Determination of the Earth's Magnetic Field With the Use of Electromagnetism and Vector Analysis

Determination of the Earth’s Magnetic Field with the use of Electromagnetism and Vector Analysis J. Hernandez, A. Orden, M. Perez and J. Trinidad National Institute of Physics, University of the Philippines Diliman, Diliman, Quezon City

Abstract The magnetic field of the earth can be decomposed into its vertical and horizontal components, where the vertical causes the magnet to dip below the horizontal surface and the horizontal which is primarily the directional force of the total magnetic field. The experiment aims to calculate the horizontal component by inducing a magnetic field in a solenoid and taking the angle of deflection of a compass in the center of the solenoid, and relate the trigonometric tangent with the current values. The horizontal component of the magnetic field was calculated to be 5.09 x 10-5 T, which deviates by about 30% from the theoretical data produced from the National Geophysical Data Center of America.

1. Introduction

Magnetism is not limited to small-scale objects such as permanent magnets and solenoids. It is also present on very large scales – a concrete example of which is the Earth. The most commonly accepted explanation for the source of magnetism of the Earth is the circulation of electric current in the molten metals of the outer core. The properties of the outer core (iron at extremely high temperatures) by itself do not produce a magnetic field. However, since the Earth rotates around its axis, the molten metals have a circulating electric current which then correspondingly produce a magnetic field with a magnitude of 0.3 to 0.6 G, depending on the surface of the Earth where the field was measured [1]. Upon inspection of the magnetic field lines of the Earth, one can see that there are areas wherein the field is parallel or perpendicular to the Earth’s surface. Imagine that a freely-suspended bar magnet is suspended on the surface of the Earth. This bar magnet would tend to adjust such as to align itself parallel to the Earth’s magnetic field. Thus, when the field is perpendicular to the surface, the bar magnet would point down. On the other hand, when the field is parallel to the surface, the bar magnet would not point down.

Figure 1. A graphic representation of the magnetic field lines of the Earth [2]. The magnetic field lines of the Earth are perpendicular to the North and South poles and parallel near the equator. The magnetic field of the Earth can essentially be likened to a bar magnet located inside a sphere [1].

The angle between the axis of a bar magnet and the horizontal surface of the Earth is referred to as the magnetic dip or magnetic inclination. The magnetic inclination is zero (i.e. field lines are parallel to surface) near the equator and 90° (i.e. field lines are perpendicular to the surface) on the poles. This then indicates that the field can be decomposed into two separate components: a vertical component, which causes a magnet to dip below the horizontal surface, and a horizontal component, which is the primary directional force of the total magnetic field [3].

Previous experiments in measuring the magnetic field of the Earth used an external magnetic field to detect the deflection in a compass [4, 5]. The main concepts involved in the experiment are as follows: the magnetic field of the Earth Be can be related to its horizontal component Bh; the introduction of a perpendicular external magnetic field Bx to a compass which indicates Bh will cause the compass to deflect at an angle θx which can then be related with Bx to obtain Bh; if Bh and the magnetic inclination θi in the vicinity is known, then Bh can be used to compute for Be. Figures 2 and 3 as well as equations (1), (2), (3) and (4) further illustrate these concepts.

Figure 2. A graphical representation of the relationship between the Earth’s magnetic field Be, the horizontal component of the Earth’s magnetic field Bh and the magnetic inclination θi [4].

Upon inspection, it can then be stated that Be and Bh are related by ,

𝐵 =

(1)

where θi is the magnetic inclination of Be.

Figure 3. A graphical representation of the relationship between the horizontal component of the Earth’s magnetic field Bh, the external magnetic field Bx and the deflection angle θx [4].

Furthermore, should a known external magnetic field Bx act perpendicularly to Bh, a compass needle in the vicinity of Bx will deflect an angle θx away from magnetic south. Upon inspection, it can then be stated that Bh and Bx are related by .

𝐵 =

(2)

Using a wounded coil of N turns and diameter D as the source of Bx, the magnitude of Bx at the center of the coil is 𝐵 =

.

(3)

where μ0 = 4π×10−7 (V·s)/(A·m) [4]. The experiment done by the researchers consisted of a variation in the method found in the literature; the researchers approximated equation (1). Due to the unavailability of data for θi and the relative proximity of the Philippines to the equator, equation (1) can be expressed as 𝐵 ≈𝐵 ,

(4)

since θi ≈ 0 near the equator. Through solving equation (3) and subsequently substituting equation (3) into equation (2), the relationship stated in equation (4) would approximately yield Be.

The experiment is mainly limited by: the validity of the results only in the vicinity of the venue of the experiment (National Institute of Physics, University of the Philippines Diliman, Diliman, Quezon City); and the exclusion of the magnetic inclination in the computation. The objectives of the experiment are: to apply concepts from electromagnetism and vector analysis in determining the Earth’s magnetic field in the vicinity; to effectively utilize error analysis and data analysis tools in accurately computing the Earth’s magnetic field; and to compare the computed magnetic field with the theoretical magnetic field of the Earth.

2. Methodology A solenoid was made by wounding nine loops of cooper wire with a radius of 9.5 cm. After which, the south magnetic pole of the earth was identified using a compass. The solenoid was then placed in line with the north direction the compass points with its surface perpendicular to the surface of the table. The compass was then placed in the center of the solenoid using a cardboard. The solenoid was then connected to a power supply to induce magnetic field in the center of the solenoid. A resistor was used to obtain data for small angles, 𝜃 < 30°. The data collected is the magnitude of the current along with its corresponding angle deflection of the compass due to the magnetic field induced in the center of the solenoid. A linear regression was then used to further describe the outcome and observations.

Figure 4. The diagram on the left (circuit diagram) and on the right (top view of the set-up of the wound coil and the compass) illustrate the proper set-up for the experiment. At the start of the experiment, the compass must point to the north and the wound coil must be in line with the compass. By doing this, the plane of the wound coil is parallel (i.e. Bx is perpendicular to Bh) to the horizontal component of the Earth’s magnetic field [5].

3. Results and Discussion

Recalling the magnetic field formula for solenoid, or equation (3), 𝐵 =

𝑁

𝐼.

The direction of deflection of the compass needle is also the direction of the vector sum of magnetic fields acting on the needle. Assuming that only the Earth’s and the solenoid’s magnetic fields are acting on the needle, we have equation (2) which is 𝑡𝑎𝑛 𝜃 = . where Bh is the horizontal intensity of Earth’s magnetic field and θx is the angle between the compass needle and the magnetic north-south line. Substituting equation (3) into equation (2) makes a linear equation: =[

𝑡𝑎𝑛 𝜃

] 𝐼.

(5)

In equation (5), tan (θx) and I are the dependent and independent variables respectively; the remaining factors are constant variables. The combination of these constants can be written as a single constant λ, where 𝜆=

.

(6)

λ can then be substituted into equation (5) 𝑡𝑎𝑛 𝜃

= 𝜆𝐼.

(7)

Therefore, we have a more explicit linear equation where λ is the slope. Table 1 shows gathered data for values of the angle of deflection as well as their corresponding trigonometric tangents and current values used.

Table 1. Summary of corresponding values for the experimental data obtained. Angle of deflection (degrees) 10 20 30 40 50 60 70 80

Trigo. tangents of angles of deflections 0.18 0.36 0.58 0.84 1.2 1.7 2.7 5.7

Current (A) 0.050 0.11 0.22 0.40 0.54 0.70 1.2 2.4

used

Current vs. trigonometric tangent Trigonometric tangent

6 5 4

y = 2.3419x + 0.026 R² = 0.9983

3 2 1 0

0

0.5

1

1.5 Current (A)

2

2.5

3

Figure 5. The graph for the current vs. the trigonometric tangent has an R 2 value of 0.9983.

Using Microsoft Excel, we can plot these data in a current vs. trigonometric tangent coordinate system. Figure 1 shows the graph of these data with a linear trend line, y = mx + b, where y and x correspond to tan (θx) and I, respectively. The linear trend line has a R2 value equal to 0.998. Thus, the angle of inclination’s trigonometric tangent is evidently linearly dependent to the current used. The equation generated by the used linear regression is (8)

𝑦 = 2.341𝑥 + 0.026. Comparing equation (8) to equation (7), the researchers obtained tan 𝜃

= 2.341𝐼 + 0.026.

(9)

From the equation of the linear regression, the slope is equal to 2.341. Note that the generated y-intercept, which is equal to 0.026, is an experimental error (equation (7) has no y-intercept). Since λ is the slope of equation (7), then λ = 2.341. Using equation (6), one can see that 𝜆 = 2.341 =

,𝑒 𝑝

.

(10)

The experiment used N = 9 turns and D = 0.095 m. Isolating unknown variable for Bh, exp, we get 𝐵

,

𝑝

=

𝜆

=

(4π x 10−7 ) 9 2.341 0.095

= 5.09 × 10−5 𝑇.

(11)

Figure 5 shows the Earth’s magnetic field at Quezon City (latitude: 14o 39’ 0’’ N, longitude: 121o 2’ 0’’ E). Comparing Bh,exp to the literature value for the horizontal intensity of Earth’s magnetic field, 3.91427 x 10-5 T [6], the experimental value has a percent error equal to %𝑒𝑟𝑟𝑜𝑟 =

|

,𝑙 𝑡 − ,𝑙 𝑡

,𝑒 𝑝 |

=

|3.91427 x 10−5 T−5.09×10−5 𝑇| 3.91427 x 10−5 T

× 100% = 30.04%.

(12)

Such a large error can be attributed to the rapid rate of deflection with respect to current, and the small size of

the compass, leading to imprecise readings. Incidentally, the reason that the measurements vary only from 10 to 80 degrees is a steep rise in the current required to increase the angle of deflection past 80 degrees. Even raising the angle to 89.9 degrees would require a current increase of over 9000%, due to the nature of vector addition and the tangent function.

4. Summary, Conclusion and Recommendations

The calculated horizontal component of the magnetic field of the earth is 5.09 x 10-5 which deviates about 30% from the theoretical value of 3.91 x 10-5. This was expected due to relatively small values involved, while the devices used were not very specific. In order to increase the precision of measurements: increase the number of trials, decrease the rate of current and induced magnetic field by decreasing the number of loops or increasing the loop size, decrease the rate of current and voltage by adding resistance to the circuit, or use a larger compass to make deflection measurements more precise.

References 1. 2.

3.

4.

5.

6.

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