Download Magnetic Induction Field of the Earth

MAGNETIC INDUCTION FIELD OF THE EARTH
The earth possesses a magnetic field, as anyone who has used a compass knows. Although
the existence of the field is no doubt related to the presence of ferromagnetic iron in the earth's core,
the mechanism by which the field arises is imperfectly understood. The symmetry of the field is
similar to that of a bar magnet or to the field generated by current flowing in a circular coil. the
object of this experiment is to measure the magnitude the direction of the earth's field at one station
in the laboratory. The device used is called a rotating coil magnetometer and is based on the
principle of electromagnetic induction.
Theory and Apparatus
Definitions
By definition, the direction of a magnetic induction field, B, is the direction in which the
north pole of a freely-suspended compass needle points when at rest in the field. Thus B is a vector
quantity. The magnitude or strength of the field is defined in terms of the force which the field
exerts on a charged particle moving in it. This definition, however, is rather awkward to apply in
actual measurements of B. In this experiment, B will be obtained from a measurement of the emf
induced in a rotating coil.
From experiments with magnetic compasses and magnetometers, the large scale symmetry
of the earth's field is known to resemble the sketch below in Fig. 1.
At some arbitrary location on the earth's surface B has both a horizontal and a vertical
component as shown in Fig. 2. The angle % which B makes with the horizontal is called the angle
of dip. At any point on the equator the field is horizontal (i.e., Bv = 0). At the poles it is vertical
(i.e., Bh = 0).
The Magnetometer and Principles of Operation
Magnetic field will be measured with a device called a rotating coil magnetometer. Examine
the instrument and identify the two large Helmholtz coils - coils C - and the smaller,
centrally-located search coil - coil S. Coil S is rotatable, being attached to a shaft.
Fig. 1 Symmetry of earth's
magnetic field.
Fig. 2 Angle of dip of earth's
magnetic field.
driven by a small electric motor. The magnetometer you are supplied with is only for student use,
having much larger dimensions than the ones manufactured commercially, but the principles are the
same.
When a coil of wire, say coil S, is placed in a magnetic field of strength B, say the earth's
field, an emf can be induced in the coil by changing the flux through the coil. If iB is the magnetic
flux through the coil, Faraday's law gives the induced emf as
d ÆB
,
e=dt
The flux can be changed in a number of different ways. Two are of interest to us:
1.
2.
by varying the magnetic induction, B, with time, when the coil is held stationary, and
by rotating the coil about an appropriate axis when the field is constant.
Method 1 is employed in the experiment, "Magnetic Induction Field of a Circular Coil". Method 2
is used here.
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Let Fig. 3 on the next page represent coil S. Suppose B is perpendicular to the axis of
rotation, and at an angle q with respect to the normal to the plane of the coil. Then by
definition, the magnetic flux through the coil is given
by the area of the coil and B is the absolute value of
the field strength. For a coil with n turns of wire,
Æ B = nBa cos q,
Fig. 3 The Rotating Coil.
If the motor rotates the coil at an angular velocity/frequency of ω rad sec.-1, then θ = ωt and
from eqs (1) and (2),
e(t) = naBw sin w t = e o sin w t
This emf, appearing across the terminals labelled `rotating coil' on the magnetometer, can be
measured. It is large enough not to require amplification.
Note that the apparatus can be laid flat or tipped up on end. This enables the search coil to
be rotated about either a horizontal or a vertical axis - and hence sample either Bv or Bh.
Peak-to-Peak and RMS Detection
For the most part, the emf will be measured with the multimeter. But an oscilloscope can
also be used. When the oscilloscope is used, peak-to-peak (pp) values are obtained as shown
below in Fig. 4. The multimeter, however,
displays a root-mean-square (rms) value. This
is because when an AC voltage is applied to
the multimeter set to the AC volts
mode the signal is first rectified and then
filtered. The result is a DC voltage equal to
the root-mean-square of the AC input.
Fig. 4 Relationship between sinusoidal
signal and its DC root-mean-square.
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That is,
e rms =
1
2 2
e pp
and
e pp = 2 e o = 2 2 e rms ,
The Oscilloscope - An Introduction
Fig. 4 shows a front view of the Philips model PM3232 oscilloscope used in this experiment.
It is a dual beam device, but you will use only one beam - beam B. As can be seen from the figure
there are too many controls for each one of them to be discussed separately here.
Fig. 5
Front view of Philips oscilloscope
The following familiarization routine is intended only for those students who have not used a scope
before and is designed to yield a Lissajous figure - a necessary condition for measuring frequency
in this experiment.
Before switching on, carry out the following checks:
1.
intensity control mid-range,
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2.
3.
4.
5.
6.
A-input and B-input lever switches in AC position,
A- and B-amplification controls fully counter-clockwise, variable controls in `cal' position,
A and B beam selectors `in'.
Time base control fully counter-clockwise, in `via YA' position,
X-position and YB-position midrange.
Now switch on the scope and allow 15 sec. for warm up.
7.
8.
9.
Apply the signal to be studied to the B-input and the standard signal (from the signal
generator) to the A-input. Note that the A-input signal provides the time base.
Rotate the A- and B-amplification controls clockwise until you observe a figure of useful
size.
Finally, vary the frequency of the standard signal until you obtain a Lissajous figure which
is recognizable.
Measuring Frequency
In this experiment the frequency of a signal is measured using the technique of Lissajous
figures. The signal whose frequency is to be measured is applied to the YB-input of the scope and
the output of a signal generator is applied to the YA-input (switched to time base). The two signals
thus superimposed at right angles produce a Lissajous figure. Fig. 6 shows two useful ones.
Obviously, if a circle is obtained, the signals are of the same frequency and the value can be read off
the dial of the signal generator.
The Experiment
Assembly and Trial Run
Begin by assembling the apparatus. Plug the motor which rotates coil S into the variac.
Connect the output of coil S - `Rotating Coil' - to the B-input of the scope and put a Philips
multimeter in parallel with the input. Connect the HP model 200AB signal generator to the A-input
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of the scope. Get the coil spinning at a modest rate and vary the frequency of the generator until you
obtain a Lissajous figure. if the controls on the scope are confusing, try following the familiarization
routine in the previous section.
The Lissajous figure is not always stationary. Why?
The Field to be Measured
Your job is to measure the magnitude and direction of the earth's magnetic field where your
apparatus happens to be located. To get a feeling for the direction of the field use a compass and dip
needle. What is the angle of dip? Note carefully the direction of Bh.
There are two methods by which the magnetometer can be used to measure the earth's field.
One is a direct method, the other, a null method.
Direct Method
Use your knowledge of the direction of Bh to orient the apparatus so as to sample Bh and Bv
separately. How must the apparatus be oriented so that only one component is sampled at a time?
Make sure to measure ε and ω (=2π f) simultaneously.
Null Method
Null methods generally yield more precise results than direct methods. In this null method
a DC current (from the U. of T. power supply) is passed through the pair of Helmholtz coils to
produce a magnetic field in opposition to the component of the earth's field under study. Once the
two fields are equal and opposite the output ε should read zero for any value of ω. The component
being measured must then equal the field produced by the coils at their geometrical centre, given by
8 m o NI
B=
(5)3/2 R
where
mo = 4π x 10-7 Weber Ampere-1 meter-1
N = number of turns in a single coil
I = current in the coils
R = radius of the coils
Measurements of Bh and Bv at Agincourt, Ontario, have yielded 0.154 and 0.563 x 10-4 Tesla,
respectively (Clark's Tables, 1965).
SQ78
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