Extended Cases of Laboratory Generated Gravitomagnetic Field Measurement Devices

A method is described for creating a measurable unbalanced gravitational acceleration using a gravitomagnetic field surrounding a superconducting toroid. A gravitomagnetic toroid for unbalanced force production has been experimentally realized as quite impractical. However recent advances in nanorod superconducting wire technology has enabled a new class of SMES devices operating at current densities and magnetic field strengths sufficient to develop measurable gravitomagnetic fields, while still maintaining mechanical integrity. It is proposed that an experimental SMES toroid configuration uses an absolute quantum gravimeter to measure acceleration fields along the axis of symmetry of a toroidal coil, thus providing experimental confirmation of the additive nature of the gravitomagnetic fields, as well as the production of a linear component of the overall acceleration field. In the present paper relativistic enhancement of this effect is also explored, as well as alternating current (AC) operations of the superconducting toroid to create gravitational waves. Lorentz force concerns are also addressed in Appendix.

• Gravitational
• Gravitomagnetic
• Lense-Thirring
• Superconducting Magnetic Energy Storage
• SMES
• Nanorods
• Nanowires

1. Introduction

When Forward [1] first proposed a gravitomagnetic toroid for unbalanced gravitational force production in 1962 any experimental realization was quite impractical. However recent advances in high temperature superconducting (HTSC) nanorod wire (nanowire) technology, described recently by Rieken and Bhargava et al. [2] , have enabled a new class of superconducting magnetic energy storage (SMES) devices operated at current densities and magnetic field strengths sufficient to develop measurable gravitomagnetic fields, while still maintaining mechanical integrity. In the present study, it is proposed that an experimental SMES toroid configuration uses a set of standard accelerometers to measure acceleration fields along the axis of symmetry of a toroidal coil, thus providing experimental confirmation of the additive nature of the gravitomagnetic fields, as well as the production of a linear component of the overall acceleration field. See Figure 1 for details.

In the instantiation of Forward’s gravitational generation coil described in this paper superconducting electron flow provides the change in mass current in the toroid.

2. Background

In this section, we provide summaries of enabling developments in high current density nanorod conductors, as well as the overall design and use of Superconducting Magnetic Energy Storage (SMES) devices, which are emerging as an alternate approach to energy storage that does not require chemical energy technologies.

2.1. Superconducting Nanorods

New developments in nanomaterial processing of superconductors [2] [3] have led to the discovery of nano-tubular superconductors Rieken and Bhargava et al. which have a Tc at 92 K. A uniqueness of the nano-tubular and other geometric structures of high temperature superconducting (HTSC) makes for a practical wire form without using the melt texturing techniques which make for brittle

thin films that are also difficult to shape into wire. Another aspect of these new HTSC materials is negation of post oxygenation at high temperatures. The elimination of this requirement makes room temperature forming and application of HTSC materials practical. The process has been demonstrated to be a low-cost and mass production method of superconductors which is scalable and without vacuum or cleanroom requirements, Rieken and Bhargava et al. [2] . These developments have led to the commercialization by True 2 Materials PTE., LTD (T2M) in Singapore, of a new HTSC wire using standard wire making practices.

Although the critical temperature of the wire is 92 K, operation at 77 K in liquid nitrogen is more reasonable due to safety issues with gases and nitrogen’s inertness, non-explosive and non-flammable, as a cryogenic liquid. Currently T2M prototype wire is in the millimeter range and approaching the micron range. However, development of an HTSC wire, or filament at nanometer scale is on the roadmap [3] of T2M. The estimated diameter of the wire used in this study, currently theoretical, is 200 nm O.D. including insulation and a 30 nm O.D. HTSC core, with the total weight of the wire at approximately 0.001 g/m.

Individual wires make up a 19 nano filament cable of 1 micron diameter as shown in Figure 2(a) illustrate the compactness of the nano cable design. This allows the scaling up of the critical current limit, quenching aside, without adding significant weight to the toroidal coil. The individual nano filament as described in Figure 2(b) as illustrated is composed of a core (a), core sleeve (b), a highly-insulated sleeve (c) with good heat transport properties and a high strength giga-pascal (GPa) outside sleeve (d) which also possesses good heat transport properties at low temperatures. The main consideration for a candidate of the materials used in this study would be of carbon composition.

To meet the 250 MA/m² required in this study thousands of filaments may be required, even with current flow between 2000 A/mm² to as high as 20,000 A/mm², although higher currents should be obtained by decreasing temperatures from 77 K to 4 K. This would allow many different cable packing designs giving designers greater freedom in energy flow design. As such the description above may be possible within practical limits due to the size and weight of each filament. Progress and work on development of a geometrical superconductor has been accomplished, however still under study, which may have high enough current densities to overcome thousands of Tesla of internal magnetic field, thereby overcoming the problems of quenching. This is partially accomplished due to the geometry of the material and near perfect crystal alignment enhancing flux trapping due to a ferromagnetic component. It is also considered and still under study that vortex quantum effects, i.e. quantum entanglement, are also playing a role in reducing or elimination of quenching.

2.2. Superconducting Magnetic Energy Storage

Superconducting Magnetic Energy Storage (SMES) devices are an emerging battery replacement technology [4] . A typical application is shown in Figure 2(c).

The device is fed by a DC current, developing a magnetic field, typically in a toroidal geometry coil. When the need for emergency power is detected, an output switch is activated that provides DC current out, which may be converted to AC power by a power inverter.

Given the recent advances in nanowire as described in Section 2.1, these devices are poised for remarkable improvements in capability in the very near term. With these coming improvements in this technology, and the similarity in geometry with the Forward design of Figure 1, the present paper will study this technology at its limits for possible reapplication as a DC or low frequency gravitomagnetic generator.

3. Gravitomagnetic Force Equation for Toroid Mass Flow

As developed in Forward 1962 Ref. [1] the linear force G_f developed by gravitomagnetic force in the mass flow toroid of Figure 1 is given by Equation (1):

G f = ( η 4π ) ( N T ˙ r 2 / R 2 ) (1)

where η is gravitomagnetic permeability, η = η o η r .

Single electron mass flow shown in Figure 3 is given by mass momentum in Equation (2):

T e = p e = ( Ω × r ) m e (2)

where Ω = angular rate, angular velocity is v = Ω × r in the classical case [5] .

Change in mass flow for the single electron flow shown in Figure 3 is given by Equation (3):

T ˙ e = p ˙ e = a ⋅ m e = ( Ω × v ) m e (3)

This is equivalent to centripetal force shown in Equation (4):

T ˙ e = F e = m e v 2 / r = m e a e = m e ( ω 2 r ) (4)

where ω is the angular rate, given in Equation (5):

ω = 2 π / t p = d θ d t (5)

We now consider the same circular motion, but in a relativistic regime. Relativistic circular motion [6] can be described by Equation (6):

u ¯ ⋅ a ¯ = 0 → α 2 = γ 4 ⋅ a 2 (6)

which reduces to Equation (7) for the relativistic acceleration of circular motion:

α = γ 2 ⋅ v 2 / r (7)

From Equation (4), mass flow change (force) for a relativistic electron is therefore expressed as follows:

T ˙ e = m e ( v 2 / r ) γ 2 (8)

where m_e is rest mass and gamma γ is defined [7] as Equation (9):

γ = 1 1 − v 2 / c 2 (9)

Effects go as gamma squared. Numerical solutions are described in Table 1.

Relativistic cases will be explored further in Section 7.

4. Current in Idealized SMES

We now attempt to estimate the possible currents enabled by the emerging technology of Section 2.1 as it relates to the core geometry constraints described in Figure 4, a toroid with torus geometry. We start with the assumptions needed to calculate the number of turns N.

From Equation (1) the torus assumptions made in Figure 4 can be factored into Equation (10) as follows:

G f = ( η o η r 4 π ) ( N T ˙ r 2 / R 2 ) (10)

where:

G_f = gravitomagnetic force;

η o = absolute gravitomagnetic permeability;

η r = relative gravitomagnetic permeability;

N = number of turns in the coil of the torus;

T ˙ = change in mass flow;

r = cross section radius of torus;

R = centerline radius of torus.

For the purposes of describing an idealized case with a realistic geometry we develop a description of a device bounded by a 10 meter toroid centerline diameter, shown in Figure 4 and Figure 5, and with a cross-sectional diameter of 1 meter, as shown in Figure 4. We furthermore define 16 sectors as shown in Figure 5 for possible AC operation described further in Section 8.

Table 1

Relativistic parameters for uniform circular motion of an electron (SI units)

Velocity (relative to speed of light)	Gamma squared, γ2 (unitless)	Relativistic acceleration, α (m/s2)	Per electron predicted force developed, T ˙ e (N)
0.5c	1.3	5.9(10)16	5.4(10)−14
0.9c	5.3	7.7(10)17	7.0(10)−13
0.99c	50	9.1(10)18	8.3(10)−12
0.999c	500	9.0(10)19	8.2(10)−11
0.9999c	5000	9.0(10)20	8.2(10)−10
0.99999998c	25,000,000	4.5(10)24	4.1(10)−6

We further add additional assumptions regarding to what extent conductors are wrapped around the toroid shaped device to determine constraints on the number of conductive loops that can be accommodated using the technology described in Section 2.1. As shown in Figure 6 via cross section we assume here a conductor winding depth of 0.5 m.

The segmented share of the inside of cross section C_s is 1/16^th of the overall inner circumference as given in Equation (11) as follows:

C s = 2 π r i / 16 = 1.57   m (11)

with a depth D the minimum inner loop cross sectional area can be described in Equation (12) as follows:

A s e c = D ⋅ C s = 0.5   m × 1.57   m = 0. 785   m 2 (12)

This area is shown packed with conductors in Figure 7. Windings are depicted as packed in depth and along sector circumference.

Assuming each nanowire conductor has a diameter d_c of 100 μm, then the cross-sectional area of each conductor will be given by Equation (12) as follows:

A c = π r 2 = 7.854 × 10 − 9   m 2 (13)

For packing the conductors in a cross sectional area described in Figure 7, assume as a worst case rectangular area described by the shortest edges such that a number of conductors in depth, N_d, may be packed in one dimension, with the number of conductors, N_cs, packed in the other dimension. These packing counts may be calculated in Equation (14a) and Equation (14b) as follows:

N d = D c / d c = 0.5   m / 100   μ m = 5000 (14a)

N c s = C s / d c = 1.57   m / 100   μ m = 15700 (14b)

The total number of windings by sector will therefore be the product of N_d and N_cs:

N s e c = N d ⋅ N c s = 5000 × 15700 = 78500000 (15)

And with 16 sectors the total number of windings for the entire toroid will be:

N = 16 N c s = 1.256 × 10 9 (16)

what is T ˙ with the forgoing assumptions?

In this idealized case electrons circulate about a coil of circumference c_r, or slightly larger, as described by Equation (17):

c r = 2 π r = 3.14   m (17)

Non-relativistic case

Consider two cases, the first a non-relativistic case. Assume further a supply voltage of 16 KV, resulting in 16 KeV of kinetic energy for each electron, which corresponds with the upper limit of a non-relativistic case, where v = 0.25c, so that γ = 1.06 ~ 1.0.

Then from Equation (4) for non-relativistic circular motion, the vector change in DC current flow is:

T ˙ e = m e v 2 / r (18)

which for a single electron has the following values:

m_e = mass of the electron = 9.11 × 10⁻³¹ kg;

v = velocity of the electron = 0.25c = 0.75 × 10⁸ m/s;

r = 0.5 m for the assumed geometry.

And where the angular acceleration of the electron is:

a e = v 2 / r = 1.125 × 10 16   m / s 2 (19)

Thus, change in mass flow represents centripetal acceleration in the case of circular motion:

T ˙ e = m e ⋅ a e = 10.25 × 10 − 15   N (20)

Equation (20) corresponds to the change in mass flow for one electron in one loop of coil. Total mass flow change is therefore the mass flow change per electron times the number of electrons:

T ˙ = T ˙ e ⋅ N e (21)

what is the number of electrons N_e in one loop in motion (part of the mass flow) at a given time for an assumed velocity of v = 0.25c? N_e in one loop can be described by the current I times the period of a single loop circulation Δ t :

N e = I ⋅ Δ t (22)

where the period of an orbit can be described by:

Δ t = c r / v = 2 π r / v = 41.89   ns (23)

what is the possible current inside the idealized device for the case where the entire winding is in series? We make the assumption about max current to stay below critical current density of 250 MA/m² as described in Section 2.1.

Current is limited by the maximum permissible current density and the cross section of the conductor t:

I = J ⋅ A c (24)

where J is material dependent. For the nanowire assumed in Ref. [2] , J = 250 MA/m². Cross sectional area A_c = 7.854 × 10⁻⁹ m² as given in Equation (13). Therefore, maximum current for this conductor diameter is I = 1.96 Amps.

Expanding on Equation (22) the number of electrons N_e in circulation in one loop may be calculated by noting that there are 6.2415 × 10¹⁸ electrons per Coulomb:

N e = ( electrons Coulomb ) I ( Coulombs sec ) ⋅ Δ t = 5.12 × 10 11   electrons (25)

5. Forces in Idealized SMES

Expressing Equation (21) as force per electron times the number of electrons in motion in one loop:

T ˙ = T ˙ e ( Newtons electron ) ⋅ N e ( # electrons ) = 5 . 248   mN (26)

Thus, each loop experiences about 5 mN of integrated centripetal force ( T ˙ ) due to the electrons in circulation within.

We now describe the scale factor to couple this force to the gravitomagnetic effect.

Revisiting Equation (10) which describes the overall linear force developed at the center of the toroidal coil, total gravitomagnetically developed force will be:

G f = ( η o η r ) ( N T ˙ r 2 / 4 π R 2 ) = ( η o η r ) ( 5248   N ) (27)

where known variables have been grouped on the right and unknown variables have been collected on the left. This raises the question what are the correct values for η o and η r ?

If η o goes as G/c as does gravitomagnetic potential (Ref. [8] , Equation (1.5)), then:

η o = − G / 2 c = 1.11 × 10 − 19 (28)

In this case G f = ( η r ) 5.8 × 10 − 16 . Values of η r are experimentally unknown at this time. However if values of η r track values of μ r then values as high as η r = 10 6 may be possible, yielding G f = 5.8 × 10 − 10   N = 0.58   nN .

Even with very sensitive measurement apparatus this would be a very difficult measurement. However, with additional current or winding count a device scaled up from the idealized case considered in this paper may someday achieve a measurable DC gravitational field, even in the non-relativistic case considered in this section.

6. Stored Energy in an Idealized SMES

What is the stored energy in the idealized torus for the non-relativistic case of Sections 4 and 5? The total energy is the sum of E_k, the kinetic energy of all of the particles in motion within the coil, and the contained magnetic field energy E_m:

E T = E k ⋅ E m (29)

where kinetic energy E_k in all loops from all electrons in each loop is:

E k = N N e ( 1 2 m e v 2 ) = 1.65   MJ (30)

where winding count N is defined in Equation (16) and electrons per loop N_e is defined in Equation (25). Velocity assumed here is the essentially non-relativistic case of v = 0.25c.

Magnetically stored energy is the primary purpose of SMES devices and is where most of the energy is contained. For a coil the magnetically stored energy E_m is related to the inductance of the coil [9]:

E m = 1 2 L ( I 2 ) (31)

For a circular cross sectioned torus shaped toroidal coil with an air core ( μ r = 1 ) the inductance may be approximated as follows [10]:

L ≅ 0.007975 d 2 D N 2   μ H = 1.26 × 10 15   μ H = 1.26 × 10 9   H (32)

which yields a stored magnetic energy E_m of 2.4 GJ. Thus E_T ~ E_m since E m ≫ E k . For SMES applications this is equivalent to E_m = 2.4 × 10⁶ kW seconds = 670 kW hr.

However, in Equation (31) this calculation assumes that μ = μ o , i.e. μ r = 1 . An alternate formulation may be used for μ ≠ μ o of the form:

L = N 2 μ A l (33)

which is depicted in Figure 8. Where in this case [11]:

μ = μ o μ r (34)

A = π r 2 = 0.785   m 2 (35)

Assuming a modest μ r = 600 (equivalent to soft iron) then L = 3.0 × 10¹³ H. Assuming a current I = 1.96 A to keep below the critical current density of the material of Reference [2] then from Equation (31) the contained energy of the coil will be:

E m = 1 2 L ( I 2 ) = 5.76 × 10 13   J = 5.76 × 10 8   kW ⋅ hr (36)

Therefore:

E T = E k + E m ~ E m = 5.76 × 10 8   kW ⋅ hr (37)

For comparative purposes, an electric auto battery requires an energy storage of 120 kW hours for a range of 320 miles. Therefore idealized non-relativistic reference device under analysis is some 6 orders larger than required for use as an automotive battery. Thus an automotive scale device could be sized at 10 cm × 10 cm × 2 cm.

Table 2 covers a range of other permeability cases for the SMES application.

Table 2

SMES Inductance and Contained Magnetic Energy as a function of core magnetic permeability

Core Permeability (μr vs μo)	Inductance (H)	Stored Magnetic Energy (J)
1	1.26(10)11	2.4(10)11
600	3.0(10)13	5.76(10)13
100,000	5.0(10)15	9.6(10)15
1,000,000	5.0(10)16	9.6(10)16

In the case of the ~10 PetaJoule (PJ) example, this instantiation of the device could supply 1.0 Gigawatt (GW) of power for 10 million seconds, or over 115 days, providing substantial mission power between charges.

However, these numbers also point out that such devices should be operated with the greatest of care and concern for safety. This is mitigated by the very gradual quenching characteristics of the nanowire materials described in Section 2.1 [2] .

7. Relativistic Operation in an Idealized SMES

We now consider an idealized case of relativistic motion for the electrons within the SMES torus to examine possible improvements in performance for this case. What is a reasonable relativistic energy limit for electrons in this application? For present purposes, we use Lawrence Berkley National Labs recently developed Laser Plasma Accelerator, which accelerates electrons to 4.25 GeV within a length of only 9 cm [12] .

Converting to Joules, for this case the energy of a single electron E_e would be [13]:

E e = 4.25 × 10 9   eV × 1.6 × 10 − 19   C / e × 1   J / C = 6.8 × 10 − 10   J (38)

This represents the total kinetic energy of the electron:

E e = m e c 2 ( γ − 1 ) (39)

where m_e is the rest mass of an electron. If solved for γ Equation (39) gives γ = 8293.

Because the electrons are traveling in matter, albeit superconducting matter, even for our idealized case we may assume some energy losses due to domain boundary scattering and Bremsstrahlung radiation. For the purposes of our idealized case we will assume average γ = 5000 < 8293.

We now revisit the gravitomagnetically induced DC gravitational force developed at the center of the SEMS torus as given in Equation (1):

G f = ( η 4 π ) ( N T ˙ r 2 / R 2 ) (1)

where η is gravitomagnetic permeability, which is defined as η = η o η r .

And from Equation (28) we have:

η o = − G / 2 c = 1.11 × 10 − 19 (28)

And where η r may take values over a range yet to be experimentally determined, but if similar to μ r may range in value from 1 to 1,000,000 [14] .

Here N = 1.256 × 10⁶ turns, r = 0.5 m, and R = 5.0 m, all of which are the same geometry assumptions as the non-relativistic case. What is not the same will be the change in mass flow T ˙ :

T ˙ = T ˙ e ⋅ N e (21)

T ˙ will be different because both T ˙ e and N e will change due to the relativistic motion of the electrons.

N_e will be decreased because the faster circulation time results fewer electrons in motion within each loop at the same time due to their higher velocity:

N e = I ⋅ Δ t (22)

where the period of an orbit can be described by:

Δ t = c r v = C r C (40)

since in this case v ~ c. Therefore Δ t will be reduced from 41.89 ns for the non-relativistic case to 10.47 ns in this relativistic case. This reduces N_e from 5.12 × 10¹¹ electrons as calculated in Equation (25) to just 1.28 × 10¹¹ electrons in this relativistic case.

This reduction in N_e will be more than compensated by the γ² growth in the change in mass flow per electron as measured from the rest frame. Because we are assuming γ = 5000 the increase in measurable force will be a factor of 25,000,000, over 7 orders of magnitude. This is expressed in Equation (41) as follows:

T ˙ e = m e a e = m e ( γ 2 v 2 / r ) = m e ( γ 2 c 2 / r ) (41)

for a relativistic case of circular motion. This yields a T ˙ e of 4.1 × 10⁻⁶ N for a single electron, a force (or change in mass flow) of a T ˙ = N e T ˙ e of 5247 N for all the mass flow change in an entire loop, and a gravitational force developed from the gravitomagnetic effect in the center of the SMES torus of G_f = 5.8 × 10⁻¹⁰ N if η r = 1 , or G_f = 5.8 mN if η r = 1000000 . For a 1 kg test mass in the center of the torus an acceleration of 0.057 m/s² would be observed, or 5.8 mG. This is well within the measureable range of absolute quantum gravimeters [15] and it should be possible to establish an upper bound for η_r using a device similar to this idealized relativistic case.

Thus, although relativistic operation does nothing to improve current or inductance, and therefore accrues no benefit for storing energy, it would theoretically provide quite a dramatic improvement in gravitational force generation.

8. Alternating Current (AC) Operation in an Idealized SMES

We now return to the notion that our idealized torus may be built in sectors as seen in Figure 9, and while in the cases of DC operation it is possible to place all

sectors in series, in order to create a quadrupole moment of mass flow these sectors, shown in Figure 10, the sectors must be powered in parallel. This means that 32 Amps will be required to provide 2 Amps to each of 16 sectors.

Returning for a moment to the DC case, from Equation (27) we have shown when all sectors are energized in the same rotational direction, that G f = ( η r ) 5.8 × 10 − 16   N , where worst case is η_r = 1. In this case G f = ( η r ) 5.8 × 10 − 16   N . In this case for a 1 kg test mass m_t from Newton’s 2^nd law:

a f = G f / m t = 5.8 × 10 − 16   m / s 2 (42)

In Earth’s gravitational field g of 9.8 m/s² this is equivalent to a relative strain of:

h f = a f / g = 5.9 × 10 − 17 (43)

For AC (Alternating Current) operation, powering sectors 15, 16, 1, and 2 as well as 7 through 10 with “inspiralling” current, while simultaneously powering sectors 3 - 6 and 11 - 14, with “outspiralling” current, as shown in Figure 10, and then periodically reversing, will create a quadrupole modulated gravitational field as depicted in Figure 11.

From Figure 9 and Figure 10 only half of the coil is energized in one direction at a time, so +a_h ~ 1/2a_f. Therefore, in the non-relativistic case total amplitude of the quadrupole GW wave is:

a g w = | − a h | + | + a h | = 1 2 a f + 1 2 a f ~ a f (44)

Then by similarity h g w = 5.9 × 10 − 17 for the non-relativistic case.

From Section 7 for the relativistic case:

h f = a f / g = 5.9 × 10 − 11 η r (45)

Even assuming a worst case η_r = 1, this value of h_f should be detectable by

LIGO if operated at a 1 Hz quadrupole oscillation rate. In accordance with Martynov et al. [16] , LIGO sensitivity at 1 Hz is h > 10⁻²³ Hz.

9. Conclusion

An argument is made for using SMES to gravitomagnetically create an unbalanced force, possibly of measurable amplitude. Sector partitioned actuation of a similar SMES device may also be used to generate gravitational waves. Further research would be required to determine to what extent SMES devices could be operated into relativistic regimes to enhance relative mass flow change in the rest frame.

Acknowledgements

The authors wish to acknowledge the Nara Institute of Science and Technology for their support of HTSC nanowire technology. The financial support of Seculine Consulting is also gratefully acknowledged.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

Appendix, Lorentz Forces inside an Idealized SMES

Due to the large magnetic fields developed in SMES devices Lorentz forces are a major design constraint. In this appendix, we explore how nanowires based on weaving together the nanotube fibers described in reference [2] should have sufficient tensile strengths to allow some of the energy storage levels described in Table 2 of this paper without the need for additional structural elements.

In accordance with reference [17] Lorentz forces developed in the idealized SMES device can be described on a per electron basis as shown in Equation (A1):

F e = q v × B (A1)

where:

q = q e = e = 1.6 × 10 − 19   C = elementary charge, the charge per electron;

v = 0.25 c = 0.75 × 10 8   m / s for non-relativistic case (v ~ 1c = 3 × 10⁸ m/s in relativistic case);

B = magnetic field in the core of an idealized SMES device.

See Figure A-1 for a depiction of the Lorentz force developed by the motion of a single electron in one loop of an SMES device.

What is B in this case? As described by reference [18] magnetic field strength inside a torus coil is given by (A2):

B = ( μ N I 2 π R ) (A2)

where here we will assume values earlier developed for the idealized SMES:

μ = μ_rμ₀ magnetic permeability;

μ_ = 1.26 × 10⁻⁶, vacuum magnetic permeability;

μ_r = 600, a moderate value for relative permeability, the equivalent of an iron core;

N = 1.256 × 10⁹, the number of loops given in Equation (16);

I = 1.96 Amps in the idealized case of Equation (24);

R = radius of torus, assumed to be 5.0 m, from Figure 4.

This gives a magnetic field of B = 600 × 1.26 × 10 − 6 × 1.26 × 10 9 × 1.96 / 10 π = 59000   T in the core of the SMES. The inner most winding loop of the SMES device will experience this field strength. Ideally this field will drop linearly until the field is zero at the outside of the torus. Field orientation inside the core of a torus is depicted in Figure A-2 for in spiraling current with velocity v_e:

With the worst case magnetic field the electron Lorentz force will be:

F e = q v × B = 1.6 × 10 − 19 × 0.75 × 10 8 × 5.9 × 10 4 = 7.1 × 10 − 7   N / electron

In a non-relativistic case the number of electrons per loop is given in Equation (25) as N_e = 5.12 × 10¹¹ electrons/loop.

Lorentz Force developed in a single loop due to the electrons in motion within that loop is given as follows for a worst case magnetic field:

F = F e N e = 7.1 × 10 − 7   N × 5.12 × 10 11 = 363500   N (A3)

As shown in Figure A-3 this force is spread over the entire conductor loop circumference and across its cross-sectional area. The force application area can therefore be given as:

A f a = C r d c = 3.14 × 10 − 4   m 2 (A4)

where:

C_r = loop circumference, which on average is 3.14 m from Equation (17);

d_c = conductor diameter assumed to be 100 μm.

Lorentz pressure, or Lorentz force per unit area using the above values is given by:

P L = F e N e / A f a = 363500   N / 3.14 × 10 − 4   m 2 = 1.16 × 10 9   Pa / loop (A5)

Equation (A5) gives Lorentz pressure for a worst case magnetic field, but the average pressure on a loop will be half of that, or 1.16 × 10⁹ Pa. Over a depth of N_d = 5000 loops the average Lorentz pressure per loop P_L = 0.58 × 10⁹ Pa, zero

on the outside, 1.16 × 10⁹ Pa on the inner loop, and shrinking linearly going from inside to out.

For the materials in reference [2] , the actual containment tensile strength of the conductors T_sn = 260 GPa > 1.16 GPa, so for the reference architecture T_sn > P_L. For an improved performance case where a d_c = 1 μm diameter nanowire conductor composed of woven nanotube fibers as shown in Figure 2(a), there would be more pressure per unit area, however if total current in all 19 filaments is limited to the original 1.96 A, then

T s n = 260   GPa > 116   GPa = P L (A6)

Thus there would still be better than a factor of two margin for tensile strength, even in this more tightly packed case. It is therefore possible to build the SMES using this dense packing factor to withstand Lorentz forces at 1.96 A without the need for supplemental structural elements.

Nomenclature

h = Planck’s constant

N = total loop count

Ω = angular rotation rate vector (rad/sec)

η_r = gravitomagnetic permeability

r = toroid cross section radius (m)

R = toroid radius (m)

v_e = velocity, electron

μ = magnetic permeability

Acronyms

HTSC―high temperature superconductor

LIGO―laser interferometer gravitational-wave observatory

LN2―liquid nitrogen

SMES―superconducting magnetic energy storage

SC―superconductor

T2M―True 2 Materials PTE, LTD

References

1. Forward R. ,et al. (1962)Guidelines to Antigravity American Journal of Physics 31, 166-170.

2. Rieken, W., Bhargava, A., et al. (2018) YBa2Cu3Ox Superconducting NanoRods. Jpn. J. Appl. Phys. 57 023101. https://doi.org/10.7567/JJAP.57.023101

3. http://www.true2materials.com

4. https://en.m.wikipedia.org/wiki/Superconducting_magnetic_energy_storage?wprov=sfti1

5. Knudsen, J.M. and Hjorth, P.G. (2000) Elements of Newtonian Mechanics: Including Nonlinear Dynamics. 3rd Edition, Springer, Berlin, 96. https://doi.org/10.1007/978-3-642-57234-0

6. https://en.wikipedia.org/wiki/Circular_motion#Relativistic_circular_motion

7. Forshaw, J. and Smith, G. (2014) Dynamics and Relativity. John Wiley & Sons, Hoboken.

8. Mashhoon, B. Gravitoelectromagnetism: A Brief Review. arXiv:gr-qc/0311030v2.

9. Jackson, J.D. (1998) Classical Electrodynamics. 3rd Edition, John Wiley & Sons, New York, 213.

10. http://www.nessengr.com/technical-data/toroid-inductor-formulas-and-calculator/

11. http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/indcur.html

12. Bonatto, A., et al. (2015) Passive and Active Plasma Deceleration for the Compact Disposal of Electron Beams. Physics of Plasmas, 22, Article ID: 083106. https://doi.org/10.1063/1.4928379

13. CODATA Value: Electron Volt. The NIST Reference on Constants, Units, and Uncertainty, Gaithersburg, June 2015. https://physics.nist.gov/cgi-bin/cuu/Value?tevj|search_for=electron+Volt

14. Metglas Magnetic Alloy 2714A. https://metglas.com/wp-content/uploads/2016/12/2714A-Technical-Bulletin.pdf

15. Freier, C., et al. (2016) Mobile Quantum Gravity Sensor with Unprecedented Stability. Journal of Physics: Conference Series, 723, Article ID: 012050. https://doi.org/10.1088/1742-6596/723/1/012050

16. Martynov, D.V., et al. (2016) Sensitivity of the Advanced LIGO Detectors at the Beginning of Gravitational Wave Astronomy. Physical Review D, 93, Article ID: 112004. https://doi.org/10.1103/PhysRevD.93.112004

17. https://en.wikipedia.org/wiki/Lorentz_force

18. http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/toroid.html