magnetic field. Driver voltage and current Use the above equation in the mathematics section to calculate the coil current for a desired magnetic field, . where is the permeability of free space or = coil current, in amperes, = coil radius, in meters, n = number of turns in each coil. Then calculate the required Helmholtz coil driver amplifier voltage: where is the peak current, is the angular frequency or , and are the inductances of the two Helmholtz coils, and and 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. 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, , and capacitor value, C, are given below. 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 to 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 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. Helmholtz coils were designed and built for the Army Research Laboratory's electromagnetic composite testing laboratory in 1993, for testing of composite materials to low-frequency magnetic fields. References External links On-Axis Field of an Ideal Helmholtz Coil Axial field of a real