Helmholtz coil

Last updated July 28, 2024

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 electromagnets 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.

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 in the same direction.^[1]

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 $\partial ^{2}B/\partial x^{2}=0$ ^[2] (meaning that the first nonzero derivative is $\partial ^{4}B/\partial x^{4}$ 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 $\partial ^{2}B/\partial x^{2}$ .^[3]

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, producing a region with a magnetic field intensity much closer to zero.^[4]

Mathematics

The calculation of the exact magnetic field at any point in space is mathematically complex 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 arranging the coils so that the origin $x=0$ is an inflection point 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={\left({\frac {4}{5}}\right)}^{3/2}{\frac {\mu _{0}nI}{R}},

where $\mu _{0}$ is the permeability of free space ( $4\pi \times 10^{-7}{\text{ T}}\cdot {\text{m/A}}$ ).

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:^[5]

B_{1}(x)={\frac {\mu _{0}IR^{2}}{2(R^{2}+x^{2})^{3/2}}}=\xi (x){\frac {\mu _{0}I}{2R}}.

Here

\mu _{0}\;

= the permeability constant =

4\pi \times 10^{-7}{\text{ T}}\cdot {\text{m/A}}=1.257\times 10^{-6}{\text{ T}}\cdot {\text{m/A}},

I\;

= coil current, in amperes,

R\;

= coil radius, in meters,

x\;

= coil distance, on axis, to point, in meters,

\xi (x)=[1+(x/R)^{2}]^{-3/2}\;

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 {\mu _{0}nI}{2R}}.

For $x\ll R$ , the distance coefficient $\xi (x)=[1+(x/R)^{2})]^{-3/2}\;$ can be expanded in Taylor series as:

$\xi (x)=1-{\frac {3}{2}}(x/R)^{2}+{\mathcal {O}}((x/R)^{4}).$

In a Helmholtz pair, the two coils are located at $x=\pm R/2$ , so the B-field strength at any $x$ would be:

{\begin{aligned}B(x)&={\frac {\mu _{0}nI}{2R}}\left[\xi (x-R/2)+\xi (x+R/2)\right]\\&={\frac {\mu _{0}nI}{2R}}\left([1+(x/R-1/2)^{2}]^{-3/2}+[1+(x/R+1/2)^{2}]^{-3/2}\right)\\\end{aligned}}

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:

$(16{\sqrt {5}})/25-(x/R)^{4}(2304{\sqrt {5}})/3125+{\mathcal {O}}((x/R)^{6})\approx 1.43-1.65(x/R)^{4}+{\mathcal {O}}((x/R)^{6})$ .

In an anti-Helmholtz pair, the B-field strength at any $x$ would be:

{\begin{aligned}B(x)&={\frac {\mu _{0}nI}{2R}}\left[\xi (x-R/2)-\xi (x+R/2)\right]\\&={\frac {\mu _{0}nI}{2R}}\left([1+(x/R-1/2)^{2}]^{-3/2}-[1+(x/R+1/2)^{2}]^{-3/2}\right)\\\end{aligned}}

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

$I=\left({\frac {5}{4}}\right)^{3/2}\left({\frac {BR}{\mu _{0}n}}\right)$

Use the above equation in the mathematics section to calculate the coil current for a desired magnetic field, $B$ .

where $\mu _{0}$ is the permeability of free space or $4\pi \times 10^{-7}{\text{ T}}\cdot {\text{m/A}}=1.257\times 10^{-6}{\text{ T}}\cdot {\text{m/A}},$

$I\;$ = coil current, in amperes,

$R\;$ = coil radius, in meters,

n = number of turns in each coil.

Using a function generator and a high-current waveform amplifier driver to generate high-frequency Helmholtz magnetic field Helmholtz-Coil-Series-Resonant.png — Using a function generator and a high-current waveform amplifier driver to generate high-frequency Helmholtz magnetic field

Then calculate the required Helmholtz coil driver amplifier voltage:^[6]

V=I{\sqrt {{\bigl [}\omega {\bigl (}L_{1}+L_{2}{\bigr )}{\bigr ]}^{2}+{\bigl (}R_{1}+R_{2}{\bigr )}^{2}}}

where

$I$ is the peak current,
$ω$ is the angular frequency or $ω = 2 πf$ ,
$L 1$ and $L 2$ are the inductances of the two Helmholtz coils, and
$R 1$ and $R 2$ 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.^[7] 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.^[6]

f_{0}={\frac {1}{2\pi {\sqrt {\left(L_{1}+L_{2}\right)C}}}}

C={\frac {1}{\left(2\pi f_{0}\right)^{2}\left(L_{1}+L_{2}\right)}}

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 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 ${\sqrt {3}}R$ 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.

Related Research Articles

<span class="mw-page-title-main">Fourier transform</span> Mathematical transform that expresses a function of time as a function of frequency

In physics, engineering and mathematics, the Fourier transform (FT) is an integral transform that takes a function as input and outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches.

The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields. This density and flux of energy and momentum are the sources of the gravitational field in the Einstein field equations of general relativity, just as mass density is the source of such a field in Newtonian gravity.

Inductance is the tendency of an electrical conductor to oppose a change in the electric current flowing through it. The electric current produces a magnetic field around the conductor. The magnetic field strength depends on the magnitude of the electric current, and follows any changes in the magnitude of the current. From Faraday's law of induction, any change in magnetic field through a circuit induces an electromotive force (EMF) (voltage) in the conductors, a process known as electromagnetic induction. This induced voltage created by the changing current has the effect of opposing the change in current. This is stated by Lenz's law, and the voltage is called back EMF.

A solenoid is a type of electromagnet formed by a helical coil of wire whose length is substantially greater than its diameter, which generates a controlled magnetic field. The coil can produce a uniform magnetic field in a volume of space when an electric current is passed through it.

In electromagnetism, skin effect is the tendency of an alternating electric current (AC) to become distributed within a conductor such that the current density is largest near the surface of the conductor and decreases exponentially with greater depths in the conductor. It is caused by opposing eddy currents induced by the changing magnetic field resulting from the alternating current. The electric current flows mainly at the skin of the conductor, between the outer surface and a level called the skin depth.

In physics and engineering, a constitutive equation or constitutive relation is a relation between two or more physical quantities that is specific to a material or substance or field, and approximates its response to external stimuli, usually as applied fields or forces. They are combined with other equations governing physical laws to solve physical problems; for example in fluid mechanics the flow of a fluid in a pipe, in solid state physics the response of a crystal to an electric field, or in structural analysis, the connection between applied stresses or loads to strains or deformations.

In probability theory and statistics, the generalized extreme value (GEV) distribution is a family of continuous probability distributions developed within extreme value theory to combine the Gumbel, Fréchet and Weibull families also known as type I, II and III extreme value distributions. By the extreme value theorem the GEV distribution is the only possible limit distribution of properly normalized maxima of a sequence of independent and identically distributed random variables. Note that a limit distribution needs to exist, which requires regularity conditions on the tail of the distribution. Despite this, the GEV distribution is often used as an approximation to model the maxima of long (finite) sequences of random variables.

In physics and astronomy, Euler's three-body problem is to solve for the motion of a particle that is acted upon by the gravitational field of two other point masses that are fixed in space. This problem is exactly solvable, and yields an approximate solution for particles moving in the gravitational fields of prolate and oblate spheroids. This problem is named after Leonhard Euler, who discussed it in memoirs published in 1760. Important extensions and analyses were contributed subsequently by Lagrange, Liouville, Laplace, Jacobi, Darboux, Le Verrier, Velde, Hamilton, Poincaré, Birkhoff and E. T. Whittaker, among others.

In orbital mechanics, mean motion is the angular speed required for a body to complete one orbit, assuming constant speed in a circular orbit which completes in the same time as the variable speed, elliptical orbit of the actual body. The concept applies equally well to a small body revolving about a large, massive primary body or to two relatively same-sized bodies revolving about a common center of mass. While nominally a mean, and theoretically so in the case of two-body motion, in practice the mean motion is not typically an average over time for the orbits of real bodies, which only approximate the two-body assumption. It is rather the instantaneous value which satisfies the above conditions as calculated from the current gravitational and geometric circumstances of the body's constantly-changing, perturbed orbit.

In mathematics, Mathieu functions, sometimes called angular Mathieu functions, are solutions of Mathieu's differential equation

In probability theory, the Rice distribution or Rician distribution is the probability distribution of the magnitude of a circularly-symmetric bivariate normal random variable, possibly with non-zero mean (noncentral). It was named after Stephen O. Rice (1907–1986).

The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation. The homogeneous form of the equation, written in terms of either the electric field $E$ or the magnetic field $B$ , takes the form:

Oblate spheroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinate system about the non-focal axis of the ellipse, i.e., the symmetry axis that separates the foci. Thus, the two foci are transformed into a ring of radius $in the x - y plane. Oblate spheroidal coordinates can also be considered as a limiting case of ellipsoidal coordinates in which the two largest semi-axes are equal in length.$

An LC circuit can be quantized using the same methods as for the quantum harmonic oscillator. An LC circuit is a variety of resonant circuit, and consists of an inductor, represented by the letter L, and a capacitor, represented by the letter C. When connected together, an electric current can alternate between them at the circuit's resonant frequency:

Static force fields are fields, such as a simple electric, magnetic or gravitational fields, that exist without excitations. The most common approximation method that physicists use for scattering calculations can be interpreted as static forces arising from the interactions between two bodies mediated by virtual particles, particles that exist for only a short time determined by the uncertainty principle. The virtual particles, also known as force carriers, are bosons, with different bosons associated with each force.

<span class="mw-page-title-main">Magnetic levitation</span> Suspension of objects by magnetic force.

Magnetic levitation (maglev) or magnetic suspension is a method by which an object is suspended with no support other than magnetic fields. Magnetic force is used to counteract the effects of the gravitational force and any other forces.

Calculations in the Newman–Penrose (NP) formalism of general relativity normally begin with the construction of a complex null tetrad $, where is a pair of real null vectors and is a pair of complex null vectors. These tetrad vectors respect the following normalization and metric conditions assuming the spacetime signature$

In superparamagnetism, the Néel effect appears when a superparamagnetic material in a conducting coil is subject to varying frequencies of magnetic fields. The non-linearity of the superparamagnetic material acts as a frequency mixer, with voltage measured at the coil terminals. It consists of several frequency components, at the initial frequency and at the frequencies of certain linear combinations. The frequency shift of the field to be measured allows for detection of a direct current field with a standard coil.

In mathematics, Wiener's lemma is a well-known identity which relates the asymptotic behaviour of the Fourier coefficients of a Borel measure on the circle to its atomic part. This result admits an analogous statement for measures on the real line. It was first discovered by Norbert Wiener.

In physics and mathematics, the Klein–Kramers equation or sometimes referred as Kramers–Chandrasekhar equation is a partial differential equation that describes the probability density function $f$ of a Brownian particle in phase space $(r, p)$ . It is a special case of the Fokker–Planck equation.

References

↑ Ramsden, Edward (2006). Hall-effect sensors : theory and applications (2nd ed.). Amsterdam: Elsevier/Newnes. p. 195. ISBN 978-0-75067934-3.
↑ Helmholtz Coil in CGS units Archived March 24, 2012, at the Wayback Machine
↑ "Electromagnetism". Archived from the original on 2011-06-03. Retrieved 2007-11-20.
↑ "Earth Field Magnetometer: Helmholtz coil" by Richard Wotiz 2004 Archived June 28, 2007, at archive.today
↑ "Magnetic Field of a Current Loop".
1 2 Yang, KC. "High frequency Helmholtz coils generate magnetic fields". EDN. Retrieved 2016-01-27.
↑ "High-Frequency Electromagnetic Coil Resonant". www.accelinstruments.com. Retrieved 2016-02-25.
↑ "ログイン - ASACUSA MUSASHI group".
↑ J, DeTroye, David; J, Chase, Ronald (Nov 1994). "The Calculation and Measurement of Helmholtz Coil Fields". Archived from the original on June 2, 2018.{{cite journal}}: Cite journal requires |journal= (help)CS1 maint: multiple names: authors list (link)

External links

On-Axis Field of an Ideal Helmholtz Coil
Axial field of a real Helmholtz coil pair
Helmholtz-Coil Fields by Franz Kraft, The Wolfram Demonstrations Project.
Kevin Kuns (2007) Calculation of Magnetic Field inside Plasma Chamber, uses elliptic integrals and their derivatives to compute off-axis fields, from PBworks.
DeTroye, David J.; Chase, Ronald J. (November 1994), The Calculation and Measurement of Helmholtz Coil Fields (PDF), Army Research Laboratory, ARL-TN-35, archived (PDF) from the original on April 18, 2013
Magnetic Fields of Coils Archived 2015-04-30 at the Wayback Machine
http://physicsx.pr.erau.edu/HelmholtzCoils/

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] Ramsden, Edward (2006). Hall-effect sensors : theory and applications (2nd ed.). Amsterdam: Elsevier/Newnes. p. 195. ISBN 978-0-75067934-3.

[2] Helmholtz Coil in CGS units Archived March 24, 2012, at the Wayback Machine

[3] "Electromagnetism". Archived from the original on 2011-06-03. Retrieved 2007-11-20.

[4] "Earth Field Magnetometer: Helmholtz coil" by Richard Wotiz 2004 Archived June 28, 2007, at archive.today

[5] "Magnetic Field of a Current Loop".

[:0-6] 1 2 Yang, KC. "High frequency Helmholtz coils generate magnetic fields". EDN. Retrieved 2016-01-27.

[7] "High-Frequency Electromagnetic Coil Resonant". www.accelinstruments.com. Retrieved 2016-02-25.

[8] "ログイン - ASACUSA MUSASHI group".

[9] J, DeTroye, David; J, Chase, Ronald (Nov 1994). "The Calculation and Measurement of Helmholtz Coil Fields". Archived from the original on June 2, 2018.{{cite journal}}: Cite journal requires |journal= (help)CS1 maint: multiple names: authors list (link)

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]