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Transcript
Measurement of the Horizontal Component (H) of Earth's Magnetic Field
Dr. Tim Niiler, WCU
based on lab by Dr. Harold Skelton
Background
The Earth's magnetic field is of
interest to scientists due to its
interaction with the Sun, its ability to
protect us from harmful
extraterrestrial radiation, its effect on
compasses, and many other reasons.
The magnitude of the field depends
greatly on solar activity. When solar
activity is high, the field lines are
compressed and the field strength
increases. The Earth's field is three- Illustration 1: Earth's magnetosphere. Image
dimensional: magnetometer stations under GPL license from Wikipedia.org.
about the globe typically report on the
declination, the inclination (or dip angle), and horizontal component of the field.
The declination, D, is the angle the field varies from true north. The inclination,
I, is the angle (downwards) from the horizontal, and the horizontal component,
H, is the projection of the field on the surface of the Earth. Today, we will be
trying to measure the horizontal component of the Earth's magnetic field at our
latitude and compare it to data from a nearby geomagnetic observatory.
Theory
Unlike our professional counterparts, we will not be using a magnetometer to
determine H. Rather, we will be using a combination of two methods involving
torque balance and harmonic motion. The strategy will be to determine H, and
/H separately, and then by solving the two simultaneous equations, be able to
determine H of Earth, and (by extension), the magnetic moment of the magnet.
This section will present some of the background theory behind the experiment.
a) Determining H, the torque on the magnet
In this section of the lab, a cylindrical magnet will be allowed to dangle
from a string and oscillate in the H component of Earth's field. This field creates
a torque on the hanging magnet:
1)
=I =− H sin 
where I is the moment of inertia of the magnet,  is the angular acceleration,  is
the magnetic moment of the magnet, and  is the angle the magnet makes with
respect to H. This is, in actuality a differential equation:
d2
2) I 2 =− H sin 
dt
In the case of small oscillations, sin≅, so this equation simplifies (with some
rearrangement to:
3)
I
d2 
 H =0
dt 2
or
4)
d2   H

=0
I
dt 2
Presuming that  = osint, we find that:
5)
d
= 0 cos t
dt
and
6)
d2 
=−2 0 sin  t
2
dt
Substitution of equation 6) into equation 4) gives us:
7)
− 2 0 sin  t 
H
 sin  t =0
I 0
For this to be true:
8)
2
 =
H
I
The quantity  is the angular frequency which is related to the period of
oscillation by:
9)
T=
2

By combination of equations 8) and 9), one can see how to determine the torque,
H:
10)
 
2
 B= I
T
2
The period of oscillation can be easily measured. The moment of inertia of the
cylindrical magnet is given by:
11)
I=
1
1
mL 2 mR2
12
4
where m is the mass of the magnet, L is its length, and R is its radius.
b) Determining /H
Using torque balance, it is possible to determine the ratio /H. At a large
distance from the magnet, its magnetic field may be approximated by that of the
axial field of a current loop:
12)
B=
0 
2 z
3
where  is the magnetic moment, and z is the distance (on axis) from the center
of the magnet.
Illustration 2: Setup for part b) of experiment. The distance, z, is from the
center of the magnet to the center of the compass.
The torque from H on the compass needle is:
13)
1=compass H sin 
The torque from the magnet on the compass needle is:
14) 2=−compass B cos
If the needle is in stasis the torques balance:
15)
compass H sin −compass B cos=0
The magnetic moment of the compass cancels, and with some rearrangement we
have:
16)
Hsin =Bcos 
Substituting equation 12) in for B in equation 16) we get:
17)
Hsin =
0 
2  z3
cos 
or solving for /H:
18)
 2 3
=
z tan 
H 0
Procedures
For part a) suspend a magnet from a string tied at its center and record the time
it takes for it to oscillate 10 times. Repeat this 10 times and calculate the
average period of oscillation. Your uncertainty in period will be the standard
deviation of this result. Use equation 10) to calculate H.
For part b) arrange a compass on the center of a ruler such that when no magnet
is near, magnetic north is at right angles to the ruler. Start with the magnet at
15 cm from the center of the compass, and pull the magnet back from the
compass in 5 cm increments until your magnet is 40cm away. Repeat this on the
far side of the compass. Using a plot of tangent of the angle (y axis) vs. z-3 (x
axis), determine the slope of this graph [*** Important! If you plot this
incorrectly, your results will be wildly inaccurate! ***]. The slope of your graph
is equal to:
19)
slope=
0 
2 H
Use this result to calculate /H. Then using your results from part a) as well,
solve for  and H individually. Estimate your uncertainty in these values.
Analysis
In addition to the above calculations, you should compare your values to the
current Fredricksburg value for H (http://geomag.usgs.gov/wwwplots/frdt.gif )
and save this graph. Read off the value that corresponds to the time you
collected your data. Your lab should include this graph with a demarcation of
what value of H you are using. This is your gold standard. Is this value of H
within your experimental uncertainty? What sources of error contribute to
inaccurate results. DO NOT STATE HUMAN ERROR OR CALCULATION
ERROR!