Magnetic Field ILAP
for Calculus III
(MATH 2421, Section 00x) [Modify your section #.]
Spring 2007
[Your name here.]
Introduction
“ILAP” stands for “Interdisciplinary Lively Application Project”. In this case, the
“Interdisciplinary” portion is taken from Physics. This project will focus on the vector
field generated by a magnetic dipole (bar magnet). We will sample the Earth’s magnetic
field here at UCDHSC, and then measure the magnetic field associated with a relatively
strong permanent magnet. Finally, we will compare our measurements to the theoretical
model.
The Earth’s Magnetic Field
According to historical measurements, the magnetic field which can be measured near the
Earth’s surface is primarily due to its molten iron core. Effectively, there is a large bar
magnet running through the center of the Earth. Oddly enough, the magnetic North Pole
(of the bar magnet) roughly corresponds to the Earth’s geographic South Pole. We note
that magnetic field lines (force lines) always form closed loops and that these lines
typically exist inside the bar magnet.
[Find a sketch of how the field lines are situated on the Earth with respect to its magnetic
core. Include it as an insert and reference it here. This will be Insert #1.]
The field lines generally run from the magnetic North Pole to the magnetic South Pole.
Some field lines form closed loops relatively far away from both magnetic poles. This
type of loop is called an eddy.
When we measure the Earth’s magnetic (B) field near the Earth’s surface, the field
vectors often have large normal components (with respect to the Earth’s surface) and a
small parallel component. That small parallel component is still strong enough to
influence the orientation of a magnetized compass needle!
[Cite some specific places on the Earth where compasses historically go crazy due to this
feature.]
Measuring the Earth’s Magnetic Field
The unit of magnetic field strength is the tesla (T). Our measuring device, the
magnetometer, gives us the value of (B dot n) where n is the unit vector normal to the
surface of the detector plane. If the magnetometer gives us a positive reading, then the B
field vector is pointing out of the plane of the detector plane surface. A negative reading
means that the B field vector is pointing into the plane. Ideally, we would like the
magnetometer to read zero whenever the B field vector is parallel to the plane.
We “zeroed” out the magnetometer using the following procedure. Remember that the
magnetometer can only give us magnitude information. We must keep track of the
direction angles associated with the B field vectors.
a) Turn on the calculator and start the EasyData application with the magnetometer
probe plugged into the interface. For this portion of the experiment, select
“0.3 mT” as the range on the probe (small black switch).
b) On a non-metallic flat surface, turn the detector surface until it gives the
maximum (positive) value. Compare the direction angle with the angle associated
with the magnetic South Pole (the red needle on the compass). This is
approximately geographic North.
Our best reading pointed almost directly to the magnetic South Pole – perhaps off
by 2 degrees at most. The reading was 0.022 mT.
c) Turn the detector surface in the opposite direction and verify that the minimum
value is achieved there.
Yes, it worked. The reading in the opposite direction was -0.031 mT.
d) Point the detector so that the direction angle is perpendicular (90 degrees off)
from either pole. Zero the magnetometer on the calculator application.
e) Carefully point the surface of the detector downward and try to find the angle
which maximized the reading. Since we are in the northern hemisphere, the
magnetic field vectors here are on eddy field lines and thus, we should expect the
direction of the B field vector to point into the ground.
f) Here is a diagram which shows which way the probe was pointed when the
maximum reading of 0.049 mT (in South Classroom 209) was achieved:
[Insert a picture of the right triangle on your engineering pad paper. It should be
labeled with the magnitude of the hypotenuse equal to 0.049 mT and the
magnitude of the horizontal leg equal to 0.023 mT. State that the horizontal
reading (after the zeroing) was 0.023 mT. Use trigonometry to find the angle of
depression and label that on your diagram also. This will be Insert #2.]
Magnetic Dipole
If a bar magnet is centered at the origin and fixed on the x-axis, then the theoretical
model for the field vectors is
B( x, y ) 
k
(3(m  r̂)r̂ - m),
r3
where m is the dipole moment vector, and
r̂ 
x
x y
2
,
y
x y
2
and r  x  y 2 .
In the diagram on Insert #3, the dipole moment is m = 1i + 0j and k = 1.
The equation of the family of field lines is derived on Insert #4.
[Neatly write up all the of Calc. II items for solving the first-order differential equation
on one page of engineering pad paper. Your final answer should be that lovely polar
equation cited in the handout. This will be Insert #4.]
Measurements
The curious thing about the magnetic dipole field (in this particular configuration) is that
if m = mi + 0j, then the constant m can be factored out in front of the remaining vector
expression. Thus, all magnetic dipole fields are equivalent, except for strength. The field
lines should always match, but the magnitudes of the vectors along those field lines
depend on m. Thus, we realize that we cannot determine the value of m by itself, without
some absolute calibration value for k. We need only take a few measurements to estimate
the value of mk.
In our demonstration, we measure x and y in meters and the magnitude of B in teslas.
a) On the magnetometer probe, we change the range setting switch to “6.4 mT”.
The strength of the permanent magnet’s field is much stronger than the Earth’s
magnetic field here in Denver. Zero the detector using the same procedure as
before.
b) Tape down the magnet on the x-axis of the worksheet. Center it at the origin.
c) The denomonstrator will choose a few (x,y)-coordinates in Quadrant I. Measure
the strength of the magnetic field by rotating the probe surface until you obtain
the largest magnitude (negative or positive is okay) reading. The maximum
reading should be no larger than 3.0 mT. Anything stronger than that is suspect.
The maximum semi-accurate reading is 6.4 mT. At that level, however, the errors
can easily approach 20%! For lower readings, the typical error is about 5%.
d) The data is presented in spreadsheet form on Insert #5.
For example, when we choose the point (x,y) = (0.07, 0.02), the maximum
reading at the point is about +2.00 mT = 0.002 T. Since the maximum B-value at
that point should be mk times the column marked “Quotient”, our point estimate
for mk is located in the column marked [complete this!].
e) Our best estimate for mk is approximately [fill this in] m/T.
Conclusions
We address some interesting queries:
a) Based on the theoretical model of the magnetic field of a dipole, should every
closed field line (loop) pass through the origin? [Look at the polar equation…]
b) Since more accurate measurements confirm that most permanent magnets have
closed field loops which do not pass through the origin (eddy loops) should we
discard the dipole model completely? [The answer is NO, of course. You do not
need to show this graphically, but you should have some idea what causes this. It
is NOT because the magnetic dipole moment increases.]
c) We have shown that for the region where we measure the strength of permanent
magnet, the accuracy of the model probably falls within the error range of the
magnetometer readings. In which regions (in Quadrant I) would you say the
readings would be most chaotic/difficult to measure B? [Consider the fractions
involved. What happens when sqrt(x^2 + y^2) is a small number?]