Magnetic field of the earth
I. Objective: Measure the magnetic field of the earth
II. Equipment. Magnetic compass, magnetic dip compass , Helmholtz coils, HP 6212 A
power supply, Keithley model 169 multimeter
III Introduction.
IIIa. Magnetic field of the earth. The earth behaves like a large magnet which creates a
magnetic field around the planet. This large magnet has its south pole approximately at
the geographic north pole of the earth. A schematic of the earth magnetic field is given
in Fig.1. The magnetic field lines emerge from the south geographic pole, surround the
earth, and sink into the earth north geographic pole. The red arrow indicated the
direction of the magnetic field at a location in the north hemisphere. The presence of the
earth magnetic field is attributed to the presence of a molten iron core at the center of the
earth. From Fig.1 it can be seen that the magnetic field can be decomposed into a
horizontal and a vertical component at any location on the earth such as Buffalo.
In Fig.2 we show the top view of a magnetic compass. This is a permanent magnet in
the form of a needle that can rotate about a vertical axis. The magnetic needle will align
itself and become parallel with the horizontal component of the earth magnetic field Bhe .
Thus the axis of the needle points along the south-north axis. The edge of the needle
marked “N” points to the geographic north pole of the earth. This property was
discovered by the Chinese, was used for navigation and contributed to the great
discoveries of the 15th century. A second type of compass is called a “dip needle” and it
is a permanent magnet in the form of a needle that can rotate around a horizontal axis
which is perpendicular to the earth magnetic field. The side view of such a dip needle is
shown in Fig.3. Under these circumstances the dip needle with align itself with earth
magnetic field Be as shown in Fig.3. The angle  between the magnetic field Be and the
horizontal is known as “field inclination” or more simply as “dip angle”. In Fig.4 we
show a schematic of the decomposition of the earth magnetic field Be into a horizontal
component Bhe and a vertical component Bve
IIIb. Helmholtz coils. These are shown in Fig.5. Each has a diameter D and consists
of N turns of wire. They are positioned so that they have a common axis and the
distance between the coil centers is adjusted so that it is equal to half the diameter D .
Under these conditions the Helmholtz coils generate a magnetic field along the common
axis of the coils which is uniform in the vicinity of the midpoint between the coil centers.
The magnetic field B is given by the equation:
B
160 Ni
D 125
(eqs.1)
1
Here the current i is measured in Amps, B is measured in tesla, D  0.136 m, N  320 ,
and 0 is a constant equal to 1.256 106 T.m/A
If we substitute the numerical values for 0 , D , and N we get the equation:
B  in Tesla   4.23 106  i  in mA 
(eqs.2)
The Tesla is a very large unit, much larger than the magnetic field of the earth. For this
reason in this experiment we will use the smaller unit of Gauss bearing mind that 1 Tesla
= 10,000 Gauss
B  in Gauss   4.23 102  i  in mA 
(eqs.3)
IIIc Measurement of the horizontal component Bhe of the earth magnetic field and
the total magnetic field of the earth Be .
The setup for the measurement of Bhe is shown in Fig.6. A magnetic needle is placed at
the center of a pair of Helmholtz coils. The Helmholtz coils are connected to a power
supply which provides the current. The coils current i is measured by the ammeter in the
circuit. The Helmholtz coils are oriented so that the planes of the two coils are parallel to
the north-south direction. The magnetic field Bcoil generated by the Helmholtz coils and
the horizontal component of the earth’s magnetic field Bhe , as well as the net magnetic
field Bnet are shown in Fig.6. These fields are also shown separately in Fig.7. The
magnetic needle at the center of the Helmholtz coils always aligns its axis with the net
magnetic field Bnet . When the coils current is zero we have that Bcoil  0 and thus
Bnet  Bhe . As a result, the magnetic needle is aligned along the north-south direction. If
we pass a current i through the coils the needle will align its axis with Bnet and the needle
axis will now form an angle  with the north-south direction , as shown in Fig.6.
B
If we refer to Fig. 7 we have that: tan   coil . If we solve this equation for Bhe we get:
Bhe
Bhe 
Bcoil
tan 
(eqs.4)
Because equation 4 involves the tangent of the angle  , the setup of Fig.6 is known as
B
“the tangent compass” . Refer now to Figure 3. In this case we have: cos   he , where
Be
Be is the net magnetic field of the earth. If we solve this equation for Be we get:
2
Be 
Bhe
cos 
(eqs.5)
IV. Procedure
IV1. Align the planes of the Helmholtz coils so that they are parallel to the north-south
direction as shown in Fig.6.
IV2. With zero current in the coils, rotate the case of the compass so that the needle point
to the zero reading.
IV3. In this section we will measure the average coil current that results in a deflection of
30 degrees.
a. Choose the current polarity so that the needle deflects to the right by 30 degrees. Bring
the coil current to zero. Repeat 5 more times. Record your data in table 1
b. Reverse the current direction so that the needle deflects to the left by 30 degrees. Bring
the coil current to zero. Repeat 5 more times. Record your data in table 2.
IV4. In this section we will measure the average coil current that results in a deflection of
60 degrees
a. Choose the current polarity so that the needle deflects to the right by 60 degrees. Bring
the coil current to zero. Repeat 5 more times. Record your data in table 3
b. Reverse the current direction so that the needle deflects to the left by 60 degrees. Bring
the coil current to zero. Repeat 5 more times. Record your data in table 4.
IV5. Align the dip needle so that it points along the north-south direction. Measure the
dip angle  .
V. For the report
V1. From tables 1 and 2 find the average value of the current i that produces a deflection
of 30 degrees. Use equation 3 to calculate the coil magnetic field Bcoil . Use equation 4
to determine the horizontal component of the earth’s magnetic field Bhe .
V2. From tables 3 and 4 find the average value of the current i that produces a deflection
of 60 degrees. Use equation 3 to calculate the coil magnetic field Bcoil . Use equation 4
to determine the horizontal component of the earth’s magnetic field Bhe .
V3. Use equation 5 to determine the net magnetic field of the earth Be .
3
Fig. 1
Be
Fig.2
N
Bhe
S
Magnetic compass
(top view)
4
Fig.3
S

Be
horizontal
Dip magnetic needle
(side view)
N
Fig.4
Be
Bve

Bhe
5
Figure 5
Fig.6
north
Bnet
(top view)
θ
Helmholtz
coil
Helmholtz
coil
Bhe θ
needle
Bcoil
south
Power supply
Ammeter
6
Fig.7
Bnet
Bhe

Bcoil
Top view
7