# A Quantitative Study of Helmholtz Coils

These coils come in pairs with the same number of windings of wire on each of the two coils. In "true Helmholtz" configuration: (1) the coils are wired in series with identical currents in the same direction in each coil, and (2) the coils are placed a distance apart that is equal to the radius of each coil. When in this configuration, they produce a very uniform magnetic field that is directed along their common central axis. One of the most common uses for such coils in physics education is in determining the charge to mass ratio of electrons, accomplished by immersing an electron tube in the central region of the coils and measuring the resultant curvature of the electron beam.

The two photos directly below show a self-contained e/m apparatus manufactured by Daedalon, with a light blue electron beam that is visible due to a small quantity of helium in the otherwise evacuated bulb. The accelerating potential of the electron beam and coil current are shown by digital displays at the bottom left and right of the apparatus, respectively.

The purpose of this experiment, however, is not to determine the e/m ratio for electrons. Rather, the purpose is (1) to investigate the magnitude of the magnetic field produced at varying distances from the center point of the two coils along the axis and (2) to see how this field is affected by coil separation distance. While Helmholtz coils can cost thousands of dollars, there is a nice pair including a base track available from Sargent-Welch for only \$99.90 (Cat# WLS1804-47 and price as of October 24, 2016). The photo below shows these coils, with PocketLab's magnetometer centered on the common central axis of the coils. The X on the top PocketLab shows the location of the magnetometer within PocketLab.

The complete setup for this experiment is shown in the photo below. A power supply provides a constant current, shown as 0.35 amp. The two coils, on the provided track, are separated by a distance equal to their radius with current in the same direction for each coil. PocketLab is taped to a 5/8" diameter wood dowel rod, with PocketLab's magnetometer at the center point on the axis of the coils.The dowel is in turn attached to a burette clamp, which is then attached to a ring stand. The ring stand can then be easily moved back and forth along the meter stick to provide known distances from the center point along the axis of the coils.

With the power supply off, the magnetometer is zeroed. By moving the ring stand, PocketLab is then moved to a distance 10 cm to the left (negative position) of the zero position. The power supply is turned on and the PocketLab app is set to begin recording data. Data is recorded for 10 seconds, and the the ring stand is moved 1 cm to the right. Data is again recorded for ten seconds. This process continues until the PocketLab magnetometer is 10 to the right of the center point. The result is a graph similar to that below, which shows the X magnetometer readings collected by the PocketLab app for the 210 seconds of data collection. The numbers above each of the "steps" is the magnetic field in microTesla. These values can be obtained easily by examination of the data file, or by using Excel to find averages for each of the steps. Finding averages is not really necessary, however, as the readings are quite stable as long as PocketLab is kept stationary.

The experiment is repeated a second time with the coils closer together than the radius of the coils, and a third time with the spacing of the coils greater than the radius of the coils. The graph below summarizes the results of the three experiments. Red represents data in which the coils are too close together, green for data in which the coils are spaced "just right" at coil radius, and blue for data in which the coils are too far apart. The vertical colored lines on the graph are at the location of the coils for each of the three experiments. The graph clearly demonstrates that when the coils are separated by a distance equal to the coil radius, then the magnetic field is most uniform within the coils, particularly in the central region equal to roughly 1/2 the radius of the coils.

(Note that the range for the PocketLab magnetometer is plus or minus 2000 microTesla.  If you find that you are exceeding this range, simply lower the current in the coils until you are within range.)

For anyone who might be interested, the video below shows the author quickly moving PocketLab from the far left of the coils to the far right of the coils a distance of 20 cm, with coil separation equal to the coil radius. The superimposed X magnetometer graph clearly shows the uniformity of the magnetic field near the center of the coils.

Optional Investigations

1. Investigate off-axis magnetic field strength, in an effort to determine the uniformity of the magnetic field on a plane perpendicular to the axis and centered between the coils.

2. Compare the experimental results with theoretical equations that predict the magnetic field strength along the axis of a pair of Helmholtz coils. This can be accomplished by using the equation for the axial magnetic field strength B at a distance x from a single coil of radius R with N turns of wire. Many calculus-based physics textbooks derive this equation (multiply the right side by N for N turns of wire):

Using this equation, a spreadsheet can be developed to produce theoretical graphs of magnetic field along the axis of the coils. The formulas become a bit tedious, but do show that experiment and theory are in good agreement.  See the graph below for a comparison of theory and experiment for the case in which the coils are spaced a "true Helmholtz" distance (i.e., separation equals radius).

3. Set up the coils so that the currents are in opposite directions. Such coils are sometimes referred to as being "reverse Helmholtz coils". The result will be a "magnetic quadrupole", with zero magnetic field strength at the axial center point between the coils, with polarity changing either side of the center point.

4. Have your students do a Web exercise investigating graphite levitation, an application of quadrupoles produced by strong neodymium magnets. Images Scientific Instruments (www.imagesco.com) sells a Pyrolytic Graphite Levitation Kit for \$49.95 (as of October 25, 2016).

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### Comments (2)

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Clif posted:

Rich-

Awesome lab again! You probably noticed that there was change in the field readings along the y and z-axis even though the field should be really uniform in the x-axis between the coils. There are some hard iron and soft iron distortions of the magnetic field because of the coin cell battery, the mounting screws, and errors in the sensor component. We will eventually be able to calibrate the magnetic field readings and compensate for the distortions.

Here's some calibration data that I thought you would appreciate. We obtained magnetic field data by doing a series or rotations of the PocketLab relative to the Earth's magnetic field. It would also be interesting to calibrate with the Helmholtz coil.

The first top row plot shows the deviations from the fit to a sphere with an offset center and the magnitude for all of the points. The second top row plot shows the residual variations. So in the second row, we use an ellipsoid fit to correct the data.  The result is much better and only suffers from drift and a small residual error.

Clif -- Thanks for the compliment on the lab and for the detailed stats on the magnetometer.  Appreciated. -- Rich

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Rich-

Awesome lab again! You probably noticed that there was change in the field readings along the y and z-axis even though the field should be really uniform in the x-axis between the coils. There are some hard iron and soft iron distortions of the magnetic field because of the coin cell battery, the mounting screws, and errors in the sensor component. We will eventually be able to calibrate the magnetic field readings and compensate for the distortions.

Here's some calibration data that I thought you would appreciate. We obtained magnetic field data by doing a series or rotations of the PocketLab relative to the Earth's magnetic field. It would also be interesting to calibrate with the Helmholtz coil.

The first top row plot shows the deviations from the fit to a sphere with an offset center and the magnitude for all of the points. The second top row plot shows the residual variations. So in the second row, we use an ellipsoid fit to correct the data.  The result is much better and only suffers from drift and a small residual error.

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