Measuring magnetic field of different magnet configurations
DAYÁN PÁEZ – DEPARTMENT OF MECHANICAL ENGINEERING, MIT.
Abstract
Magnetic fields are difficult to visualize. Current
pedagogical tools rely on iron filings in a plane with
one, or possibly two magnets. This experiment
attempts to render a qualitative picture of the
magnetic field outside of the plane of the magnets. In
addition, a 3D vector field is plotted with the
acquired data from a single magnet.
Introduction
A relative simple arrangement of no more than nine
permanent earth magnets creates a magnetic field
complex enough to exhibit chaos. Because of the
non-linearity of these three-dimensional vector fields,
most current depictions are just artistic
representations. This experiment attempts to
empirically determine the magnetic field of an
arrangement of two magnets on a plane parallel to,
and above the magnets. In addition, the magnetic
field of a single earth magnet is measured at various
points in three-dimensional space and compared with
expectations.
Experimental Setup
For this experiment small (8mm diameter by 3mm)
permanent earth magnets were used. A magnetic
field sensor from Vernier attached to a Vernier
GoLink! and a computer sampled snapshots of the
magnetic field in a given orientation at a given point.
Field of two magnets
To map the magnetic field due to two magnets, the
magnets were placed on a 75 mm by 75 mm steel
plate that (a) fixed their location and (b) enhanced the
effect of the magnetic field. The measurement was
taken on a plane 75 mm above the plate, using a grid
with 5 mm spacing to mark the locations. A typical
setup is shown in Figure 1.
Because the magnetic field sensor can only measure
one direction of the magnetic field at a time, three
sweeps of the grid were required to measure the full
magnetic field vector. The sweeping action for this
setup was done manually, moving from point to point
once every second and setting the computer to
sample at the same rate. Great care was taken to
sample every point in the right sequence with each
sweep to ascertain that each point in the grid had its
corresponding three magnitudes in each of the three
spatial directions. A metronome was used to keep the
manual and computer sampling rates in sync. The
arrangement used is shown in Figure 2.
3D field of one magnet
In order to speed up data acquisition for the case of a
single magnet, a linear potentiometer from the MIT
2.671 lab was used to precisely locate the magnet as
it swept across the sensor fixed in space. For a given
y- and z-location of the sensor, the sweep of the
magnet provided 100 samples of the magnetic field
along the x-direction, from -100 mm ≤ x ≤ 100 mm.
A voltmeter (also from Vernier) sampling at the same
rate measured the position of the magnet.
The height of the plane of the measurement was
controlled discretely using a stack of five books, each
10-13 mm thick. This stack of books was placed
inside a drawer that moved perpendicular to the
linear dimension of the potentiometer, providing
discrete control along the y-direction. For a given
height, the drawer was displaced -80 mm≤ y ≤80 mm
by intervals of 10 mm. For each interval in the y-
Sensor
Grid
y
z
Figure 1: Typical setup for an arrangement of magnets.
Magnets
x
Figure 2: Layout for experiment 1.
Computer
9V Battery
Potentiometer
10 kΩ
10 kΩ
POT
Drawer
GoLink!
Mag. Sensor
(a)
To computer
Stack of
books
(b)
Figure 3: (a) Setup for experiment 2. Drawer opens perpendicular to plane of page. Sensor taped onto stack of
books. (b) Circuit for potentiometer. Extra 10 k resistor added to minimize current when potentiometer set to 0.
direction, the potentiometer sweep provided the data
in the x-direction. Because only one linear
potentiometer was available, the sampling in the ydirection had to be discretely done with the rather
rudimentary technique of the drawer.
After data was acquired for all five plane heights with
the sensor in one direction (z), the procedure was
repeated with the sensor facing right, to acquire the xcomponent of the magnetic field. Because of the
expected symmetry, the y-component of the single
magnet was not measured. The full setup for
Experiment 2 is depicted in Figure 3.
Data Analysis
An advantage of the method of data acquisition used
in the first experiment is that every point sampled had
its unique magnetic field magnitude in each of the
three spatial orientations. However, the process was
hindered by a slow rate and human error as the sensor
was swept across the measuring plane. In addition,
total synchronization between the computer sampling
and the human sweeping could not be guaranteed.
In contrast, the second method reduced sources of
human error by taping the sensor down, thereby
keeping it stationary, while the magnet was swept
across in the x-direction and its position recorded
with a potentiometer. This yielded in relatively faster
sampling time (1 sweep in 1 second) and a large
amount of data.
Nevertheless, the second method had one major
drawback compared to the first. Because there was no
guarantee that the points sampled in the x-direction
were the same every time, attempts to plot a surface
yielded mismatched graphs. In order to create a
uniform grid of x- and y- values for surface plotting, a
fixed interval of points for the x-axis was imposed,
and values surrounding that point were averaged.
This results in more reliable data as it reduces the
effect of noise through averaging.
Results and Discussion
Arrangement of Magnets
The x-, y-, and z-components of the magnetic field for
the arrangement of two magnets are shown in Figure
4. The corrugated appearance of the z- and xcomponents suggests error in data collection, where a
location on the grid was possibly skipped or
resampled. Notice how the y- component has the
expected smooth curve. We can tell from this graph
what the locations of the magnets were, namely,
Magnet 1
x = 7.5 mm
Magnet 2
y = 7.5 mm
x= 67.5mm
y= 67.5mm
Also notice how the z-component is considerably
greater in magnitude than the other two components.
This is due to the presence of the steel plate, which
enhances the magnetic field by increasing the size of
the effective south pole of the magnet.
This experiment could be much improved by
increasing the sampling rate and doing away with the
steel plate. This was the reason behind the second
experiment.
(c)
(b)
(a)
Figure 4: Magnitude of the magnetic field in (a) the direction, (b) the direction, and (c) the direction.
3D magnetic field of a single magnet
magnetic field of one magnet or an arrangement of
magnets. These data can be useful as pedagogical
tools for the visualizations of complex fields. In
addition, with the collected data from this experiment
and possible future ones, it would be possible to
empirically determine the falloff rate of magnetic
fields in general.
A graph of the magnetic field in the z- and xdirection for a single magnet is shown in Figure 5a,
5b. Because only one magnet was used symmetry
was assumed in the x- and y- directions, and so only
the x- component was calculated. The magnitudes are
shown by height. Comparing the z-component of the
field for both the two magnets and the single magnet
demonstrates qualitatively the “enhanced” field in the
former. In addition, Figure 5 shows visually the
falloff rate of the magnetic field. Note that the
“amplitude” of the magnetic field decreases about
95% from a depth of 100 mm to 200 mm. A 3D
vector field is shown in Figure 5c.
Acknowledgements
The author would like to thank Dr. Barbara Hughey
of Mechanical Engineering Department at MIT, for
lending the sensor and measuring equipment, as well
as for the potentiometer. In addition, the first
experiment could not have been completed without
the supervision of Alexis Dale.
Conclusion
With the use of two potentiometers and a Vernier
magnetic field sensor, it is feasible to plot the
(a)
(b)
(c)
Figure 5: Magnetic field vector by height in the (a) direction and the (b) , directions. (c) A visual
representation of the magnetic field, where the size of the arrow is proportional to the magnitude of the field.