Electric Power Generation from Earth’s Rotation through its Own Magnetic Field?

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Electric Power Generation from Earth’s

Rotation through its Own Magnetic

Field?

September 4, 2017

Rick Rosenboom - 10715576

Report Bachelor Project Physics and Astronomy, size 12 EC,

conducted between 01-04-2017 and 04-09-2017

Faculty of Physics and Astronomy

Department of Physics of Energy

University of Amsterdam, Vrije Universiteit Amsterdam

Supervisor: Rinke Wijngaarden

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1 Abstract

This thesis was inspired by the article “Electric Power Generation from Earth’s Rotation Through its own Magnetic Field” (Chyba and Hand, 2016). In this article, Chyba and Hand describe the option to generate electric energy through the rotational movement of the Earth and its own magnetic field. However, there does not seem to be full consensus in the literature on how to describe electrodynamics in a rotating frame. Surprisingly, there is a discussion in the literature whether the magnetic field of a magnet (and in particular of the Earth) co-rotates with the magnet or not. In this thesis, it was tried to get a clearer view on electricity and magnetism in a rotating system. This was reached through an experimental setup, which modeled for the Earth and its magnetic field. After many different experiments which in the meantime had led to improvements of the experimental setup, it was found that some of the predictions of Chyba and Hand (2016) were right. Anyhow, some of them were contradicted by small but interesting results. Unfortunately more experiments are needed to form a clear statement about whether it is possible to generate electric power from the Earth’s rotation through its own magnetic field.

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2 Popular Summary

De elektro motorische kracht (emk) is een voltage dat ontstaat indien een elek-trisch circuit en een magneetveld bewegen ten opzichte van elkaar. Het is bek-end dat de aarde draait ´en dat deze zich in een magneetveld bevindt. Zou het dan wellicht mogelijk zijn om middels deze draaiing van de aarde energie op te wekken? Na vele experimenten vonden we resultaten die ons wellicht meer kunnen vertellen over deze kwestie. Voor dit onderzoek werd een opstelling gemaakt die model stond voor de aarde en haar magnetisch veld. Gedurende

Figure 2: Zijaanzicht van de experimentele opstelling.

dit onderzoek werd ondervonden dat er verschillende variabelen zijn die de re-sultaten be¨ınvloeden. Zo werd telkens de opstelling aangepast zodat we geen last hadden van een bepaalde verstoring, maar enkel het effect waar we naar op-zoek waren. Het uiteindelijke resultaat was echter erg klein en niet te verklaren aan de hand van berekeningen. Daarom is het twijfelachtig of het mogelijk is om energie op te wekken middels de draaiing van de aarde. Verder onderzoek binnen dit specifieke gebied zou in de toekomst kunnen leiden tot het ontstaan van een nieuw soort duurzame energie!

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3 Introduction

In 2016 Chyba and Hand published their article about the possibility to gener-ate electric power from the Earth’s rotation through its magnetic field (Chyba and Hand, 2016). This new way of generating energy could, if proved right, contribute to a more sustainable Earth, since it gradually gets clear to everyone that burning fossil fuels is not the most accountable way to generate energy and that we have to switch to a more sustainable method of generating our energy. In this article Chyba and Hand state that it would theoretically be possible to create electric power through an e.m.f. that is described by changes in mag-netic flux, which stands for the change in amount of magmag-netic field lines passing through a surface. It seems useful to provide a summary of Chyba and Hand’s article, which is the motivation of this thesis. First of all, Chyba and Hand tell us about the non-rotation of Earth’s magnetic field. This is based on experiments carried out by Barnett (1908), Kennard (1917) and Pegram (1917). It could be suggested from these experiments that the components of Earth’s magnetic field, which are axisymmetric about Earth’s rotation axis, are stationary with respect to the rotating Earth. Later on in this thesis will be explained precisely why this is right. The rotation of the Earth creates a v × B force, with the non-rotating component of Earth’s magnetic flux density B and the azimuthal speed v = 465sin(θ)m/s (Lorrain, 2007). This force will generate position-dependent volume charge densities that generate an electric field, which perfectly cancels v × B (Davis, 1947; Hones and Bergeson, 1965; Davis, 1948). This immedi-ately tells us that it would be impossible to generate electric energy by this method, because the electric field formed as a result of the position-dependent volume charge densities instantly cancels the created e.m.f. However, Chyba and Hand tell in their article that they found a loophole in the proof. Later in this thesis will be explained what Chyba and Hand precisely proposed, but in summary it came down to a modification in the setup, by including materi-als with different magnetic properties, that would lead to a situation in which the position-dependent volume charge densities could not cancel the generated e.m.f. anymore. This all seems to form quite a clear story, but unfortunately it is not as easy as it seems. The first argument about the non-rotating mag-netic field of the Earth seems logical. The Earth is a big dipole, with its north pole at the geographic south pole and vice versa. Chyba and Hand talk during their explanation about field lines, but field lines do not physically exist. They are made up by mankind to make thinking about electromagnetism easier. In this particular case, it’s better to think about the magnetic field as a density distribution. The further you get to one of the poles, the bigger the magnetic density will get. Rotation of the Earth around its axis will not influence this density, because the axis is parallel to the magnetic dipole: The poles do not move while the Earth’s surface does. This makes it very plausible to say that Earth’s magnetic field does not rotate. The next conclusion mentioned in the article written by Chyba and Hand (2016) about the forming of an e.m.f. is con-troversial. Within the scientific world there is not yet an unanimous agreement about how to describe electrodynamics correctly in a rotating system. Even

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with an apparently analogous experiment, researchers do not seem to find the same results. Leus (2011) and Mueller (2014) did apparently the same experi-ment, but found different results. During this experiment will be tried to figure out what really happens in the field of electromagnetism in a rotating system. Thereafter experiments will be done to confirm the hypothesis, based on the articles of Mueller and Leus. Finally, the third point of Chyba and Hand, about the magnetically permeable material forming a barrier for the magnetic field, is also rather doubtful. Therefore experiments will as well be done with parts of the electric circuit shielded from the magnetic field by a magnetic shielding. The answering of all these question could perhaps lead to a clear statement about the possibility to generate electric energy from the Earth’s rotation through its magnetic field.

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4 Theoretical Framework

4.1 Maxwell equations

The Maxwell equations are

∇ · E = ρ ε0

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∇ × E = −∂B

∂t (2)

which is the basis of the Induction Law of Faraday. Magnetic monopoles do not exist, so

∇ · B = 0 (3)

∇ × B = µ0J + µ0ε0

∂E

∂t (4)

where the last term is negligibly small for the situation that will be discussed below. Hence the simplified version will be used:

∇ × B = µ0J (5)

The Maxwell equations are a set of partial differential equations which form together with the Lorentz force law the foundation of classical electromagnetism. These Maxwell equations describe how electric and magnetic fields are generated by charges, currents and changes of each other.

4.2 Lorentz force

The Lorentz force is given by the formula

FL= q (E + v × B) (6)

where the first term arises from the Coulomb force on a charge in an electric field E, while the second term describes the interaction between moving charges and a magnetic field. A Lorentz force is a result of the combination of a magnetic force and an electric one on a point charge. This point charge in the presence of both an electric and a magnetic field, with a speed v and charge q experiences a force given by the formula mentioned above.

4.3 e.m.f

The work done on a charge that traverses the loop of an electrical circuit is W =

I

C

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If this work is non-zero (such that positive work is done on the charge), then this work is converted into electrical energy, just like a battery delivers electrical energy. E.m.f. is defined as the ratio W_q , hence

e.m.f. = I

C

(E + v × B) · ds (8)

The magnetic force component (qv × B) of the Lorentz force equation, equation 6, is responsible for the electromotive force. The e.m.f. is the voltage produced by a source of electrical energy. Besides, when a conductor is moved through a magnetic field an e.m.f. is created as a result of the magnetic field exerting opposite forces on electrons and nuclei in the wire. This kind of e.m.f. is called ‘Motional e.m.f.’, since the e.m.f. is due to motion is the wire. The other form of e.m.f. is called the ‘Induced e.m.f.’. In this case the magnetic field moves while the conductor does not (Griffiths, 2005).

4.4 The link between Magnetism and Electricity

Electricity and magnetism are fundamentally interlinked as a consequence of Einstein’s theory of special relativity. These two phenomena on their own are inconsistent with the special relativity, but in contrary, if both electricity and magnetism are taken into account, the resulting theory, which is called elec-tromagnetism, is fully consistent with special relativity (Jackson, 1975). The relative contributions of both magnetism and electricity are dependent of the frame of reference.

4.5 Maxwell equation, Faraday law of induction

In case there is a changing flux through the contour C, ∇ × E 6= 0 and ir-respective of v × B there may be an electric field induced in the loop. Irre-spective of v × B, was written, because for a loop in a uniform magnetic field H

Cv × B ds = 0, as shown by calculations in the appendix. So it was

investi-gated whether it would be possible to have a non-zero e.m.f. from e.m.f. = I C E ds = Z Z S ∇ × E·da (9)

where S is the surface enclosed by C and where Stokes’ theorem was used. Using the Maxwell equation eq. 2

e.m.f. = Z Z s −∂B ∂t ·da = − dΦ dt (10) with Φ :=RR

SB·da the total flux treading trough S.

For a loop with area A, rotating with angular velocity ω in a uniform static magnetic field B, the flux is

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which leads to

e.m.f. = −dΦ

dt = N ABωsinωt (12)

i.e. a rotating loop in a steady magnetic field yields an alternating current voltage. This is in any case happening in the experiment if a coil/loop would rotate in the Earth’s field.

4.6 e.m.f. if

dΦ dt

= 0

Chyba and Hand argue that even if dΦ_dt = 0, one can have still have an e.m.f. For a non-zero e.m.f. a necessary condition is

e.m.f. = I

C

(E + v × B) · ds 6= 0 (13)

or using Stokes’ theorem e.m.f. =

Z Z

S

∇ × E+∇ × (v × B) ·da 6= 0 (14)

or in case for a loop of fixed size dΦ_dt = 0 (and thus ∇ × E = 0) that

Z Z

S

∇ × (v × B) ·da 6= 0 (15)

clearly if ∇ × (v × B) = 0 everywhere, we cannot satisfy this, hence we need at least at some space within S with

∇ × (v × B) 6= 0 (16)

Chyba and Hand want to arrange that with a cylindrical shell of high perme-ability. Namely, they tell us that “magnetically permeable materials channel magnetic flux and can be used to alter B to give 5 × (v × B) 6= 0”. This inequality guarantees that the electrons in such a conductor cannot rearrange themselves to generate an electrostatic field E = −5V that satisfies E = v ×B” (Chyba and Hand, 2016).

4.7 Ohm’s Law

Ohm’s Law states that if there is an electric field E0 in a moving system, then a current density J will flow according to

E0 = J/σ (17)

with σ the conductivity. Combining with E0 = E + v × B for a contour in a moving frame of reference it can be found that

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or

∇ × E+∇ × (v × B) = (∇ × J) /σ (19)

with the Maxwell equations eq. 5 and eq. 2 it results in −∂B

∂t +∇ × (v × B) = 1

µ(∇ × (∇ × B)) /σ (20)

using the vector identity (for any B)

∇ × (∇ × B) = ∇ (∇ · B) − ∇2_B ₍₂₁₎

and the Maxwell equation eq 3 it is found that −∂B

∂t+∇ × (v × B) = − 1 µσ∇

2_B ₍₂₂₎

which is eq. 7 of Chyba and Hand: −∂B

∂t+∇ × (v × B) = −η∇

2_B ₍₂₃₎

with

η = (µσ)−1 (24)

using eq. 14 it is possible to state e.m.f. = Z Z S ∇ × E+∇ × (v × B) ·da = Z Z S −η∇2_B·da ₍₂₅₎

Chyba and Hand use a slightly different form of this equation: after substituting B =∇ × A (with the A vector field) they write in their eq. 63

e.m.f. = Z Z S −η∇2_{B·da = −η}Z Z S ∇2_{(∇ × A) ·da = − η}I C ∇2_A·ds ₍₂₆₎

Here the last equality holds because of Stokes theorem, while the penultimate equal sign holds because (in reverse order)

∇ × ∇2_Aeq.21_{= ∇ × (∇ (∇ · A) − ∇ × (∇ × A)) = ∇ × ∇ (∇ · A) − ∇ × ∇ ×}

(∇ × A)∇×∇P =0= −∇ × ∇ × (∇ × A) =: −∇ × ∇ × Q = −∇ (∇ · Q) + ∇2_{Q = −}

∇ (∇ · ∇ × A) + ∇2_{(∇ × A)}∇·∇×A=0₌ _∇2_{(∇ × A) .}

In a long derivation (eq. 9-62) Chyba and Hand study the properties of ∇2_A

for a cylindrical shell (tube) in static and dynamic circumstances. Using these results, they finally arrive at their eq. 63:

e.m.f. = −η I C ∇2_{A·ds = − 2R} mvβ2` (a/b) 2 sin φ0cos2φ0+ O (Rm) 2 (27)

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with Rm= σµvξ = σµrµ0vb (28) and β2= 2B∞µr(µr− 1) (µr+ 1)2− (a/b)2(µr− 1)2 (29) Here a and b are the inner and outer radii of the shell, which has conductivity σ and permeability µ = µrµ0 and is moving at velocity v in a static magnetic

field B∞.

4.8 Note on the parameter φ

0

In eq. 27 there is a factor sin φ0cos2φ0 which now is considered. A plot is

shown below.

Figure 3: Plot of the factor sin φ0cos2φ0.

The first maximum occurs for _dxd sin x cos2_{x = 0 or if x = arccos}1 3

√ 6. The maximum value of sin φ0cos2φ0 is thus sin arccos1₃

√ 6 cos2 _arccos1 3 √ 6 = 0.384 9

The optimal angle φ0in radians is thus arccos1₃

√

6 = 0.615 48 that is 0.615 48 ×

180

π = 35. 264 degrees. Note that φ0 is the angle between ˆx and one wire. The

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4.9 Calculation of e.m.f. due rotation with respect to the

permanent magnet for the setup used

The e.m.f. that would be generated during the experiment is given by: e.m.f.= −2Rmvβ2` (a/b)

2

sin φ0cos2φ0= −2×σµrµ0vb×v×_(µ 2B∞µr(µr−1)

r+1)2−(a/b)2(µr−1)2×

` (a/b)2sin φ0cos2φ0' −2 × σµrµ0v2b ×

2B∞µ2r

µ2

r−(a/b)2µ2r

× ` (a/b)2sin φ0cos2φ0 =

−4v2_B

∞× σµrµ0× b (a/b)

2

1−(a/b)2 × ` sin φ0cos2φ0.

In the optimal situation, sin φ0cos2φ0 = 0.384 9 (see section 4.8). For the

geometry used during the experiment _1−(a/b)(a/b)22 =

(10/18)2

1−(10/18)2 = 0.446 43. Under

those circumstances approximately e.m.f.= −0.513 19v2_B

∞b`σµrµ0. From that

it can be found that an e.m.f. due rotation with respect to the strong magnet used during the experiment.

e.m.f.= −0.687 32 × 1. 885 0m_s2

× 0.1T × 9.3mm × 28mm × 0.33Sm−1_{× 5000 ×}

4π × 10−7Hm−1= −0.132µV with a rotation of 10 rps.

The conclusion is that it is needed to rotate the components faster to be able to measure the voltages with the voltmeter that was used. 100 rps would result in 13µV . Unfortunately these more precise calculations were done when the setup was already built.

4.10 Calculation of e.m.f. due the Earth’s rotation of the

geometry of the setup

The influence of the magnetic field of the Earth is given by: Rm = σµvξ =

σµrµ0vb = 0.33Sm−1 × 5 · 103 × 4π × 10−7Hm−1 × 345ms−1 × 1cm = 7. 153 4 × 10−3. (eq. 13 & 15) β2= 2B∞µr(µr−1) (µr+1)2−(µr−1)2 = 2B∞ 5·103₍_5·103₋₁₎ (5·103₊₁₎2_{−(1.0cm/1.8cm)}2_(5·103₋₁₎2 = 2.8901 B∞ e.m.f. = −2Rmvβ2` (a/b) 2 sin φ0cos2φ0 = −2 × 7.153410 × 5 × 345ms−1 × 2.8901 × 50 · 10−6T × 2.8cm(1.0cm/1.8cm)20.3849 = −2.4µV

According to Chyba and Hand (2016) this result will change sign if the magnetic shielding component is rotated by 180 degrees. Later on in this article this prediction will be tested as well.

4.11 Finding an approximate expression for the e.m.f. if

a/b = 1 (very thin shell)

In this next subsection a geometry is proposed that would ensure to give results that could be measured more easily, since the voltages generated are higher. This could come in handy during future experiments.

For a/b = 1, e.m.f.=−4 × 0.384 9B∞σµrµ0v2b`

µr(µr−1) (µr+1)2−(µr−1)2 ' −4 × 0.384 9B∞σµrµ0v2b` µ2_r 4µr = −0.384 9B∞σµ 2 rµ0v2b`

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In the case during the experiment −0.384 9B∞σµ2rµ0v2b` =

−0.384 9 × 50 · 10−6_{T × 0.33Sm}−1_{× 5 · 10}32

× 4π × 10−7_Hm−1_{× 345ms}−12 × 1cm × 2.8cm = 6. 649mV

The same was done with the exact equations:

Calculation of e.m.f. due the Earth’s rotation for our geometry

Rm= σµvξ = σµrµ0vb = 0.33Sm−1× 5 · 103× 4π × 10−7Hm−1× 345ms−1× 1cm = 7. 153 4 × 10−3 (eq. 13 & 15) β2= 2B∞µr(µr−1) (µr+1)2−(µr−1)2 = 2B∞ 5·103₍_5·103₋₁₎ (5·103₊₁₎2_−(5·103₋₁₎2 = 2499. 5B∞

e.m.f. = −2Rmvβ2` (a/b)2sin φ0cos2φ0=

= −2×7. 153 4×10−3×345ms−1_{×2499. 5×50·10}−6_{T ×2.8cm (1.0cm/1.8cm)}2

0.384 9 = 2.05mV

In conclusion: a/b = 1 (very thin shell) yields an e.m.f. that is 103_{times larger}

than in the situation of a/b ≈ 0.5

4.12 Enhancement factors for realistic shells: a/b = 0.9, 0.95, 0.99, 0.999

For a/b = 0.9 this results in β2=

2B∞µr(µr−1)

(µr+1)2−(a/b)2(µr−1)2 = 2B∞

5·103₍_5·103₋₁₎

(5·103₊₁₎2_−(0.9)2_(5·103₋₁₎2 =

10. 484B∞

For a/b = 0.95 this results in β2=_(µ 2B∞µr(µr−1)

r+1)2−(a/b)2(µr−1)2 = 2B∞

5·103(5·103−1)

(5·103₊₁₎2_−(0.95)2_(5·103₋₁₎2 =

20. 350B∞

For a/b = 0.99 this results in β2=_(µ 2B∞µr(µr−1)

r+1)2−(a/b)2(µr−1)2 = 2B∞

5·103(5·103−1)

(5·103₊₁₎2_−(0.99)2_(5·103₋₁₎2 =

96. 636B∞

For a/b = 0.999 this results in β2= _(µ 2B∞µr(µr−1)

r+1)2−(a/b)2(µr−1)2 = 2B∞

5·103₍_5·103₋₁₎

(5·103₊₁₎2_−(0.999)2_(5·103₋₁₎2 =

714. 6B∞

5 Induction in a rotating loop.

It seems useful to figure out what exactly happens in terms of electromagnetism in a rotating object to determine if it would be possible to generate electric energy from the rotation of the Earth and its magnetic field. Therefore three different articles will be reviewed with the purpose to explore the characteris-tics of electromagnetism in rotating reference frames. The first one will be an article of Barnett (1908). Barnett took a cylindrical condenser and placed it in an approximately uniform magnetic field parallel to its axis. The magnets, which produced the field, were rotated while the condenser was short-circuited by a wire at rest like itself. While the magnets were still rotating, the connec-tion between the armatures of the condenser was broken and the condenser was tested for charge after the field was annulled, the rotation stopped or the con-denser removed. Barnett’s hypothesis was that if the lines of induction remained fixed the capacitor would become charged by the motional electromotive forces

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within it. When the connection between the circuit and the condenser was bro-ken, the condenser would not get the chance to lose its charge and would remain charged. Eventually the charge could be measured by connecting the capaci-tor to an electrometer after the rotation was stopped. Barnett argued (based on arguments involving aether) that he should not measure a voltage after the condensers were rotated, but he did however. Nowadays this contradiction is still unsolved. Leus (2011) used a rotating magnetic circuit comprising a per-manent magnetic ring, yoke and a stationary conductor in the air gap between the ring and yoke. The yoke was placed as a magnetic shielding. During the measurements both the magnet and the yoke were rotated and a voltage was measured across the stationary probe. The magnitude of the measured voltage was proportional to the speed of the rotation of the magnet (Leus, 2011). The induced e.m.f. could, according to Leus, not be described by a application of Faraday’s flux law, e.m.f. = −∂ϕ_∂t, because the magnetic flux in the air gap and the area of the air gap both remained constant. Leus therefore came up with the controversial explanation that the magnetic field itself would rotate along with the rotating magnet. Earlier in this paper was concluded that the magnetic field would not move along with the magnet, because of the non-existence of the magnetic field lines. This does of course not make this article less interesting, but before experiments will be done it seems clever to have a look at the article Muller published in 2014 called ”Unipolar Induction Revisited: New Experi-ments and the “Edge Effect” Theory”. Muller (2014) refers to this subject as ”Faraday’s paradox” which states that the results of Barnett even nowadays are seen as a kind of oddity. Because of this Muller tries in his experiment to find a more ”down-to-earth” solution, by contrasting the case of rotational unipolar induction with the regular case of rectilinear motional induction. In his experi-ment ring ceramic magnets were stacked together to mimic a cylindrical magnet with a central hole and an gap through which a rectangular circuit is inserted. Because of this configuration the circuit is able to move relative to the magnet. Besides, a system was made whereby the whether or not shielded parts of the circuit could move independent of each other. An important remark is that the circuit is only able to move in slight oscillations relative to the frame. The whole system is enclosed within a ferromagnetic frame which shields the branch of the circuit which lies inside the frame. This would ensure that different generated e.m.f.’s would not cancel each other during the measurement. The results of this experiment show a relativistic symmetry. That means that moving the circuit relative to the magnet gives identical results as moving the magnet relative to the conductor, but with opposite signs. Whenever the circuit moved, indepen-dent of whether or not the magnet was moving, Muller measured a rotational e.m.f. Movement of the magnet only resulted in the measurement of only a translational e.m.f. Muller also comes with the conclusion that the magnetic field of an axisymmetric magnet does not rotate when the magnet rotates about its axis of symmetry. This conclusion is based on the result that whenever the part of the circuit that lies outside the frame is moved it never results in any e.m.f. The difference between the rotational and translational motional induc-tors resides, according to Muller (2014), not so much in the nature of the motion,

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but in the fact if the magnet has edges that move perpendicularly or longitudi-nally to themselves. Unfortunately the conclusions are not as ”down-to-earth” as Muller hoped them to be. Combined with the other articles mentioned above it seems clear that the opinions about electromagnetism in rotating objects are quite divided. This fact was the inspiration for designing a new experimental setup which could give a broader insight in what is happening within the field of electromagnetism in rotating systems.

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6 Earth’s magnetic field

Earlier this article the non-rotation of the magnetic field of the Earth was men-tioned, but since the non-rotation of the magnetic field of the Earth is a principle during this article, it seems useful to give a more extensive explanation. The Earth is approximated with a uniformly magnetized sphere during this exam-ple. This approximation is correct, since the Earth appears to be a dipole and within a dipole the magnetization (M ) is uniform. If the direction of the magnetization would be chosen along the z-axis, it would lead to the following statements: Jb = 5 × M = 0 and Kb = M × ˆn = M sin(θ) ˆφ. Where Jb stands

for the volume current and Kb stands for the surface current (Griffiths, 2005).

A rotating spherical shell with a uniform surface charge σ corresponds with the surface density K = σv = σωR sin(θ) ˆφ. Therefore it follows that the field of a uniformly charged sphere is identical to the field of a spinning spherical shell, because σωR → M . This is also plausible if you would think of the magnetic field as a magnetic distribution rather than a collection of field lines, as men-tioned before. It can thus be seen that the surface current density Kb lacks a

dependence of time. Therefore whether or not rotating the current density K generates the same magnetic field.

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7 Experimental setup

The requirements for the experimental setup were as follows: A magnetic dipole was needed as well as a electric circuit wherein the e.m.f. could occur. A volt-meter would also come in handy for measuring the e.m.f.’s. Finally both the electric circuit and the magnetic dipole had to be movable independent of each other. At first there was tried to develop a setup that resembled Barnett’s setup, but after calculating what kind of materials were needed to generate a measurable voltage the decision was made to try another setup. Barnett used a cylindrical capacitor to measure the e.m.f. he generated during his experiment. A realizable cylindrical capacitor would have an inside radius a of 0, 05m and an outside radius of 0, 10m. With the dielectric constant k = 1 (vacuum), the formula C_L = _ln(b/a)2πk , would lead to a value of 0, 80259 ∗ 10−10F/m. Such a cylindrical capacitor of length L = 1m would be needed to achieve a capacity of 80, 2590pF . This would, combined with a potential in mV lead to a very small charge which would be very hard to measure. The next setup that was considered was eventually built. It consists of all needed parts and it has a volt-meter in the same frame of reference as the electric circuit, which can interact wirelessly with the receiver that is connected to a computer. Both the magnetic dipole and the electric circuit can be rotated relatively to each other and both parts are connected on a rail where the relative distance between both parts can be altered. The magnetic dipole is a magnet with the following specifics: S-70-35-N Disc magnet Ø 70 mm, height 35 mm, neodymium, N45 and nickel-plated. N45 indicates the remanent field (www.supermagnete.de, 2017). The different components were rotated by a drilling machine which operated at an approximate speed of 10 rps.

(a) Front view of the setup. (b) Side view of the setup

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8 Hypothesis

8.1 Calculating the e.m.f. of a rectangular circuit rotating

in a magnetic field

Before measurements were done with the setup, calculations were made to pro-vide a hypothesis about what voltages could be measured with the setup men-tioned above. The e.m.f. is given by summation of the integrals of each line of the electric circuit (equation 8). The first electric circuit of which the e.m.f. was calculated was the most simple one possible: A rectangular one.

Figure 5: The electric circuit used to calculate the e.m.f.

For the calculation of the e.m.f. generated by rotating this circuit in a magnetic field, an expression for the magnetic field was needed. As mentioned before, a magnetic dipole was used during this setup. The magnitude of the magnetic field of a dipole is B = µ0

4π

3(m·ˆr)ˆr−m

r3 with m = mˆz and ˆr = ˆz cos θ + ˆx sin θ.

It seems useful to decompose this term in terms of the coordinates x and z (y is ignored because the velocity is in the y direction. The cross product between vy and by will always result in 0):

˜

B : = _m1 3(m·ˆr)ˆ_r3r−m

=3(ˆz·(ˆz cos θ+ˆx sin θ))(ˆ_r3z cos θ+ˆx sin θ)−ˆz

=3 cos θ(ˆz cos θ+ˆ_r3 x sin θ)−ˆz

=(3ˆz cos θ cos θ+3ˆ_r3x sin θ cos θ)−ˆz

˜ Bz= (3 cos2θ−1) r3 = 2z2−x2_−y2 (x2_+y2_+z2₎5/2 = 2z2−x2 (x2_+z2₎5/2 ˜ Bx=3 cos θ sin θ_r3 = 3xz (x2_+y2_+z2₎5/2 = 3xz (x2_+z2₎5/2.

Since the magnetic field term was decomposed in terms of coordinates x and z it is possible to calculate the e.m.f. generated by the rotating of the rectangular electric circuit in a magnetic field. This can be done by integrating all four sides of the circuit:

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On the axis v = ωs = 0, since s = 0, so the integral from point d to a turns out to be 0. Rb avyBzds + Rc b −vyBxds + Rd c −vyBzds = Rb aωsBzds + Rc b −ωwBxds + Rd c −ωsBzds

Filling in the magnetic field terms of coordinates x and z: Rb aωsBzds = ω Rw 0 x 2p2_−x2 (x2_+p2₎5/2dx = ω w2 (w2_+p2₎32 Rd c ωsBzds = −ω Rw 0 x 2(p+h)2_−x2 (x2_+(p+h)2₎5/2dx = −ω w2 (w2+h2+p2+2ph)32 Rc b −ωwBxds = ωw Rp+h p − 3wz (w2_+z2₎5/2dz = ωw 2(w2+p2) 3/2 −(w2_+p2_+2ph+h2₎3/2 (w2+p2+2ph+h2)32(w2+p2)32

it can be found that: e.m.f. =H = ω w2 (w2_+p2₎32 −ω w2 (w2_+h2_+p2_+2ph)32 +ωw2(w 2_+p2₎3/2 −(w2_+p2_+2ph+h2₎3/2 (w2_+p2_+2ph+h2₎32(w2_+p2₎32 e.m.f. =H = ωw2(w2+h2+p2+2ph) 3 2₋₍w2_+p2₎32 (w2_+p2₎32(w2_+h2_+p2_+2ph)32 +ωw2(w2+p2) 3/2 −(w2_+p2_+2ph+h2₎3/2 (w2_+p2_+2ph+h2₎32(w2_+p2₎32 = 0

Based on the calculations above, it should not be possible to measure an e.m.f. in this particular circuit. However, it could be possible to generate voltages through the influence of the magnetic field of the Earth. This is calculated among other things in the next section.

8.2 Expected results

8.2.1 Classical induction in a loop

Of course there is an influence from Earth’s magnetic field beside the influence from the magnet in the setup. It was expected that this would result in an alternating current. The magnetic field of the Earth has components perpen-dicular to the rotation of the electric circuit and can therefore result in classical induction. It seemed useful to calculate what voltage could be measured by the setup used during the experiment:

Induction in a rotating loop in Earth’s field (formula 11): e.m.f. =N ABω = 1 × 10 cm2 × 0.5 mT × 2π

0.1s= 3.1416 × 106 µV

Induction in a rotating loop in the field of our permanent magnet e.m.f. = N ABω= 1 × 10 cm2 × 0.1 T ×_0.1s2π = 6.2832 mV

8.2.2 Crude estimate from Chyba and Hand’s figure 3

In Chyba and Hand’s article a figure was used to give an idea of the voltages he expected. The following calculations were based on this figure.

e.m.f. =R v × B = R ωrB∂r = 1 2Br

2_{ω for one leg and 2the net e.m.f. for our}

field and geometry is thus 1 ₁₀₀1 Br2_ω 1

100· 2T (2cm) 2 2π

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estimation is way off from what is obtained by Chyba and Hand’s equations. See below for further results.

8.3 Calculation of Chyba and Hand’s equations

For this calculation Chyba and Hand’s equations were used (see section 4.9) which gave the values that were expected for the e.m.f. generated in the setup

mentioned in their article: e.m.f.(x0, y0) = −ηH₀52Azz·∂l = −2Rˆ mvβ2l(a/b)2sin(ϕ0)cos2(ϕ0)+

O(R2

m). Chyba and Hand assume the following: σ = 0.5Sm−1, µr= 6.5·103, v =

345ms−1, ξ = b = 1cm, l = 20cm.

Note on the parameter ϕ0: In Chyba and Hand’s equation 63 there is a

fac-tor sin(ϕ0)cos2(ϕ0) which now is considered. The first maximum occurs for ∂

∂xsin(x)cos

2_{(x) = 0 or if x = arccos}1 3

√

6. The maximum value of sin(ϕ0)cos2(ϕ0)

is thus sin(arccos1₃√6)cos2_(arccos1 3

√

6) = 0.3849 The optimal angle ϕ0 in

ra-dians is thus arccos1₃√6= 0.61548 that is 0.61548 ×180_π = 35.264 degrees. Note that this is the angle between ˆx and one wire. The opening angle of the circuit is thus 2 × 35.264 degrees = 70.528 degrees. This gives the angle in which the contact points should be connected to the Mn-Zn Ferrite.

The first measurements were based on the crude approximation of the results (see section 8.2.2) that were expected to measure. However, calculations that were done more precisely during the experiment proved that the effect would be too small to be measured accurately by the setup that was designed for this experiment. See the section 4.9 for these calculations.

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9 Results

9.1 Improving the setup

From the first measurements on the rotating loop as discussed in section 8.1, it could be seen that as expected, the voltage increased as the distance between the magnet and the circuit decreased. After the first improvement of the setup the results mentioned before turned out to be outdated. Because of this reason these results are not presented. The voltage increased as well if the rotational speed of one of the components was increased. But what seemed to be really strange, was that only positive values were found during most of the measurements. As described before, it was expected that these measurements would result in approximately zero, because the different e.m.f.’s would cancel each other. It could be possible that the voltage meter used was not working properly and that therefore these strange results were measured. To investigate the capability of the voltmeter the following setup was conceived:

Figure 6: Setup for evaluating the capability of the voltmeter to measure both positive and negative voltages.

The waveform generator in this setup was set to generate a sinusoidal voltage. Two resistors were placed in this alignment to decrease the voltage given by the waveform generator by a factor of 1000. This was done to feed the voltmeter voltages within the range that was expected to be produced during the exper-iment. The waveform generator gave a wave function with an amplitude of 10 mV which thus was decreased to a amplitude of 10 µV. This was approximately in the expected range of voltages generated by the setup. It was expected that the result of this experiment would be 0, but on the contrary only positive volt-ages were measured. This leaded to the conclusion that the voltmeter was not working properly. After talking to the employees of the electronics department it became clear what precisely went wrong: For the setup a wireless connection was needed between the voltmeter and the data receiver, so both the magnet and the circuit could be rotated without having struggles with the wiring. This wireless connection worked fine, but was only able to send about 3 data points per second to the receiver. With the relative small speed the components were

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rotated this was to slow to mean the results and measure 0 at the end. It was concluded that alternating voltages came from the influence of the magnetic field of the Earth but that the experiment would lead to a DC voltage if it would be measurable. The alternating voltage was not the result that had to be measured during this experiment, so a different setup was proposed to filter the alternating voltage and leave the direct voltage untouched:

Figure 7: Setup to filter the alternating voltages out of the signal to be able to measure the direct voltage only.

The first part of this setup is exactly the same as used before to determine the capability of the voltmeter to measure both positive and negative voltages (Figure 12). The part in between these two resistors and the voltmeter is called a second order low pass filter. A low pass filter is an alignment of electric components that passes signals with a frequency lower than a certain cutoff frequency. With the components used in this setup the cutoff frequency became 0.04 Hz. This cutoff frequency is given by the formula 1/RC, with R(Ω) as the resistance of the resistor used and C(F ) as the capacity of the capacitor. Within the setup used, the value for the capacity became 500 µF , as a result of two in series connected capacitors shown in the circuit with each a capacity of 1000µF . The resistor used had a resistance value of 50(kΩ). This time constant was checked by connecting the setup to a waveform generator.

To determine which values should be given to the waveform generator first a small experiment with an oscilloscope was done. The oscilloscope was directly connected to the circuit and during the experiment the magnet was rotated. The oscilloscope showed a graph with a maximal amplitude of 5mV. It could be possible that this voltage was a result of the nearby computers or the not completely ideal magnet, but at this time this didn’t really matter. By all means it was known that the waveform generator should generate a wave with an amplitude of 5V which would be decreased by the resistor setup to 5mV. If the voltmeter would function in this setup, it would be sure that it would function as well in the magnet-circuit setup. In summary, the amplitude was set on 5V, the frequency on 10Hz and during the experiment the offset was varied to see if the voltmeter would measure the correct direct voltage. The alternating voltage should thus be filtered out of the circuit by the second order low pass filter and the only thing that should be measured was the direct voltage which was given by the offset. The offset was varied between -2V and 2V in

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steps of 0,5V.

Figure 8: Results of the experiment to determine the capability of the second order low pass filter setup to measure only the direct current given by the offset. As can be seen, a nice linear correlation was found between the measured voltage and the given offset. Unfortunately the measured voltage is not exact the same as the offered offset, but qualitatively could be said that the filter setup works. After talking to the electronics employees is was concluded that the setup needed a resistor of 50 Ohm. After including this resistor in the circuit the voltmeter gave the expected results.

The next step was to connect the filter setup with the magnet-circuit setup that was used before. Of course, the resistor setup that was used for evaluating the capability of the voltmeter to measure both positive and negative voltages was left out of this new setup, since the waveform generator was replaced by the circuit-magnet setup. Only the second order low pass filter was placed in between the voltmeter and the circuit to filter out the alternating voltage that disturbed the results.

It should be added that even when a short circuit was caused, values around 20 microvolts were measured. This turned out to be the offset of the voltmeter that was used during the experiment. 20 microvolts should thus be seen as the zero point during measurements.

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9.2 Measurement of the expected value 0 in a rectangular

loop

The following measurements supported the hypothesis and calculations that the rotation of this particular circuit did not generate any voltage at all:

Table 1: Summary of the experiments to measure 0, offset around 20 microvolts.

Configuration of the setup 1 2 3 4 5

Mean voltage measured (microvolts) 20 22 22 21 21

Rms noise (microvolts) 4 6 2 1 1

1:Rotating the electric circuit clockwise with respect to the magnet 2:Rotating the electric circuit counterclockwise with respect to the magnet 3:No rotation with respect to each other

4:Rotating the magnet clockwise with respect to the electric circuit 5:Rotating the magnet counterclockwise with respect to the electric circuit At this point it seemed clear that the results of the hypothesis were measured: All the e.m.f.’s of the different sides of the circuit that were generated as a result of a relative rotation of one of the components of the setup used cancel and leave us with no measurable voltage. Graphs of some of these configurations can be found in the appendix. The fact that the values did not vary around approximately 20 microvolts is due to the offset of the voltmeter which was around 20 microvolts. Subsequently is was found that the thermo-electric effect could also have influenced this value.

9.3 Electric circuit partly made of iron wire

During all the previous measurements the circuit was entirely made of copper wire. After measuring that in just an ordinary electric circuit no voltages were generated it was time to test something different: one side of the circuit was removed and replaced by a different kind of metal. By now, one side was made of iron wire. This change did not lead to any sensational results: No difference in voltages was measured between the different experiments.

9.4 Magnetic shielding

9.4.1 Magnetic shielding placed around a segment of the electric

circuit

According to Chyba and Hand (2016) there exists a loophole within the theories about electromagnetism such that it would be possible to generate an electric force within a system. This could be done by implementing magnetic shielding in the electric circuit, which would move along with the Earth’s surface and

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therefore through its magnetic field. This, according to Chyba and Hand, re-sults in an electromotive force. This electromotive force would be the result of a changing magnetic field (formula 12) caused by the magnetic shielding inte-grated in the circuit. For this experiment a Mn-Zn Ferrite was used as shielding. This was chosen based on the article of Chyba and Hand (2016), who suggested this kind of material for this particular experiment. To filter out the alternat-ing current, the same second order low pass filter was used as before. With the components used in this setup the cutoff frequency became 4,237 Hz. See section 9.1 for the calculations of this value.

First the Mn-Zn Ferrite was placed around a segment of the electric circuit. The magnetic shielding of a segment of the electric circuit did not result in any re-markable results. The direction of rotation did not influence the measurements. The same applies to rotation of one of the components.

9.4.2 Magnetic Shielding integrated into the electric circuit

Chyba and Hand (2016) proposed in his article a different solution to gener-ate an e.m.f. through magnetic shielding. The Mn-Zn Ferrite was integrgener-ated into the electric circuit. This was done by connecting the wires of the electric circuit via silver paste. The connection were made on opposite sides as pro-posed by Chyba and Hand in figure 1 (2016). Once again all the measurements were done, but before and afterwards the resistance value of the Mn-Zn Ferrite was measured. This value could namely influence the 1/RC value, where the denominator stands for the time that had to be bridged before a trustworthy measurement could be done. Before and after the measurements, the resistance values were respectively 5,95(KΩ) and 6,16(KΩ). These values are lower than the resistance of the resistor in the low pass filter, so the time constant did not change as a result of the integration of the Mn-Zn Ferrite in the circuit. After executing some measurements it seemed that every time different results were measured and that the previous results were not reproducible.

Figure 9: Measurement of the direct voltage through a second order low pass filter with magnetic shielding integrated into the circuit.

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By accident the Mn-Zn Ferrite was touched and this resulted in a huge increase of the voltage. The following figure was made of the results:

Figure 10: Measurement of the direct voltage through a second order low pass filter when the Mn-Zn Ferrite was touched.

Based on this figure it was concluded that there was a huge influence of tem-perature, most probably due to the thermo-electric effect, and that this was precluding the experiment from getting reproducible results. Therefore the Mn-Zn Ferrite was isolated by a layer of sealing foil, toilet paper and another layer of sealing foil, so the temperature between the attachment point on the Mn-Zn Ferrite would not differ enormously during the experiment as a result of air flow. This immediately led to a voltage that was constant and did not change over time if noting was changed to the setup. This also influenced the offset that was mentioned before. In between the measurements shown in table 2 and table 3, the magnetic shielding was isolated again, which led to a different off-set. The offsets of the experiments shown in table 2 and table 3 are respectively 16 microvolts and 6 microvolts. During these measurements only the electric circuit was rotated, since there appeared to be no difference between rotating the magnet or the circuit. Of course only the circuit could be rotated when the magnet was removed from the setup. The following results were measured:

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Kind of rotation: 1:No rotation. 2:Clockwise rotation. 3:Counterclockwise rotation.

Table 2: Summary of the results of the measurements while magnetic shielding was integrated into the electric circuit, without presence of a magnet, offset around 16 microvolts.

Kind of rotation, without presence of a magnet 1 2 3

Mean voltage measured (microvolts) 16 16 16

Rms noise (microvolts) 1 1 1

Table 3: Summary of the results of the measurements while magnetic shielding was integrated into the electric circuit, in the presence of a magnet, offset around 6 microvolts

Kind of rotation, in the presence of a magnet 1 2 3

Mean voltage measured (microvolts) 6 8 8

Rms noise (microvolts) 1 1 1

It seems clear that no difference is measured when the magnet is moved away from the setup. On the other hand, when the magnet is placed back into the setup a small but clear difference can be seen between the measurements of the rotating circuit and the measurements of the stationary circuit. The magnet thus has to have an influence on the results. The difference in voltages between the two non rotating experiments is probably the results of a different temperature near the Mn-Zn Ferrite. In between these experiments some other experiments were done, so the Mn-Zn Ferrite had to get isolated again.

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Figure 11: Measurement of the di-rect voltage through a second order low pass filter when the Mn-Zn Ferrite was isolated, without the presence of a magnet, no rotation.

Figure 12: Measurement of the di-rect voltage through a second order low pass filter when the Mn-Zn Ferrite was isolated, without the presence of a magnet, clockwise rotation.

Figure 13: Measurement of the di-rect voltage through a second order low pass filter when the Mn-Zn Ferrite was isolated, without the presence of a magnet, counterclockwise rotation.

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Figure 14: Measurement of the direct voltage through a second order low pass filter when the Mn-Zn Ferrite was isolated, in the presence of a magnet, no rotation.

Figure 15: Measurement of the direct voltage through a second order low pass filter when the Mn-Zn Ferrite was isolated, in the presence of a magnet, clockwise rotation.

Figure 16: Measurement of the direct voltage through a second order low pass filter when the Mn-Zn Ferrite was isolated, in the presence of a magnet, counterclockwise rotation.

9.5 Rotation by 180 degrees

Earlier this article in section 4.10 was mentioned that a rotation by 180 degrees of the Mn-Zn would lead to a switch of signs, if the permanent magnet would be excluded of the setup. This would thus only be a result of the influence of Earth’s magnetic field. The value of the e.m.f. would be given by the formula: e.m.f. = −2Rmvβ2` (a/b)2sin φ0cos2φ0

e.m.f. = −2Rmvβ2` (a/b) 2

sin φ0cos2φ0=

= −2×7. 153 4×10−3×345ms−1_{×2. 890 1×50·10}−6_{T ×2.8cm (1.0cm/1.8cm)}2

0.384 9 = −2.4µV (see section 4.10 for the full calculation)

However, exactly the same results were measured as before the rotation by 180 degrees. Thereby, the calculations mentioned before showed that the contact points that were made on the Mn-Zn Ferrite were maybe placed incorrectly. Another piece of Mn-Zn Ferrite was prepared with the exact configuration as proposed by Chyba and Hand (2016). Unfortunately this did not give any promising result at all. There was no measurable difference between different angles with respect to the magnetic field of the Earth, while it was suggested

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that a difference of 2µV should be observed. This would lead to a total difference of 4µV between the measurements with the contact points on opposite sides. The value of the noise was around 100nV, but no effect was measurable. This seems to contradict the calculations of Chyba and Hand mentioned in section 4.10. These results could suggest that the theory of Chyba and Hand is not perfect and should be investigated even more.

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10 Conclusion

During this experiment the setup was improved and led to more confident re-sults. After all these improvements were implemented it was possible to affirm the theory that an ordinary rectangular loop would not generate an e.m.f. when rotating. Thereby a loop which was partly made of iron would give no note-worthy result either. When the magnetic shielding was introduced the results became more interesting. The shielding integrated into the circuit led to a clear but confusing result. This same result was measured under different configura-tions, but unfortunately when the calculations were done, it appeared that the generated voltages by the setup that was used should have been too small to measure. This could suggest that the theory about this particular subject is in-correct or that there was still an inconsistency within the setup. However, it was not possible to measure the change of signs that was predicted by Chyba and Hand (2016). Summarizing it could be said that it is clear that via a rectangular loop it is impossible to generate an e.m.f. However when magnetic shielding is brought into the setup things get quite unclear. Explicit results were measured but they got contradicted by precise calculations. In the end results were found that contradicted the hypothesis opposed by Chyba and Hand that predicted a change of sign. An obvious conclusion about what happens if magnetic shielding is placed into the setup is not possible to form based on these facts and therefore further research within this field is needed to decide if it is possible to generate electric power from Earth’s rotation through its own magnetic field.

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11 Discussion

For further experiments it would be useful to check what happens if magnetic shielding is placed into the circuit. To facilitate these experiments firstly the rotational speed should be increased so if it would results in voltages, these voltages would be bigger and therefore easier to measure. It should be useful as well to keep track of the actual speed with which the setup is rotated, to be able to check the results with calculations. Secondly, the influence of the magnetic field of the Earth was quite bothersome during experiments with a permanent magnet. To securely do all these measurements one should fix a place wherein no influence of external magnetic fields were measured. For further research within this particular field it would be useful to chose some parameters in just a way that the setup generates way bigger voltages. This could for example be done by choosing a magnetic shielding that is really thin, so the factor (a/b) becomes much bigger. At last, the magnet used in this experiment is not perfect. This was measured by a Gauss meter which said that there were small fluctuations of the magnetic field on the surface of the magnet. This could naturally influence the results as well.

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12 Acknowledgements

I want to thank everyone that supported me during this experiment. Without you I would have lacked mostly the experience of what should be taken into account doing an experiment. I would like to thank Ron Lootens for building the voltmeter setup and improving it during the experiment. Last but not least I want to thank Rinke Wijngaarden from the bottom of my heart for helping me through this project. At different times I seemed to run out of ideas of what to do next, but every time Rinke inspired me to go on and brought me to new insights. For example, some of the calculations in this article seemed unusable to me until Rinke showed me the significance of them. Thank you for giving me a taste of what it’s like to do academic research!

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13 Appendix

13.1 Graphs of some measurements mentioned in section

9.2

Figure 17: Mentioned in Table 1 as configuration 3: Measurement of the direct voltage through a second order low pass filter when the magnet and the electric circuit do not rotate with respect to each other.

Figure 18: Mentioned in Table 1 as configuration 4: Measurement of the di-rect voltage through a second order low pass filter when rotating the magnet clockwise with respect to the electric circuit.

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Figure 19: Mentioned in Table 1 as configuration 5: Measurement of the di-rect voltage through a second order low pass filter when rotating the magnet counterclockwise with respect to the electric circuit.

13.2 Is the permanent magnet used during the

experi-ments far from perfect?

Using a Gauss meter, it was tried to clarify to what extent the permanent magnet used during the experiments had a homogeneous magnetic field. When the Gauss meter was nearly clamped against the magnet, only really minuscule changes were measured, varying between 0,484 and 0,490 Tesla, while the meter Gauss meter was moved from the center of the magnet to the edge. When the distance between the magnet and the Gauss meter was increased by two centimeters the fluctuations became bigger with values varying between 0,142 and 0,213 Tesla when the meter was moved along the surface of the magnet, moving from the centre of the magnet to the edge.

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14 References

-Barnett, S.F. (1908). An Investigation of the Electric Intensities and Electric Displacement produced in Insulators by their motion in a Magnetic Field, Phys. Rev. 35, 323.

-Chyba, C.F. Hand, H.P. (2016). Electric Power Generation from Earth’s Ro-tating through its Own Magnetic Field, Phys. Rev. Applied 6, 014017.

-Davis, L. (1947). Stellar electromagnetic fields, Phys. Rev. 72, 632. -Davis, L. (1948). Stellar electromagnetic fields, Phys. Rev. 73, 536. -Griffiths, D. J. (2005). Introduction to electrodynamics.

-Hones, E. W. Bergeson, J.E. (1965). Electric field generated by a rotating magnetized sphere, J. Geophys. Res. 70, 4951.

-Jackson, J. D. (1975). Electrodynamics. Wiley-VCH Verlag GmbH Co. KGaA.

-Kennard, E.H. (1917). On unipolar induction: Another experiment and its significance as evidence for the existence of the aether, Philos. Mag. 33, 179. -Leus, V., Taylor, S. (2011). On the motion of the field of a permanent magnet. European Journal of Physics, 32(5), 1179.

-Lorrain, P., Lorrain, F., Houle, S. (2007). Magneto-fluid dynamics: fundamen-tals and case studies of natural phenomena. Springer Science Business Media. -Muller, F. J. (2014). Unipolar Induction Revisited: New Experiments and the “Edge Effect” Theory. IEEE Transactions on Magnetics, 50(1), 1-11.

-Pegram G.B. (1917). Unipolar induction and electron theory, Phys. Rev. 10, 591 .

-Disc magnet Ø 70 mm height 35 mm. (2017). Retrieved from https://www.supermagnete.de/eng/disc-

Electric Power Generation from Earth’s Rotation through its Own Magnetic Field?