Craig Walsh

Department of Physics, Dalhousie University

Two coils placed a specific distance apart (Helmholtz arrangement) can be used to produce a very uniform magnetic field. The field was measured by moving a Hall Effect sensor along the axial length of the coils. A LabVIEW program collected the data from a FLUKE 45 digital voltmeter over a RS232C serial interface. Experimental data was then compared with theoretical predictions.

Spatially uniform magnetic fields are important for a variety of applications such as nuclear magnetic resonance and measurements of magnetocardiograms. A uniform field can be produced by a Helmholtz coil pair, as shown in Figure 1(a).

It consists of two identical coils, each of radius, a, separated by the same distance, a, carrying current, I. To calculate the axial field, first consider a single coil, with the origin, z = 0, defined at the centre, as shown in Figure 1(b). Its axial magnetic field, B(z), is [1]

(1)

Next, for a Helmholtz coil, consider two such coils with z = 0 defined mid-way between them. The coordinates, z₁ and z₂, of each coil must then be shifted according to

(2)

The total field, B(z), of the Helmholtz coil is then the sum of the individual fields, B(z₁) and B(z₂)

(3)

Equation 3 is plotted in Figure 2, with m ₀NI/2 = 1 and a = 1. Note that the field is approximately constant on the interval -0.5 to 0.5 (-a/2 to a/2).

The objective of the experiment was to measure the axial magnetic field of such a coil and determine whether it is uniform, as predicted by Equation 3.

The experimental apparatus is shown in Figure 3.

The Helmholtz magnetic field was measured with a Hall effect sensor that produces a voltage which depends linearly on magnetic field. The sensor was calibrated against several known fields from a solenoid to determine the calibration equation

B = 16.34(mT/V)V_H — 79.70 mT (4)

The Hall sensor was scanned through the Helmholtz coil by mounting it on a one-meter balsam stick that was attached to a chart recorder in x-t sweep mode. The purpose of the long stick was to keep the recorder away from the coils and the sensor, since its metal components would interfere with the field. The recorder had a 27 cm range of motion, which was enough to cover the interior of the coil and several centimeters on either side. The sweep rate was 4.79 mm/s.

The Helmholtz coil parameters were a = 20.0 ± 0.1 cm, N = 58, and I = 9.03 A. The coils were actually made of layered metal ribbons rather than adjacent turns. The coils were held together with an aluminum frame and the entire apparatus was held upright with wooden forms to minimize magnetic interference.

The Hall voltage was measured with a Fluke 45 digital multimeter and recorded by a LabVIEW program over an RS232C serial interface. The program's front and back panels are shown in Figures 4 and 5. It calls a Fluke 45 interface subvi to collect the desired number of data points from the Hall probe, and then uses the calibration equation (4) to convert the Hall voltages to magnetic field units. Finally, it calls a subvi to calculate the position of each measurement based on the measurement's time stamp, and the probe’s initial position and speed.

A graph of typical results is shown in Figure 6. The measured curve has the same shape as the theoretical curve, but is about 1.2 times larger than expected, probably because of an error in the Hall sensor calibration. Despite this error, the magnetic field is constant, as expected, in the region from —5 cm to 5 cm (-a/2 to a/2).

The numerical derivative of the data is plotted in Figure 7. The data was smoothed by a 3-point average before computing the derivative. The rate of change of the field is on the order of 0.1 mT/cm in the region ± 10cm to ± 5 cm, while within —5 cm to 5 cm it is essentially zero.

The measurements showed that the axial magnetic field inside a Helmholtz coil is uniform.