Helmholtz coil
Encyclopedia
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
. It is named in honor of the German physicist Hermann von Helmholtz
Hermann von Helmholtz
Hermann Ludwig Ferdinand von Helmholtz was a German physician and physicist who made significant contributions to several widely varied areas of modern science...
.
Description
A Helmholtz pair consists of two identical circular magneticcoils
Coil
A coil is a series of loops. A coiled coil is a structure in which the coil itself is in turn also looping.-Electromagnetic coils:An electromagnetic coil is formed when a conductor is wound around a core or form to create an inductor or electromagnet...
that are placed symmetrically one on each side of the
experimental area along a common axis, and separated by a distance
equal to the radius 
of the coil. Each coil carries an equalelectrical current flowing in the same direction.
Setting
, which is what defines a Helmholtz pair, minimizes the nonuniformity of the field at the center of the coils, in the sense of setting 
(meaning that the first nonzero derivative is 
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
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 
.In some applications, a Helmholtz coil is used to cancel out the Earth's magnetic field
Earth's magnetic field
Earth's magnetic field is the magnetic field that extends from the Earth's inner core to where it meets the solar wind, a stream of energetic particles emanating from the Sun...
, producing a region with a magnetic field intensity much closer to zero.
Mathematics
Bessel function
In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y of Bessel's differential equation:...
s. Things are simpler along the axis of the coil-pair, and it is convenient to think about the Taylor series
Taylor series
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
expansion of the field strength as a function of
, 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 charge
is an inflection pointInflection point
In differential calculus, an inflection point, point of inflection, or inflection is a point on a curve at which the curvature or concavity changes sign. The curve changes from being concave upwards to concave downwards , or vice versa...
for each coil separately we can guarantee that
the order
term is also zero, and hence the leading non-uniform term is of order 
. One can easily show that the inflection point for a simple coil is 
from the coil center along the axis; hence the location of each coil at
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
is the permeability of free space (
).Derivation
Start with the formula for the on-axis field due to a single wire loop http://hyperphysics.phy-astr.gsu.edu/HBASE/magnetic/curloo.html#c3 (which is itself derived from the Biot-Savart lawBiot-Savart law
The Biot–Savart law is an equation in electromagnetism that describes the magnetic field B generated by an electric current. The vector field B depends on the magnitude, direction, length, and proximity of the electric current, and also on a fundamental constant called the magnetic constant...
):
-
- Where:
= the permeability constant = 
= coil current, in ampereAmpere
The ampere , often shortened to amp, is the SI unit of electric current and is one of the seven SI base units. It is named after André-Marie Ampère , French mathematician and physicist, considered the father of electrodynamics...
s
= coil radius, in meters
= 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
= total current-
- Where:
= number of wire loops in one coilAdding this to the formula:
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:
There are also two coils instead of one, so let's multiply the formula by 2, then simplify the formula:
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 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 coilMaxwell coil
A Maxwell coil is a device for producing a large volume of almost constant magnetic field.-Description:A constant-field Maxwell coil set consists of three coils oriented on the surface of a virtual sphere...
.
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.
- 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.