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 {\displaystyle h} equal to the radius R {\displaystyle R} of the coil. Each coil carries an equal electric current in the same direction. Setting h = R {\displaystyle 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 ∂ 2 B / ∂ x 2 = 0 {\displaystyle \partial ^{2}B/\partial x^{2}=0} (meaning that the first nonzero derivative is ∂ 4 B / ∂ x 4 {\displaystyle \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 {\displaystyle 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 ∂ 2 B / ∂ x 2 {\displaystyle \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
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 {\displaystyle 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 {\displaystyle x=0} is an inflection point for the field strength due to each coil separately, one can guarantee that the order x 2 {\displaystyle x^{2}} term is also zero, and hence the leading non-constant term is of order x 4 {\displaystyle x^{4}} . The inflection point for a simple coil is located along the coil axis at a distance R / 2 {\displaystyle R/2} from its centre. Thus the locations for the two coils are x = ± R / 2 {\displaystyle 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 = ( 4 5 ) 3 / 2 μ 0 n I R , {\displaystyle B={\left({\frac {4}{5}}\right)}^{3/2}{\frac {\mu _{0}nI}{R}},}
where μ 0 {\displaystyle \mu _{0}} is the permeability of free space ( 4 π × 10 − 7 T ⋅ m/A {\displaystyle 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:
… excerpt ends here. Continue reading the full article.
