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[[Image:Hhcoil.jpg|thumb|A Helmholtz coil]]
{{Original research|date=June 2010}}
[[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 ==
[[Image:Hyperbolic coordinates.svg|thumb|400px|right|Hyperbolic coordinates plotted on the Cartesian plane: ''u'' in blue and ''v'' in red.]]
A Helmholtz pair consists of two identical circular [[coil|magnetic
coils]] that are placed symmetrically one on each side of the
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.
In [[mathematics]], '''hyperbolic coordinates''' are a method of locating points in Quadrant I of the [[Cartesian plane]]{{Why?|date=May 2010}}
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>
:<math>\{(x, y) \ :\  x > 0,\ y > 0\ \} = Q\ \!</math >.
[http://www.circuitcellar.com/library/print/0606/Wotiz191/5.htm "Earth Field Magnetometer: Helmholtz coil"] by Richard Wotiz 2004
</ref>


== Mathematics ==
Hyperbolic coordinates take values in the hyperbolic plane defined as:
[[Image:VFPt helmholtz coil thumb.svg|thumb|255px|Magnetic field lines in a plane
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
:<math>HP = \{(u, v) : u \in \mathbb{R}, v > 0 \}</math>.


:<math> B = {\left ( \frac{4}{5} \right )}^{3/2} \frac{\mu_0 n I}{R}</math>
These coordinates in ''HP'' are useful for studying logarithmic comparisons of [[direct proportion]] in ''Q'' and measuring deviations from direct proportion.


<math>\mu_0</math> is the [[permeability of free space]] (<math>1.26 \times 10^{-6} \text{ T}\cdot\text{m/A}</math>).
For <math>(x,y)</math> in <math>Q</math> take


=== Derivation ===
:<math>u = -\frac{1}{2} \ln \left( \frac{y}{x} \right)</math>


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]]):
and


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


::Where:
Sometimes the parameter <math>u</math> is called [[hyperbolic angle]] and v the [[geometric mean]].


:<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>
The inverse mapping is


:<math>I\;</math> = coil current, in [[ampere]]s
:<math>x = v e^u ,\quad y = v e^{-u}</math>.
:<math>R\;</math> = coil radius, in meters
:<math>x\;</math> = coil distance, on axis, to point, in meters


However the coil consists of a number of wire loops, the total current in the coil is given by
This is a [[continuous mapping]], but not an [[analytic function]].


:<math>nI\;</math> = total current
==Quadrant model of hyperbolic geometry==


::Where:
The correspondence


:<math>n\;</math> = number of wire loops in one coil
:<math>Q \leftrightarrow HP</math>


Adding this to the formula:
affords the [[hyperbolic geometry]] structure to ''Q'' that is erected on ''HP'' by [[hyperbolic motion]]s. The ''hyperbolic lines'' in ''Q'' are [[Line (mathematics)#Ray|rays]] from the origin or [[petal]]-shaped [[curve]]s leaving and re-entering the origin. The left-right shift in ''HP'' corresponds to a [[squeeze mapping]] applied to ''Q''. Note that hyperbolas in ''Q'' do ''not'' represent [[geodesic]]s in this model.


:<math> B = \frac{\mu_0 n I R^2}{2(R^2+x^2)^{3/2}}</math>
If one only considers the [[Euclidean topology]] of the plane and the topology inherited by ''Q'', then the lines bounding ''Q'' seem close to ''Q''. Insight from the [[metric space]] ''HP'' shows that the [[open set]] ''Q'' has only the [[origin (mathematics)|origin]] as boundary when viewed as the quadrant model of the hyperbolic plane. Indeed, consider rays from the origin in ''Q'', and their images, vertical rays from the boundary ''R'' of ''HP''. Any point in ''HP'' is an infinite distance from the point ''p'' at the foot of the perpendicular to ''R'', but a sequence of points on this perpendicular may tend in the direction of ''p''. The corresponding sequence in ''Q'' tends along a ray toward the origin. The old Euclidean boundary of ''Q'' is irrelevant to the quadrant model.


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:
==Applications in physical science==
Physical unit relations like:
* ''V'' = ''I R''  : [[Ohm's law]]
* ''P'' = ''V I''  : [[Electrical power]]
* ''P V'' = ''k T''  :  [[Ideal gas law]]
* ''f'' λ = ''c'' : [[Sine wave]]s
all suggest looking carefully at the quadrant. For example, in [[thermodynamics]] the [[isothermal process]] explicitly follows the hyperbolic path and [[work (thermodynamics)|work]] can be interpreted as a hyperbolic angle change. Similarly, an [[isobaric process#Variable density viewpoint|isobaric process]] may trace a hyperbola in the quadrant of absolute temperature and gas density.


:<math> B = \frac{\mu_0 n I R^2}{2(R^2+(R/2)^2)^{3/2}}</math>
For hyperbolic coordinates in the [[Theory of relativity]] see the History section below.


There are also two coils instead of one, so let's multiply the formula by 2, then simplify the formula:
==Statistical applications==
*Comparative study of [[population density]] in the quadrant begins with selecting a reference nation, region, or [[urban density|urban]] area whose population and area are taken as  the point (1,1).
*Analysis of the [[legislator|elected representation]] of regions in a [[representative democracy]] begins with selection of a standard for comparison: a particular represented group, whose magnitude and slate magnitude (of representatives) stands at (1,1) in the quadrant.


:<math> B = \frac{2\mu_0 n I R^2}{2(R^2+(R/2)^2)^{3/2}}</math>
==Economic applications==
There are many natural applications of hyperbolic coordinates in [[economics]]:
* Analysis of currency [[exchange rate]] fluctuation:
The unit currency sets <math>x = 1</math>. The price currency corresponds to <math>y</math>. For


:<math> B = {\left ( \frac{4}{5} \right )}^{3/2} \frac{\mu_0 n I}{R}</math>
:<math>0 < y < 1</math>


==Maxwell coils==
we find <math>u > 0</math>, a positive hyperbolic angle. For a ''fluctuation'' take a new price
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 ==
:<math>0 < z < y</math>.


* [[Maxwell coil]]
Then the change in ''u'' is:
* [[Solenoid]]
* [[Halbach array]]


== References ==
:<math>\Delta u = \frac{1}{2} \log \left( \frac{y}{z} \right)</math>.


Quantifying exchange rate fluctuation through hyperbolic angle provides an objective, symmetric, and consistent [[measure (mathematics)|measure]]. The quantity <math>\Delta u</math> is the length of the left-right shift in the hyperbolic motion view of the currency fluctuation.
* Analysis of inflation or deflation of prices of a [[basket of consumer goods]].
* Quantification of change in marketshare in [[duopoly]].
* Corporate [[stock split]]s versus stock buy-back.
==History==
While the geometric mean is an ancient concept, the hyperbolic angle is contemporary with the development of [[logarithm]], the latter part of the seventeenth century. [[Gregoire de Saint-Vincent]], [[Marin Mersenne]], and [[Alphonse Antonio de Sarasa]] evaluated the quadrature of the hyperbola as a function having properties now familiar for the logarithm. The exponential function, the hyperbolic sine, and the hyperbolic cosine followed. As [[complex function]] theory referred to [[infinite series]] the circular functions sine and cosine seemed to absorb the hyperbolic sine and cosine as depending on an imaginary variable. In the nineteenth century [[biquaternion]]s came into use and exposed the alternative complex plane called [[split-complex number]]s where the hyperbolic angle is raised to a level equal to the classical angle. In English literature biquaternions were used to model [[spacetime]] and show its symmetries. There the hyperbolic angle parameter came to be called [[rapidity]]. For relativists, examining the quadrant as the possible future between oppositely directed photons, the geometric mean parameter is [[time|temporal]].
In relativity the focus is on the 3-dimensional [[hypersurface]] in the future of spacetime where various velocities arrive after a given [[proper time]]. Scott Walter<ref>Walter (1999) page 6</ref> explains that in November 1907 [[Herman Minkowski]] alluded to a well-known three-dimensional hyperbolic geometry while speaking to the Göttingen Mathematical Society, but not to a four-dimensional one.<ref>Walter (1999) page 8</ref>
In tribute to [[Wolfgang Rindler]], the author of the standard introductory university-level textbook on relativity, hyperbolic coordinates of spacetime are called [[Rindler coordinates]].
==References==
<references/>
<references/>
*David Betounes (2001) ''Differential Equations: Theory and Applications'', page 254, Springer-TELOS, ISBN 0-387-95140-7 .
*Scott Walter (1999). [http://www.univ-nancy2.fr/DepPhilo/walter/papers/nes.pdf "The non-Euclidean style of Minkowskian relativity"]. Chapter 4 in: Jeremy J. Gray (ed.), ''The Symbolic Universe: Geometry and Physics 1890-1930'', pp.&nbsp;91–127. [[Oxford University Press]]. ISBN 0-19-850088-2.


== External links ==
{{Orthogonal coordinate systems}}
{{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:Coordinate systems]]
[[Category:Magnetic devices]]
[[Category:Hyperbolic geometry]]


[[de:Helmholtz-Spule]]
[[ar:نظام إحداثيات قطعي زائدي]]
[[fr:Bobines d'Helmholtz]]
[[pt:Coordenadas hiperbólicas]]
[[he:סליל הלמהולץ]]
[[zh:雙曲坐標系]]
[[pl:Cewka Helmholtza]]
[[ro:Bobină Helmholtz]]
[[ru:Кольца Гельмгольца]]
[[uk:Котушка Гельмгольца]]
[[vi:Cuộn Helmholtz]]
[[zh:亥姆霍茲線圈]]

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Hyperbolic coordinates plotted on the Cartesian plane: u in blue and v in red.

In mathematics, hyperbolic coordinates are a method of locating points in Quadrant I of the Cartesian planeTemplate:Why?

.

Hyperbolic coordinates take values in the hyperbolic plane defined as:

.

These coordinates in HP are useful for studying logarithmic comparisons of direct proportion in Q and measuring deviations from direct proportion.

For in take

and

.

Sometimes the parameter is called hyperbolic angle and v the geometric mean.

The inverse mapping is

.

This is a continuous mapping, but not an analytic function.

Quadrant model of hyperbolic geometry

The correspondence

affords the hyperbolic geometry structure to Q that is erected on HP by hyperbolic motions. The hyperbolic lines in Q are rays from the origin or petal-shaped curves leaving and re-entering the origin. The left-right shift in HP corresponds to a squeeze mapping applied to Q. Note that hyperbolas in Q do not represent geodesics in this model.

If one only considers the Euclidean topology of the plane and the topology inherited by Q, then the lines bounding Q seem close to Q. Insight from the metric space HP shows that the open set Q has only the origin as boundary when viewed as the quadrant model of the hyperbolic plane. Indeed, consider rays from the origin in Q, and their images, vertical rays from the boundary R of HP. Any point in HP is an infinite distance from the point p at the foot of the perpendicular to R, but a sequence of points on this perpendicular may tend in the direction of p. The corresponding sequence in Q tends along a ray toward the origin. The old Euclidean boundary of Q is irrelevant to the quadrant model.

Applications in physical science

Physical unit relations like:

all suggest looking carefully at the quadrant. For example, in thermodynamics the isothermal process explicitly follows the hyperbolic path and work can be interpreted as a hyperbolic angle change. Similarly, an isobaric process may trace a hyperbola in the quadrant of absolute temperature and gas density.

For hyperbolic coordinates in the Theory of relativity see the History section below.

Statistical applications

  • Comparative study of population density in the quadrant begins with selecting a reference nation, region, or urban area whose population and area are taken as the point (1,1).
  • Analysis of the elected representation of regions in a representative democracy begins with selection of a standard for comparison: a particular represented group, whose magnitude and slate magnitude (of representatives) stands at (1,1) in the quadrant.

Economic applications

There are many natural applications of hyperbolic coordinates in economics:

The unit currency sets . The price currency corresponds to . For

we find , a positive hyperbolic angle. For a fluctuation take a new price

.

Then the change in u is:

.

Quantifying exchange rate fluctuation through hyperbolic angle provides an objective, symmetric, and consistent measure. The quantity is the length of the left-right shift in the hyperbolic motion view of the currency fluctuation.

History

While the geometric mean is an ancient concept, the hyperbolic angle is contemporary with the development of logarithm, the latter part of the seventeenth century. Gregoire de Saint-Vincent, Marin Mersenne, and Alphonse Antonio de Sarasa evaluated the quadrature of the hyperbola as a function having properties now familiar for the logarithm. The exponential function, the hyperbolic sine, and the hyperbolic cosine followed. As complex function theory referred to infinite series the circular functions sine and cosine seemed to absorb the hyperbolic sine and cosine as depending on an imaginary variable. In the nineteenth century biquaternions came into use and exposed the alternative complex plane called split-complex numbers where the hyperbolic angle is raised to a level equal to the classical angle. In English literature biquaternions were used to model spacetime and show its symmetries. There the hyperbolic angle parameter came to be called rapidity. For relativists, examining the quadrant as the possible future between oppositely directed photons, the geometric mean parameter is temporal.

In relativity the focus is on the 3-dimensional hypersurface in the future of spacetime where various velocities arrive after a given proper time. Scott Walter[1] explains that in November 1907 Herman Minkowski alluded to a well-known three-dimensional hyperbolic geometry while speaking to the Göttingen Mathematical Society, but not to a four-dimensional one.[2] In tribute to Wolfgang Rindler, the author of the standard introductory university-level textbook on relativity, hyperbolic coordinates of spacetime are called Rindler coordinates.

References

  1. ↑ Walter (1999) page 6
  2. ↑ Walter (1999) page 8

Template:Orthogonal coordinate systems

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