Helmholtz coil

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A Helmholtz coil

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Helmholtz coil schematic drawing

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 solenoid electromagnets on the same axis. Besides creating magnetic fields, Helmholtz coils are also used in scientific apparatus to cancel external magnetic fields, such as the Earth's magnetic field.

A beam of cathode rays in a Cathode Ray Tube bent into a circle within a stabilized Helmholtz Coil.

Description

A Helmholtz pair consists of two identical circular magnetic coils (solenoids) 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.

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 $\partial^{2}B/\partial x^{2} = 0$ (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 $\partial^{2}B/\partial x^{2}$.

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.

Mathematics

File:VFPt helmholtz coil thumb.svg

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.)

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Magnetic field induction along the axis crossing the center of coils; z = 0 is the point in the middle of distance between coils.

File:Helmholtz coil, B magnitude cross section.svg

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 B₀. The eight contours are for field magnitudes of 0.5 B₀, 0.8 B₀, 0.9 B₀, 0.95 B₀, 0.99 B₀, 1.01 B₀, 1.05 B₀, and 1.1 B₀.

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 flux density B at the midpoint between the coils will be given by

$B = {\left ( \frac{4}{5} \right )}^{3/2} \frac{\mu_0 n I}{R},$

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):

$B_1(x) = \frac{\mu_0 I R^2}{2(R^2+x^2)^{3/2}}.$

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.

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:

$B_1(x) = \frac{\mu_0 n I R^2}{2(R^2+x^2)^{3/2}}.$

In a Helmholtz coil, a point halfway between the two loops has an x value equal to R/2, so calculate the field strength at that point:

$B_1\left(\frac{R}{2}\right) = \frac{\mu_0 n I R^2}{2(R^2+(R/2)^2)^{3/2}}.$

There are also two coils instead of one (the coil above is at x=0; there is a second coil at x=R). From symmetry, the field strength at the midpoint will be twice the single coil value:

$\begin{align} B\left(\frac{R}{2}\right) &= 2 B_1(R/2) \\ &= \frac{2\mu_0 n I R^2}{2(R^2+(R/2)^2)^{3/2}} = \frac{\mu_0 n I R^2}{(R^2+(R/2)^2)^{3/2}} \\ &= \frac{\mu_0 n I R^2}{(R^2+\frac{1}{4}R^2)^{3/2}} = \frac{\mu_0 n I R^2}{(\frac{5}{4}R^2)^{3/2}} \\ &= {\left ( \frac{4}{5} \right )}^{3/2} \frac{\mu_0 n I}{R} \\ &= {\left ( \frac{8}{5\sqrt{5}} \right )} \frac{\mu_0 n I}{R}. \\ \end{align}$

Maxwell coils

File:Helmholtz coils in free radical experiment NBS 1957.jpg

Helmholtz coils (hoops) on three perpendicular axes used to cancel the Earth's magnetic field inside the vacuum tank in a 1957 electron beam experiment

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

• Maxwell coil
• Solenoid
• Halbach array
• A magnetic bottle 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.

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.
• Magnetic Fields of Coils
• http://physicsx.pr.erau.edu/HelmholtzCoils/