Magnetic Field Mapping - Wabash · PDF fileMagnetic Field Mapping Badger, Bradon, Daron, Jonathan E., ... Rdimensional eld of the Helmholtz coil and it looks to be pretty constant

WJP, PHY382 (2015) Wabash Journal of Physics v4.3, p.1

Magnetic Field Mapping

Badger, Bradon, Daron, Jonathan E., and Madsen, M.J.

Department of Physics, Wabash College, Crawfordsville, IN 47933

(Dated: May 5, 2015)

We analyzed the magnetic fields of a Helmholtz coil, permanent magnet, and the

effects of ferro-fluid on the magnetic field of a permanent magnet using a triple axis

automated mobility non-magnetic sensor support structure with a magnetometer

sensor. We found the models describing the Helmholtz coil’s magnetic field and the

magnetic dipole field of the permanent magnet were in agreement with our findings.

We also confirm that the ferro-fluid is paramagnetic based on a qualitative analysis

of our data.


I. INTRODUCTION

Magnetic field mapping can be used as a quality control assessment tool in production

of complex multipole magnets or complex assemblies such as loudspeakers or photocopier

rollers. By accurately knowing the behavior of a magnetic field, we can test the effectiveness

and efficiency of different ways to produce a magnetic field[4]. A previous research group

consisting of students at Wabash College developed a method for using a three-axis scanner

in order to map out the magnetic field of a regular current-carrying wire loop. We will use

these methods to progress research of other magnetic systems to be mapped. In terms of

this experiment, we have three specific systems of interest for mapping; a Helmholtz coil[7],

a permanent magnet[1][3][5], and a small amount of ferro fluid[2][6]. The purpose of this

experiment is to accurately map the 3-Dimensional magnetic field in or around each of our

systems and compare our results with expectations.

II. MODEL

To map the magnetic field of a system we must progress through a series of steps to

accurately and effectively do so (See FIG. 1). The first step, and potentially the most

important, is to determine any symmetries, limiting regions of space, and special geometries

of the system we wish to map. This then enables us decide where, how often, and how to

move the sensor through a region of space in order to acquire data on the system’s magnetic

field. Doing so also enables us to know how to set the sensors sensitivity or measurement

range so that we can obtain the most accurate and precise measurements possible without

breaking or saturating our sensor. After considering all of these aspects, creating a path

to scan, setting the maximum and minimum ranges for each axis, and a zero point for our

system we then move on to actually scanning the path with the magnetometer mounted

on the automated mobility non-magnetic support structure. This support structure will

visit all the points and collect the magnetic field data. Lastly, the analysis of the data is

also dependent on the type of system we are mapping. If the system involves mapping the

magnetic field induced by a current then we can use a current switching technique. To do

this we take a measurement of the field then switch the direction of the current and take

another measurement of the field at the same point and average the two to eliminate the


effects of the Earth’s magnetic field and any background noise. This analysis is done while

taking the data. If the system we are considering contains any magnetic fields which we

cannot control (permanent magnets) then we have to run a data scan of the background and

Earth’s magnetic field along the same scan path that we will use for the system’s scan and

subtract it from the data scan of our system. This analysis must be done post-scanning.


𝐵-Field/ Geometry

Create Path

Move to (𝑥𝑖 , 𝑦𝑖 , 𝑧𝑖)

Switched

𝐵𝑎𝑣𝑔

Background

Subtracted 𝐵

FIG. 1: The flow chart above is a model of how we must progress to map the magnetic

field of any system. The first step, and potentially the most important, is to determine any

symmetries, limiting regions of space, and special geometries of the system we wish to map.

This then enables us decide where, how often, and how to move the sensor through a region

of space in order to acquire data on the systems magnetic field. Doing so also enables us to

know how to set the sensors sensitivity so that we can obtain the most accurate and precise

measurements possible. After considering all of these aspects and creating a path to scan

and a zero point for our system we then move on to actually scanning the path with the

magnetometer mounted on the automated mobility non-magnetic support structure. This

support structure will visit all the points and collect the magnetic field data. Lastly, the

analysis of the data is also dependent on the type of system we are mapping. If the system

involves mapping the magnetic field induced by a current then we can use a current switching

technique. To do this we take a measurement of the field then switch the direction of the

current and take another measurement of the field at the same point and average the two to

eliminate the effects of the Earth’s magnetic field and any background noise. This analysis

is done while taking the data. If the system we are considering contains any magnetic

fields which we cannot control (permanent magnets) then we have to run a data scan of the

background and Earth’s magnetic field along the same scan path that we will use for the

system’s scan and subtract it from the data scan of our system. This analysis must be done

post-scanning.

The magnetic field sensor that we are using, the HMC5883L 3-Axis digital compass IC [8],

is a solid state sensor. It uses the properties of Giant Magnetoresistance (GMR) to detect


potential differences due to any magnetic fields that the sensor may be in. Magnetoresistance

is a property that a ferromagnetic material possess. This property is the result of the

materials ability to conduct charges, specifically electrons in a conductor such as iron, based

on how much those charges scatter as they flow through the material. This sensor however

has limitations. One of these limitations is that we cannot scan any magnetic fields stronger

than about 9 Gauss or we will saturate the sensor and it may even break. So to get a scan

of a stronger magnetic field we used a Hall Probe “21E”. The hall probe is a transducer

that uses the hall effect to vary its output voltage in response to a magnetic field. The hall

effect is the natural drift of the electrons in the conductor due to the Lorentz Force when in

a magnetic field producing a separation/build up of charge.

Area of Interest

Motor

B Field Sensor (Magnetometer) Non-Magnetic Surface

CPU Input

Y

X

Z

Data Processor

200 mm

250 mm

220 mm

FIG. 2: The figure above shows our setup for the three different systems we mapped. For all

three, we used a magnetometer transported by an automated mobile non-magnetic sensor

support structure to scan the magnetic field for many points in three-dimensional space.

The data processor of this triple-axis scanner reads information such as coordinate inputs

and tells the motor to move our magnetic field sensor to these specific coordinates. It then

tells the magnetometer to scan the specific points in space. Afterwards, this information is

sent back to the computer and prepared for analysis. Notice the parameters above, but the

range of motion in the x direction is 175 mm, 150 mm in the y direction, and 180 mm in the

z direction. The accuracy of motion in all directions for the sensor is ±.5 mm Each object

of interest is set upon a non-magnetic surface and far away from any magnetic materials

that could cause interference. For 2 of the 3 setups, we accounted for the Earth’s magnetic

field by taking a background scan.


III. DATA/ANALYSIS

A. Helmholtz Coil

Our first system of interest is a Helmholtz Coil. Sending current through the coils is a

good way to create a uniform, constant magnetic field throughout the inside. Specifically

the magnetic field is given by

B =(4

5

) 32 µ0nI

R, (1)

where n is the turns per unit length of the coil, I is the current passed through the coil,

and R is the radius of the coil. For our coil n = 750 1m

turns per unit length. This model

gives that the expected magnetic field is B = (5.14 ± 0.13)Gauss(95%CI). As a group at

Nankai University in China confirms by studying its field parameters, the Coil’s magnetic

field is mainly concentrated in the axial direction (in our case, the z direction)[7]. Our

system consists of a Helmholtz Coil with the current provided by a DC current generator

and two relays to switch the direction of the current. This current switching technique was

used in order to account for background noise and the Earths magnetic field. It should be

noted that (R = 13.100± .039)cm (95% triangular pdf). We used a rectangularly symmetric

scan path taken inside the coil in order to scan as many points as possible. In the bottom

right figure of Fig.3 we can see the three dimensional field of the Helmholtz coil and it

looks to be pretty constant and uniform which, according to the group from China, is to be

expected[7]. The magnitude of each of the vectors is about 3.2 Gauss. However, performing

a more rigorous analysis of the magnetic field brings up the bottom left figure in Fig.3. In

this figure, we have the z-component of the magnetic field as a function of the z-position.

Several lines are depicted here for several different xy-positions withing the coil. This data

suggests that the Helmholtz coil we measured was not quite ideal, as the z-component of

the magnetic field should be independent of the z-position within the coil. However, overall

the magnetic field observed is pretty uniform and to god approximation can be treated as

such within the coil. It is also important to note that it does not agree with the predicted

value of B = (5.14 ± 0.13)Gauss(95%CI).


Z

Y

X

R R

Control by Arduino

(a)

-50

0

50x HmmL

-500

50y HmmL

-50

0

50

z HmmL

(b)

(c) (d)

FIG. 3: figure (a) depicts a Helmholtz Coil with the current provided by a DC current gen-

erator and 2 relays to switch the direction of the current. This current switching technique

was used in order to account for background noise and the Earths magnetic field. It should

be noted that (R = 13.100 ± .039) cm (95% triangular pdf). In figure (b), we have our

rectangularly symmetric scan path taken inside the coil. In figure (d), we can see the three

dimensional field of the Helmholtz coil and it looks to be pretty constant and uniform. The

magnitude of each of the vectors is about 3.2 Gauss.However, performing a more rigorous

analysis of the magnetic field brings up figure (c). In figure (c.) we have the z-component of

the magnetic field as a function of the z-position. Several lines are depicted here for several

different xy-positions withing the coil. This data suggests that the Helmholtz coil we mea-

sured was not quite ideal as the z-component of the magnetic field should be independent

of the z-position within the coil.


B. Magnetic Dipole

The next system that we took into consideration was the magnetic dipole, a permanent

magnet. The model for the magnetic field of a dipole is

~B =3µ

r3

[sin θ cos θ(cosφx+ sinφy) + (cos2 θ − 1

3)z], (2)

Where r is the distance from the center of the magnet, φ is the azimuthal angle, and θ is the

polar angle (NOTE: This is for use with spherical coordinates). This model can be simply

converted to Cartesian coordinates using the geometry of the axis. This becomes

Bx =3µxz

(x2 + y2 + z2)52

, (3)

By =3µyz

(x2 + y2 + z2)52

, (4)

Bz =3µ

(x2 + y2 + z2)32

( z2

x2 + y2 + z2− 1

3

). (5)

Which we can then plot a model from(See FIG. 4). We used a Neodymium Rare-Earth

magnet of about 0.2 Tesla. In FIG. 4(a) we can see the setup. We raised the magnet up

off the non-magnetic platform in order to scan as much area around the magnet as possible

and so that we could see the dipole field. When scanning this magnet we found it easiest to

visualize the data in a planar scan of the magnetic field through the xz-plane. The scan path

that we used can be seen in FIG. 4(b). When measuring the magnetic field we had to set

our measurement sensitivity/range to the maximum setting of 8.1Gauss with our sensor and

measure the magnet from about 3cm away to avoid saturation. Special attention was also

needed to create the scan path so that the planar scan never went underneath the magnet;

this was done to avoid knocking over the setup. The vector plot in FIG. 4(d) shows the data

from our planar scan of the magnetic field. In this plot the center of the magnet is positioned

at (x, y, z) = (0, 0, 0) and we can see the dipole field curving around the magnet in space.

This data qualitatively shows that the field of a magnetic dipole (permanent magnet) does

indeed match the expected field (FIG. 4(c)) for the model of this system.


Z

Y

X

Permanent Magnet

Non-magnetic Support Stand for Magnet

h

d

w

(a)

-40 -20 0 20 40

-40

-20

0

20

40

x HmmL

zHm

mL

(b)

-60 -40 -20 0 20 40 60

-40

-20

0

20

40

x(mm )

z(mm

)

(c)

(d)

FIG. 4: (a) shows our setup of a permanent magnet on top of a non-magnetic support

stand, where h = (25.00 ± .36)mm (95%CI), w = (2.400 ± .048)mm (95%CI), and d =

(2.000 ± .048)mm (95%CI). (b) is a two-dimensional plot of our scan path of an arch. (d)

shows a scan of positions in the xz plane at y = 0 around a permanent magnet centered at

(x, y, z) = (0, 0, 0). The many arrows indicate the direction and relative magnitude of the

magnetic field at the specific position. As one can see, its general appearance is that of a

typical magnetic dipole as predicted by the model(c), with magnetic field vectors downward

and out below the magnet, and inward above the magnet. Note that the magnet was on top

of a stand, so we could not scan directly below it.


C. Ferro-Fluid

The last system that we took into consideration is the ferro-fluid and permanent magnet.

To scan this system we had to be very careful not to come to close when scanning or we

ran the risk of getting the fluid on the magnetometer. To best visualize symmetry and any

effects that the ferro-fluid had on the magnetic field we decided to do a planar scan of the

space above the system in the xy-plane. Our system with the setup and scan path can be

seen in FIG. 5. The scan path was created about 5cm above the system to avoid any risk

of spilling the fluid, getting fluid on the magnetometer, and saturating the sensor. We took

three scans of this system. The first scan was of the ferro-fluid bath on top of the magnet.

Then after that we removed the ferro-fluid and just scanned the magnet along the same

path. Lastly, we again scanned the path with no magnet or ferro-fluid. We then subtracted

the background of the Earth’s magnetic field from the data with both the fluid and magnet.

This data is shown in the bottom left of FIG. 5. This data shows the magnetic field of the

entire system and we can see that from this plane the data matches with what we would

expect from a magnetic dipole (the permanent magnet, which is supplying the magnetic

field in this system). After noting this we then took the systems data (which is what we just

mentioned) and we subtracted the permanent magnet’s scan data from it so that we can see

the effect of the ferro-fluid on the system. This data is shown in the bottom right of FIG. 5.

Here we observed that the data shows a magnetic field in the opposite direction and of

greater magnitude than the system’s data. This is indicative of the fact that the ferro-fluid

we used is paramagnetic, meaning that the ferro-fluid’s internal particles will magnetize and

align with the magnetic dipole field. The effect that this has on the system is to internalize

a lot of the magnetic field within the fluid and redirect it toward the other end of the dipole

(It is effectively offering a path of least resistance for the magnetic field to return to the

dipole.). This data (Bottom right of FIG. 5) agrees with the expectations of the effect of

the ferro-fluid as the fluid is paramagnetic and will cause the magnetic field in the matter

(ferro-fluid) to increase by superposition reducing the overall field outside the fluid and still

leaving a dipole like field outside the ferro-fluid.


Z

Y

X

Cup of Ferro Fluid Placed on Top of a Magnet

Non-Magnetic Support Stand

h

(a)

-50

0

50xHmmL

-50

0

50

yHmmL

-100

-50

0

zHmmL

(b)

(c) (d)

FIG. 5: In figure (a) we see our setup consisting of a non-magnetic support stand with a

magnet on top and a small amount of ferro fluid sitting above it. Figure (b) depicts our

planar scan path. In plot (c), we see a scan in the xy plane 50 mm above a container of ferro-

fluid with a 2.0 Gauss magnet underneath. The magnet is centered at (x, y, z) = (0, 0, 0).

Notice the vectors are largest near the center of the z axis, where the x and y-positions

equal zero. This is also to be expected since one of the poles of the magnet underneath the

ferro-fluid is pointed straight upwards. In addition to a scan of the ferro-fluid above the

magnet, we also scanned the magnet by itself in the same position. After subtracting the

magnetic field data from the scan of the ferro-fluid and magnet, we are given plot (d), which

can be seen as the effect of the ferro-fluid on the total magnetic field. As one can see, the

vectors point downward and inward, opposite that of the field in plot (c). This makes sense,

as the fluid is redirecting some of the magnetic field from our sensor, effectively causing it to

read a smaller B field magnitude. Since the magnitude is smaller with the ferro fluid above

the magnet then without, the vectors describing the difference between them will appear to

point away from the direction the field is actually headed. This supports the fact that the

ferro-fluid is indeed, paramagnetic, with χm = 2.64.


(a)

(b)

FIG. 6: Figure (a) above shows a path in the xy plane 20 mm above the ferro-fluid. The

path has a 1 mm resolution. For this scan, we used a Hall probe in order to scan magnetic

fields of higher magnitude without saturating the sensor. The data from the scan can be

seen in plot (b). We see the magnitude of the z component of the magnetic field according

to Contours. The use of Hall Probes seems to be a key element in the progression of this

project as we begin to scan stronger magnetic fields.

IV. CONCLUSION

We have found by mapping the magnetic fields of a Helmholtz coil, permanent magnet

(magnetic dipole), and ferro-fluid with a permanent magnet that the models and expecta-

tions agree with what our data shows. For the Helmholtz coil we can see that the magnetic

field model prediction for a uniform magnetic field withing the coil yields a good approxima-


tion as we found that the magnetic field inside the coil is nearly uniform. For the magnetic

dipole we found that the model was in good agreeance with the data from the permanent

magnet by inspection (Note: We did not perform a quantitative analysis of the magnetic

field.). Qualitatively analyzing the data from the ferro-fluid and permanent magnet system

we found that the ferro-fluids effect on the magnetic field was to internalize the field and

greatly reduce the amount of magnetic field outside the ferro-fluid. This agrees with the

expectations of the effect of the ferro-fluid as the fluid is paramagnetic and will cause the

magnetic field in the matter (ferro-fluid) to increase by superposition reducing the overall

field outside the fluid. Overall, we have seen that the model of how to go about three di-

mensionally mapping the magnetic fields of systems must be followed closely and carefully

or it will not work.

[1] Arrott, A.S., and T.L. Templeton. “Physical Measurements in a Permanent Magnet Field

Varying Spatially.” Journal of Applied Physics 99 (2005)

[2] Berger, Patricia, Nicholas B. Adelman, Katie J. Beckman, Dean J. Campbell, Arthur B. Ellis,

and George C. Lisensky. “Preparation and Properties of an Aqueous Ferrofluid.” Journal of

Chemical Education 76.7 (1999)

[3] Jackson, John David. “Chapter 5.” Classical Electrodynamics. New York: Wiley, 1962. 180-83.

[4] Rosa, D. and Stenton, D.. “Electric and Magnetic Field Mapping.” The Physics Teacher 35,

136 (1997).

[5] Sander, M. “Novel Pulsed Magnetization Process for Cryo-permanent Magnets.” Physica C:

Superconductivity 392-396 (2003): 704-08. ScienceDirect. Web. 28 Sept. 2014.

[6] Scherer, C., and A. M. Figueiredo Neto. “Ferrofluids: Properties and Applications.” Brazilian

Journal of Physics 35.3a (2005): 718-27.

[7] Wang, Jin, Guofeng Li, Ke Liang, and Xianhu Gao. “The Theory Of Field Parameters For

Helmholtz Coil.” Modern Physics Letters B 24.02 (2010)

[8] “3-Axis Digital Compass IC HMC5883L.” Honeywell, Feb. 2013. Web. Feb. 2015.

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Magnetic Field Mapping - Wabash · PDF fileMagnetic Field Mapping Badger, Bradon, Daron, Jonathan E., ... Rdimensional eld of the Helmholtz coil and it looks to be pretty constant