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A Helmholtz coil is a device for producing a region of nearly uniform
magnetic field A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
, named after the German physicist
Hermann von Helmholtz Hermann Ludwig Ferdinand von Helmholtz (31 August 1821 – 8 September 1894) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The Helmholtz Association, ...
. It consists of two
electromagnet An electromagnet is a type of magnet in which the magnetic field is produced by an electric current. Electromagnets usually consist of wire wound into a coil. A current through the wire creates a magnetic field which is concentrated in the ...
s on the same axis, carrying an equal electric current in the same direction. Besides creating magnetic fields, Helmholtz coils are also used in scientific apparatus to cancel external magnetic fields, such as the Earth's magnetic field. When the pair of two electromagnetics of a Helmholtz coil carry an equal electric current in the opposite direction, it is known as anti-Helmholtz coil, which creates a region of nearly uniform magnetic field
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
, and is used for creating magnetic traps for atomic physics experiments.


Description

A Helmholtz pair consists of two identical circular magnetic coils that are placed symmetrically along a common axis, one on each side of the experimental area, and separated by a distance h equal to the radius R of the coil. Each coil carries an equal
electric current An electric current is a stream of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is measured as the net rate of flow of electric charge through a surface or into a control volume. The moving pa ...
in the same direction. Setting h=R, which is what defines a Helmholtz pair, minimizes the nonuniformity of the field at the center of the coils, in the sense of setting \partial^B/\partial x^ = 0 (meaning that the first nonzero derivative is \partial^B/\partial x^ as explained below), but leaves about 7% variation in field strength between the center and the planes of the coils. A slightly larger value of h 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 \partial^B/\partial x^. When a Helmholtz pair of coils carry an equal electric current in the opposite direction, they create a region of nearly uniform magnetic field gradient. This is known as anti-Helmholtz coil, and is used for creating magnetic traps for atomic physics experiments. In some applications, a Helmholtz coil is used to cancel out the
Earth's magnetic field Earth's magnetic field, also known as the geomagnetic field, is the magnetic field that extends from Earth's interior out into space, where it interacts with the solar wind, a stream of charged particles emanating from the Sun. The magnetic f ...
, producing a region with a magnetic field intensity much closer to zero.


Mathematics

The calculation of the exact magnetic field at any point in space is mathematically complex and involves the study of
Bessel function Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary ...
s. Things are simpler along the axis of the coil-pair, and it is convenient to think about the
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
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 arranging the coils so that the origin x=0 is an
inflection point In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of ...
for the field strength due to each coil separately, one can guarantee that the order x^2 term is also zero, and hence the leading non-constant term is of order x^4. The inflection point for a simple coil is located along the coil axis at a distance R/2 from its centre. Thus the locations for the two coils are x=\pm R/2. The calculation detailed below gives the exact value of the magnetic field at the center point. If the radius is ''R'', the number of turns in each coil is ''n'' and the current through the coils is ''I'', then the magnetic field B at the midpoint between the coils will be given by : B = ^ \frac, where \mu_0 is the
permeability of free space The vacuum magnetic permeability (variously ''vacuum permeability'', ''permeability of free space'', ''permeability of vacuum''), also known as the magnetic constant, is the magnetic permeability in a classical vacuum. It is a physical constan ...
.


Derivation

Start with the formula for the on-axis field due to a single wire loop which is itself derived from the
Biot–Savart law In physics, specifically electromagnetism, the Biot–Savart law ( or ) is an equation describing the magnetic field generated by a constant electric current. It relates the magnetic field to the magnitude, direction, length, and proximity of the ...
: : B_1(x) = \frac=\xi(x) \frac. Here :\mu_0\; = the
permeability constant The vacuum magnetic permeability (variously ''vacuum permeability'', ''permeability of free space'', ''permeability of vacuum''), also known as the magnetic constant, is the magnetic permeability in a classical vacuum. It is a physical constant, ...
= 4\pi \times 10^ \text\cdot\text = 1.257 \times 10^ \text\cdot\text, :I\; = coil current, in
ampere The ampere (, ; symbol: A), often shortened to amp,SI supports only the use of symbols and deprecates the use of abbreviations for units. is the unit of electric current in the International System of Units (SI). One ampere is equal to elect ...
s, :R\; = coil radius, in meters, :x\; = coil distance, on axis, to point, in meters, :\xi(x)= +(x/R)^2)\;is the distance dependent, dimensionless coefficient. 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: : B_1(x) = \xi(x)\frac. For x\ll R, the distance coefficient \xi(x)= +(x/R)^2)\;can be expanded in
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
as:
\xi(x)=1-\frac(x/R)^2+\mathcal((x/R)^4).
In a Helmholtz pair, the two coils are located at x=\pm R/2, so the
B-field A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
strength at any x would be: : \begin B(x) &= \frac\left xi(x-R/2)+\xi(x+R/2)\right\\ &=\frac\left( +(x/R-1/2)^2+ +(x/R+1/2)^2\right) \\ \end The points near the center (halfway between the two coils) have x\ll R, and the
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
of \xi(x-R/2)+\xi(x+R/2) is:
(16 \sqrt 5)/25-(x/R)^4(2304 \sqrt 5)/3125+\mathcal O ((x/R)^6)\approx 1.43-1.65(x/R)^4+\mathcal O ((x/R)^6).
In an anti-Helmholtz pair, the B-field strength at any x would be: : \begin B(x) &= \frac\left xi(x-R/2)-\xi(x+R/2)\right\\ &=\frac\left( +(x/R-1/2)^2- +(x/R+1/2)^2\right) \\ \end The points near the center (halfway between the two coils) have x\ll R, and the Taylor series of \xi(x-R/2)-\xi(x+R/2) is:
(x/R)(96 \sqrt 5)/125-(x/R)^3(512 \sqrt 5)/625+\mathcal O ((x/R)^5)\approx 1.72(x/R)-1.83(x/R)^3+\mathcal O ((x/R)^5).


Time-varying magnetic field

Most Helmholtz coils use DC (direct) current to produce a static magnetic field. Many applications and experiments require a time-varying magnetic field. These applications include magnetic field susceptibility tests, scientific experiments, and biomedical studies (the interaction between magnetic field and living tissue). The required magnetic fields are usually either pulse or continuous sinewave. The magnetic field frequency range can be anywhere from near DC (0 Hz) to many kilohertz or even megahertz (MHz). An AC Helmholtz coil driver is needed to generate the required time-varying magnetic field. The waveform amplifier driver must be able to output high AC current to produce the magnetic field.


Driver voltage and current

I=\left ( \frac \right )^\left ( \frac \right ) Use the above equation in the mathematics section to calculate the coil current for a desired magnetic field, . where \mu_0 is the permeability of free space or 4\pi \times 10^ \text\cdot\text = 1.257 \times 10^ \text\cdot\text, I\; = coil current, in amperes, R\; = coil radius, in meters, n = number of turns in each coil. Then calculate the required Helmholtz coil driver amplifier voltage: :V=I\sqrt where * is the peak current, * is the angular frequency or , * and are the inductances of the two Helmholtz coils, and * and are the resistances of the two coils.


High-frequency series resonant

Generating a static magnetic field is relatively easy; the strength of the field is proportional to the current. Generating a high-frequency magnetic field is more challenging. The coils are inductors, and their impedance increases proportionally with frequency. To provide the same field intensity at twice the frequency requires twice the voltage across the coil. Instead of directly driving the coil with a high voltage, a series resonant circuit may be used to provide the high voltage. A series capacitor is added in series with the coils. The capacitance is chosen to resonate the coil at the desired frequency. Only the coils parasitic resistance remains. This method only works at frequencies close to the resonant frequency; to generate the field at other frequencies requires different capacitors. The Helmholtz coil resonant frequency, f_0, and capacitor value, C, are given below. :f_0=\frac :C=\frac


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 James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and ligh ...
showed in 1873 that a third larger-diameter coil located midway between the two Helmholtz coils with the coil distance increased from coil radius R to \sqrtR 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 A Maxwell coil is a device for producing a large volume of almost constant (or constant-gradient) magnetic field. It is named in honour of the Scottish physicist James Clerk Maxwell. A Maxwell coil is an improvement of a Helmholtz coil: in oper ...
.


See also

*
Solenoid upright=1.20, An illustration of a solenoid upright=1.20, Magnetic field created by a seven-loop solenoid (cross-sectional view) described using field lines A solenoid () is a type of electromagnet formed by a helix, helical coil of wire whose ...
*
Halbach array A Halbach array is a special arrangement of permanent magnets that augments the magnetic field on one side of the array while cancelling the field to near zero on the other side. This is achieved by having a spatially rotating pattern of magn ...
* A
magnetic bottle A magnetic mirror, known as a magnetic trap (магнитный захват) in Russia and briefly as a pyrotron in the US, is a type of magnetic confinement device used in fusion power to trap high temperature plasma using magnetic fields. The ...
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. *Helmholtz coils were designed and built for the Army Research Laboratory's electromagnetic composite testing laboratory in 1993, for testing of composite materials to low-frequency magnetic fields.


References


External links


On-Axis Field of an Ideal Helmholtz Coil


*
Helmholtz-Coil Fields
' by Franz Kraft,
The Wolfram Demonstrations Project The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hos ...
. * Kevin Kuns (2007
Calculation of Magnetic Field inside Plasma Chamber
uses
elliptic integral In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising in ...
s and their
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
s to compute off-axis fields, from
PBworks PBworks (formerly PBwiki) is a commercial real-time collaborative editing (RTCE) system created by David Weekly, with Ramit Sethi and Nathan Schmidt, who joined shortly thereafter as co-founders. Based in San Mateo, California, United States, t ...
. *{{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://apps.dtic.mil/dtic/tr/fulltext/u2/a286081.pdf , archive-url=https://web.archive.org/web/20130418084434/http://www.dtic.mil/dtic/tr/fulltext/u2/a286081.pdf , url-status=live , archive-date=April 18, 2013
Magnetic Fields of Coils
* http://physicsx.pr.erau.edu/HelmholtzCoils/ Electromagnetic coils Magnetic devices Hermann von Helmholtz