\
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| \end{align}</math>
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| ==Maxwell coils==
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| [[Image:Helmholtz coils in free radical experiment NBS 1957.jpg|thumb|Helmholtz coils ''(hoops)'' on three perpendicular axes used to cancel the Earth's magnetic field inside the vacuum tank in a 1957 electron beam experiment]]
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| 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]].
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| == See also ==
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| * [[Maxwell coil]]
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| * [[Solenoid]]
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| * [[Halbach array]]
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| * A [[magnetic bottle]] has the same structure as Helmholtz coils, but with the magnets separated further apart so that the field expands in the middle, trapping charged particles with the diverging field lines. If one coil is reversed, it produces a [[cusp trap]], which also traps charged particles.<ref>http://radphys4.c.u-tokyo.ac.jp/asacusa/wiki/index.php?Cusp%20trap</ref>
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| == References ==
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| <references/> | |
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| == External links ==
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| {{Commons category|Helmholtz coils}}
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| * [http://www.netdenizen.com/emagnet/helmholtz/idealhelmholtz.htm On-Axis Field of an Ideal Helmholtz Coil]
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| * [http://www.netdenizen.com/emagnet/helmholtz/realhelmholtz.htm Axial field of a real Helmholtz coil pair]
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| * ''[http://demonstrations.wolfram.com/HelmholtzCoilFields/ Helmholtz-Coil Fields]'' by Franz Kraft, [[The Wolfram Demonstrations Project]].
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| * 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]].
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| *{{Citation |last1= DeTroye |first1=David J. |last2=Chase |first2=Ronald J. |date=November 1994 |title=The Calculation and Measurement of Helmholtz Coil Fields |publisher=Army Research Laboratory |id=ARL-TN-35 |url=http://www.dtic.mil/dtic/tr/fulltext/u2/a286081.pdf }}
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| * [http://course1.winona.edu/fotto/physics222/labs/Lab_10_HH_Coils.pdf Magnetic Fields of Coils]
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| * http://physicsx.pr.erau.edu/HelmholtzCoils/
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| [[Category:Electromagnetic coils]]
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| [[Category:Magnetic devices]]
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Nice to satisfy you, I am Marvella Shryock. What I adore doing is to gather badges but I've been taking on new issues recently. Managing individuals has been his day occupation for a whilst. Years ago we moved to North Dakota.
Also visit my site :: healthy food delivery - linked webpage -