Quadrupole magnetic field

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 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 ${\displaystyle h}$ equal to the radius ${\displaystyle R}$ of the coil. Each coil carries an equal electric current in the same direction.

Setting ${\displaystyle h=R}$, which is what defines a Helmholtz pair, minimizes the nonuniformity of the field at the center of the coils, in the sense of setting ${\displaystyle \partial ^{2}B/\partial x^{2}=0}$ (meaning that the first nonzero derivative is ${\displaystyle \partial ^{4}B/\partial x^{4}}$ 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 ${\displaystyle h}$ 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 ${\displaystyle \partial ^{2}B/\partial x^{2}}$.

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