Helmholtz coil

The term Helmholtz coils refers to a device for producing a region of nearly uniform magnetic field. It is named in honor of the German physicist Hermann 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 of Bessel functions. 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 $x$, 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 an inflection 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 the Biot-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 amperes:$R;$ = coil radius, in meters:$x;$ = 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

:$nI;$ = 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.]