The Giant Magnet: Measuring Earth's Magnetic Field
| Polish version is here |
The following article was originally published in the journal for educators Fizyka w Szkole (eng. Physics in School) (6/2017):
The phenomenon of magnetism is not uniform in its essence – we use this term to describe an entire set of physical phenomena related to the magnetic field. Such a field can be produced by magnetic materials, but also by the flow of electric current.
Based on their magnetic properties, all known substances, i.e., elements, chemical compounds, and composite materials, can be classified accordingly. In addition, every chemical element exhibits one of four fundamental types of magnetism: diamagnetism, paramagnetism, ferromagnetism, or ferrimagnetism.
In practice, ferromagnetic materials have found the greatest application; they can be divided into magnetically hard materials used as permanent magnets, soft materials used, for instance, in transformer cores, and semi-hard materials (magnetic data storage media).
There is no doubt that magnetic forces are among the fundamental interactions in nature. They occur via the magnetic field, which is generated on a macroscopic scale by the motion of electric charge carriers, e.g., during the flow of electric current. A constant electric current creates a magnetic field that does not change over time, whereas an alternating current produces time-varying magnetic and electric fields that are inextricably linked; together, they are called the electromagnetic field.
However, one might wonder what the source of the magnetic field is on the subatomic level.
On this scale, the field arises mainly from the orbital motion of electrons and their spin – the latter being dominant. The contribution of proton and neutron motion to the magnetic field can, in most cases, be neglected.
The resultant magnetic moment of an atom is the sum of all the magnetic moments of its electrons. Nature tends to seek the lowest possible energy state – therefore, individual magnetic moments (both spin and orbital) most often align in opposite directions. Naturally, this leads to the mutual cancellation of the magnetic moments of paired electrons. If all electrons in a given atom are paired, that atom does not exhibit an external magnetic moment. However, if an atom has a certain number of unpaired electrons, one can observe an external magnetic moment whose value depends predominantly on the number of these electrons [1].
Today, we know of many different sources of magnetic fields. One of the largest—and very close to us—is our planet: Earth.
Earth’s magnetic field occurs naturally inside and around the Earth. It can be approximated by the field of a magnetic dipole with one pole near the geographic North Pole and the other near the South Pole. The line connecting the geomagnetic poles is inclined by about 10° relative to the planet’s axis of rotation. Around Earth, the so-called magnetosphere extends tens of thousands of kilometers into space—this is the region where the Earth’s magnetic field is influential.
It should be remembered that, according to the adopted convention, the geomagnetic north pole is located near the Earth’s geographic South Pole and vice versa. We know that opposite magnetic poles attract each other, so the north end of the compass needle indeed points approximately to the geographic North Pole—that is, simultaneously the geomagnetic south pole.
In the past, it was believed that Earth’s magnetic field originated from the magnetization of deep layers of rock containing iron ores (e.g., magnetite). However, at the beginning of the 20th century, Pierre Curie demonstrated that all known substances lose their ferromagnetic properties upon heating them above a certain temperature. This temperature is characteristic of each ferromagnetic substance and has been named the Curie temperature [2]. Since the temperature of Earth’s interior is much higher than the Curie temperatures of the substances it is composed of, magnetization cannot be responsible for the magnetic field of the entire planet.
Today, it is believed that Earth’s magnetic field is most likely generated by swirling electric currents flowing in the Earth’s metallic core—the so-called self-exciting dynamo or geodynamo theory. Convection occurring in the Earth’s liquid core could be the driving force of this geodynamo.
The intensity of Earth’s magnetic field was already measured by Carl Friedrich Gauss in the first half of the 19th century. Later measurements showed that our planet’s magnetic field is dynamic—its parameters change slowly but continuously.
I believe our Dear Reader will not mind trying to measure the Earth’s magnetic flux density themselves. For that purpose, one will have to build a simple measuring device.
Apparatus for Measuring the Earth’s Magnetic Flux Density
The device is not complicated to build and can be made from commonly available materials. Its construction is schematically presented in two perpendicular projections in Fig.1.
The apparatus consists of a magnetic needle a (taken from any compass) placed on a small column b mounted on a base d. Two ring-shaped coils c, each made of a few dozen turns of thin insulated wire, are also mounted on the base. It is important that both coils have the same dimensions and that they are placed at a distance equal to their radius r. The magnetic needle must be located on the line connecting the centers of both coils, exactly halfway between them. This winding arrangement is called a Helmholtz coil.
Aside from the magnetic needle, no part of the apparatus can be made of ferromagnetic materials.
The coil size should be appropriately larger than the magnetic needle used. Round plastic lids, such as those that close coffee cans, work well as coil forms (see Photo 1).
The interior of these two lids must, of course, be cut out so that only the rim remains (see Photo 2).
Next, wind, for example, 20 turns of 0.2 mm (32 gauge AWG) diameter enamelled copper wire around the rim of each cut-out lid. The winding must be very precise, i.e., turn by turn – for both coils in the same direction (see Photo 3).
The number of turns is not critical—but it must be the same for both coils. You will need this number (denoted as n) for further calculations. Secure the winding against unwinding by coating it with a clear varnish.
The base of the device can be made from another large plastic screw cap or lid. Mark the center point where the column for the magnetic needle will be placed, and then cut out holes on both sides where the previously made coils will be inserted (Photo 4).
Next, glue the coils into the holes and connect them in series, i.e., the end of the first coil to the beginning of the second. Make the connections beneath the base, and lead the free ends of both coils outside. You can make the column from a piece of plastic tube, such as one from a ballpoint pen. Glue it into the opening in the base, and place a pivot for the magnetic needle at its top—this could be a small pin on which the needle can rotate freely. Keep in mind that you must choose the height of the column so that the needle is located at the geometric center of the two coils (Photo 5).
The final step is to mount an angular scale (e.g., printed and affixed to a cardboard disc) on the column and then attach the magnetic needle itself. It is a good idea to fix the disc so that it can rotate with slight resistance – however, it must not impede the needle’s movement. Photo 6 shows the completed apparatus from two angles.
However, to use the described device, we need to consider the theoretical foundations of its operation.
A Little Bit of Math
The mathematical foundations of the proposed measurement method are not complicated [3]. To make them easier to understand, I will present them step by step.
The acceleration a for a displacement x in oscillatory motion is given by Equation (1).
| a = -ω2x | (1) |
Knowing that angular frequency ω is related to frequency ν by ν = ω / 2π, we obtain:
| a = -4ν2π2x | (2) |
For rotational motion, the analogous equation allowing calculation of angular acceleration ε for a given angular displacement α takes the form:
| ε = -4ν2π2α | (3) |
An observant reader will certainly notice that a magnetic needle suspended on a pivot can be treated as a rigid body, and the torque M acting on it, given its moment of inertia I, has the value:
| M = Iε | (4) |
At the same time, for a given magnetic moment μ of the needle suspended in a magnetic field with flux density B, when the needle is deflected by angle α, the following holds:
| M = -μB sin α | (5) |
which, for small angles (where the sine is practically equal to that angle, sin α ≈ α), simplifies to:
| M = -μB α | (6) |
This relation demonstrates that the motion is harmonic because the torque is proportional to the deflection, with the proportionality constant k = μB.
Using Equations (3), (4), and (6), we find:
| μB = 4ν2π2I | (7) |
Thus, if we know the magnetic moment μ and the moment of inertia I of the magnetic needle, as well as measure its oscillation frequency ν in the magnetic field, we can determine the flux density B!
A reader eager to determine the Earth’s magnetic flux density might now become discouraged…
– How can one easily find the magnetic moment and moment of inertia of a magnetic needle? – they might hopefully ask.
– There is no need to do that at all! – I reply from behind my desk.
Indeed, those two quantities are not necessary. This is fortunate, because under home or school conditions it would be difficult to determine them. However, there is a trade-off – in order to eliminate these variables, we must use some more math. Besides the Earth’s magnetic flux density Bz , we need an additional field with flux density Bd. In this case, the direction of Bd must be parallel to the Earth’s field. Depending on how the flux density vector aligns, the resultant flux density B interacting with the needle is described as follows:
| B↑ = Bz + Bd (vector directions the same) | (8) |
| B↓ = Bz - Bd (vector directions opposite) | (9) |
Substituting Equations (8) and (9) into Equation (7), we get:
| μ (Bz + Bd) = 4ν↑2π2I | (10) |
| μ (Bz - Bd) = 4ν↓2π2I | (11) |
We can now divide Equations (10) and (11) and rearrange, yielding:
| Bz = Bd (ν↑2 + ν↓2/ν↑2 - ν↓2) | (12) |
As we see, we have indeed eliminated μ and I. Determining the Earth’s magnetic flux density Bz thus amounts to measuring the oscillation frequencies ν↑ and ν↓, respectively, for the cases where the flux density Bd is in the same or opposite direction as the Earth’s field.
We still need to generate the additional magnetic field and determine its flux density Bd. This is precisely why, in our measurement device, we placed the needle inside a Helmholtz coil.
A Helmholtz coil, as we can already see, is a system of two coils that provides a relatively uniform magnetic field within its interior. That means there is a relatively large region in this setup where the flux density vector is approximately constant (Fig.2) [4].
The flux density Bd of a Helmholtz coil with n turns and radius r (carrying current J) can be determined from the Biot-Savart law. It is given by:
| Bd = μ0 n J * 8/5r√5 | (13) |
Substituting this into Equation (12), we get:
| Bz = (ν↑2 + ν↓2/ν↑2 - ν↓2) * μ0 n J * 8/5r√5 | (14) |
Finally! The mathematical battle is over, and victory is ours. Only easily obtainable variables remain. You need to measure the radius r (for example, in meters; if r = 0.052 m, then it is about 2.05 in), count the number of turns n, and find the oscillation frequencies ν↑ and ν↓ when current flows in opposite directions. We measure the current J with an ammeter or a multimeter. The mysterious coefficient μ0 is the magnetic permeability of free space, a constant of 12.57 × 10-7 Vs/Am.
Recall: in the above method, it is crucial that the lines of the Earth’s magnetic field and those of the coil-generated field are parallel!
Using the described device, one can also measure magnetic flux density by another method shown schematically in Fig.3. In the undisturbed Earth’s magnetic flux density Bz , the needle aligns with the field lines (Fig.3A). If we generate an additional magnetic field Bd perpendicular to (thus different from) the Earth’s field, a resultant force acts on the needle. This force causes the needle to deflect by some angle α (Fig.3B).
Using trigonometry, we find:
| ctg α = Bz /Bd | (15) |
We are, of course, interested in the Earth’s magnetic flux density Bz , while Bd is once again provided by the Helmholtz coil. Hence:
| Bz = ctg α * μ0 n J * 8/5r√5 | (16) |
Thus, by measuring the angle α through which the magnetic needle is deflected as a result of a current J flowing through a Helmholtz coil with two n-turn windings, we can also determine the Earth’s magnetic flux density Bz .
Measurements
Using the first measurement method, the first step is to place the apparatus on a flat, leveled surface so that, with the needle pointing north, it lies along the line connecting the centers of the coils (Photo 7).
Next, assemble a simple electrical circuit by connecting in series the Helmholtz coil, a DC power source of a few volts (e.g., four 1.5 V cells in series), a potentiometer of appropriately chosen resistance (most often 1 kΩ), and an ammeter in the milliamp range. I used a multimeter that measures current to hundredths of a milliamp. A switch to reverse the coil current is handy, though you can also swap the voltage source terminals manually.
Since the vectors Bz and Bd should share the same direction once the apparatus is set up correctly, you should not see any needle deflection after switching on the coil current. The current J must not be too large; otherwise, Bd may exceed Bz , forcing the needle to settle opposite its usual direction and no longer point toward geographic north. Although this does not prevent measurement, it complicates it somewhat.
After the current settles, gently set the needle oscillating horizontally by nudging it carefully – the oscillation amplitude should be small to maintain near-harmonic motion. Next, measure the oscillation period T. A simple method is to use a stopwatch to measure, for example, 10 full oscillations, then divide the total time by 10. Then, at the same current J, reverse the current direction and measure the oscillation period again.
Repeat this for successive current values J. Naturally, multiple measurements allow averaging for improved accuracy.
Since frequency ν is the reciprocal of the period T, convert T↑ and T↓ into ν↑ and ν↓. Substitute them into Equation (14) to find Bz .
Table 1 (n = 20; r = 0.052 m, approx. 2.05 in) shows an example of my experimental data.
| J [A] | (ν↑2 + ν↓2) / (ν↑2 - ν↓2) | Bz [nT] (1 nT) |
| 0.00501 | 11.69 | 20260.45 |
| 0.00710 | 8.34 | 20484.31 |
| 0.01532 | 3.89 | 20616.04 |
| 0.01988 | 2.89 | 19875.18 |
| 0.03011 | 1.95 | 20311.51 |
As shown, the measured values of Bz are similar in all cases. On average, Bz is 20309.50 nT with a standard deviation s = 251.22 nT.
For the second method, place the apparatus on a flat surface so that, with the needle pointing north, it is parallel to the plane of the coils (Photo 8A).
As before, assemble the same electrical circuit. When you switch on the current J, the needle deflects by some angle α, depending on the current direction (Photo 8B). Repeat the measurement for various current values so you can average the result. My measurements (n = 20, r = 0.052 m) are in Table 2.
| J [A] | α [°] | Bz [nT] |
|---|---|---|
| 0.00409 | 4 | 20233.76 |
| 0.00713 | 7 | 20088.28 |
| 0.01211 | 12 | 19709.09 |
| 0.01931 | 18 | 20559.05 |
| 0.03069 | 27 | 20836.65 |
These measurements show that Bz values are also consistent across the trials. After averaging, Bz = 20285.37 nT with a standard deviation s = 387.92 nT.
The values of Bz determined by both methods are very similar – the average from both is 20297.44 nT.
Is It Definitely Correct?
To confirm the reliability of these methods for measuring Bz , one should refer to data obtained by others. Such information can be found in relevant literature and presented, for instance, in map form (Fig.4) [5].
I performed my measurements in southern Poland – the approximate value read from the map is about 49000 nT. This is more than twice our measured value! Did we make an error, even though both methods produced very similar results? To answer this, we need to understand the convention for expressing Earth’s magnetic field quantities.
On most of Earth’s surface, the magnetic field lines are not parallel to the planet’s surface. At any point, the magnetic field can be characterized by its field intensity vector. In most cases, a spherical coordinate system is used: one must specify the inclination, declination, and the magnitude of this vector. Declination is the angle between the horizontal component of the field’s magnitude and the geographic meridian, while inclination is the angle the field vector makes with the horizontal plane (Fig.5).
In our measurements, we completely ignored the field’s inclination; thus, we measured only the horizontal component of Earth’s magnetic flux density Bz . The value from Fig.4 corresponds to the magnitude of the overall field. To compare them, we need the inclination angle at the measurement site (Fig.6).
Where I took measurements, that angle is about 66°. Thus, the horizontal component of Earth’s magnetic flux density is Bz ' = 49000 nT × cos 66° ≈ 19930.10 nT.
Our measured horizontal component Bz and the value Bz ' from literature are very close. The absolute error is only 367.34 nT, and the relative error is 1.84%.
Of course, many factors affect measurement accuracy, including how precisely the apparatus is built, the presence of other magnetic fields, or large metal masses that distort Earth’s field. Yet considering the simplicity and ease of constructing the device, one can conclude that it enables remarkably accurate measurements of the Earth’s magnetic flux density. Naturally, it can also be adapted to measure the flux density of other magnetic fields.
References:
- [1] Rubakov V. A., Classical theory of gauge fields, Princeton University Press, 2002, pp. 54 – 55, 207 – 208 back
- [2] Ples M, Silnik cieplno-magnetyczny, Młody Technik, 1(2015), Wydawnictwo AVT, Warszawa, pp. 74 – 75 back
- [3] Gaj J., Laboratorium fizyczne w domu, Wydawnictwa Naukowo-Techniczne, Warszawa, 1985 back
- [4] Ramsden E., Hall-effect sensors: theory and applications (2nd ed.), Elsevier/Newnes, Amsterdam, 2006, pp. 195 back
- [5] Maus S., Macmillan S., McLean S., Hamilton B., Thomson A., Nair M., Rollins C., The US/UK World Magnetic Model for 2010-2015, NOAA Technical Report NESDIS/NGDC, 2010 back
All photographs and illustrations were created by the author.
Marek Ples