- Helmholtz coil
The term

**Helmholtz coils**refers to a device for producing a region of nearly uniformmagnetic field . It is named in honor of the German physicistHermann von Helmholtz .**Description**A Helmholtz pair consists of two identical circular magneticcoils that are placed symmetrically one on each side of theexperimental area along a common axis, and separated by a distance$h$ equal to the radius $R$ of the coil. Each coil carries an equal

electrical current flowing in the same direction.Setting $h=R$,which is what defines a Helmholtz pair, minimizesthe nonuniformity of the field at the center of the coils, in thesense of setting $d^2\; B/dx^2=0$, but leaves about 6% variation in fieldstrength between the center and the planes of the coils.A slightly larger value of $h$ reduces the difference in fieldbetween the center and the planes of the coils, at the expense of worseningthe field's uniformity in the region near the center, as measured by $d^2\; B/dx^2$. [

*[*]*http://www.lightandmatter.com/html_books/0sn/ch11/ch11.html Electromagnetism*]**Mathematics**

thumb|255px|Magnetic field vector in a planebisecting the current loops. Note the field is approximately uniform in between the coil pair. (In this picture the coils are placed one above the other: the axis is vertical)

thumb|255px|Contours showing the magnitude of the magnetic fieldnear the coil pair. Inside the central 'octopus' the field is within1% of its central value ">$B\_0$. The contours are forfield magnitudes of $.5\; B\_0$, $.8\; B\_0$, $.9\; B\_0$, $0.95\; B\_0$, $.99B\_0$, $1.01B\_0$, $1.05B\_0$, and $1.1B\_0$ .The calculation of the exact magnetic field at any point in space has mathematical complexities and involves the study ofBessel function s. Things are simpler along the axis of the coil-pair, and it is convenient to think about theTaylor series expansion of the field strength as a function of $x$, 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$x=0$ is aninflection point for each coil separately we can guarantee thatthe order $x^2$ term is also zero, and hence the leading non-uniform term is of order $x^4$. One can easily show that the inflection point for a simple coil is $R/2$from the coil center along the axis; hence the location of each coil at $x=pm\; R/2$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

:$B\; =\; \{left\; (\; frac\{4\}\{5\}\; ight\; )\}^\{3/2\}\; frac\{mu\_0\; n\; I\}\{R\}$

$mu\_0$ is the

permeability constant ($1.26\; imes\; 10^\{-6\}\; ext\{\; T\}cdot\; ext\{m/A\}$), and $R$ is in meters.**Derivation**Start with the formula for the on-axis field due to a single wire loop [

*http://www.netdenizen.com/emagnet/solenoids/ilooponaxis.htm*] (which is itself derived from theBiot-Savart law [*http://www.netdenizen.com/emagnet/thebasics/maxwelletc.htm#BiotSavart*] )::$B\; =\; frac\{mu\_0\; I\; R^2\}\{2(R^2+x^2)^\{3/2$

::Where:

:$mu\_0;$ = the

permeability constant = $4pi\; imes\; 10^\{-7\}\; ext\{\; T\}cdot\; ext\{m/A\}\; =\; 1.26\; imes\; 10^\{-6\}\; ext\{\; T\}cdot\; ext\{m/A\}$:$I;$ = coil current, in

ampere s:$R;$ = coil radius, in meters:$x;$ = coil distance, on axis, to point, in metersHowever the coil consists of a number of wire loops, the total current in the coil is given by

:$nI;$ = total current

::Where:

:$n;$ = number of wire loops in one coil

Adding this to the formula:

:$B\; =\; frac\{mu\_0\; n\; I\; R^2\}\{2(R^2+x^2)^\{3/2$

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:

:$B\; =\; frac\{mu\_0\; n\; I\; R^2\}\{2(R^2+(R/2)^2)^\{3/2$

There are also two coils instead of one, so let's multiply the formula by 2, then simplify the formula:

:$B\; =\; frac\{2mu\_0\; n\; I\; R^2\}\{2(R^2+(R/2)^2)^\{3/2$

:$B\; =\; \{left\; (\; frac\{4\}\{5\}\; ight\; )\}^\{3/2\}\; frac\{mu\_0\; n\; I\}\{R\}$

**See also***

Maxwell coil **References****External links*** [

*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 .

* [*http://plasmalab.pbwiki.com/f/bfield.pdf Excellent complete derivation for*]**OFF-AXIS**field for a single current loop. Includes reduction to on-axis field as derived from the Biot-Savart Law.**See expression on Page 8 in this paper. Uses elliptic integrals.**

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**helmholtz coil**— ˈhel]mˌhōlts, ˈheu̇] noun Usage: usually capitalized H Etymology: after Hermann L.F. von Helmholtz died 1894 German physicist : one of two equal parallel coaxial circular coils in series that are separated from each other by a distance equal to… … Useful english dictionary**Helmholtz coil**— noun one of a pair of flat coils, having equal numbers of turns and equal diameters, arranged on a common axis and connected in series; used to obtain a nearly uniform magnetic field without the use of a long solenoid … Wiktionary**Coil**— Winding redirects here. For other uses, see Winding (disambiguation). For other uses, see Coil (disambiguation). A coil is a series of loops. A coiled coil is a structure in which the coil itself is in turn also looping. Contents 1… … Wikipedia**Hermann von Helmholtz**— Infobox Scientist name = Hermann von Helmholtz |225px image width = 225px caption = Hermann Ludwig Ferdinand von Helmholtz birth date = birth date|1821|8|31|mf=y birth place = Potsdam, Kingdom of Prussia residence = Germany nationality = German… … Wikipedia**Electromagnetic coil**— An electromagnetic coil (or simply a coil ) is formed when a conductor (usually a solid copper wire) is wound around a core or form to create an inductor or electromagnet. One loop of wire is usually referred to as a turn , and a coil consists of … Wikipedia**Maxwell coil**— Magnetic field around a Maxwell coil A Maxwell coil is a device for producing a large volume of almost constant (or constant gradient) magnetic field. Description A constant field Maxwell coil set consists of three coils oriented on the surface… … Wikipedia**magnetism**— /mag ni tiz euhm/, n. 1. the properties of attraction possessed by magnets; the molecular properties common to magnets. 2. the agency producing magnetic phenomena. 3. the science dealing with magnetic phenomena. 4. strong attractive power or… … Universalium**Magnetic field**— This article is about a scientific description of the magnetic influence of an electric current or magnetic material. For the physics of magnetic materials, see magnetism. For information about objects that create magnetic fields, see magnet. For … Wikipedia**List of electronics topics**— Alphabetization has been neglected in some parts of this article (the b section in particular). You can help by editing it. This is a list of communications, computers, electronic circuits, fiberoptics, microelectronics, medical electronics,… … Wikipedia**List of physics topics F-L**— NOTOC A B C D E F G H I J K L M N O P Q R S T U V W X Y Z> F * Famous physicists; see list of physicistsFa* Faraday cage * Faraday constant * Faraday s law of electrolysis * Faraday s law of induction * Faraday, Michael * Fahrenheit * Fahrenheit … Wikipedia