Artificial Gravitational Lenses

Artificial Gravitational Lenses Fran De Aquino Maranhao State University, Physics Department, S.Luis/MA, Brazil.

Copyright © 2012 by Fran De Aquino. All Rights Reserved.

We show that it is possible to produce gravitational lenses at laboratory scale by means of a toroidal device which strongly intensifies the radial gravitational acceleration at its nucleus, and can make the acceleration repulsive besides attractive. This means that a light flux through the toroid can become convergent or divergent from its central axis. These lenses are similar to optical lenses and can be very useful for telescopes, microscopes, and for the concentration of solar light in order to convert solar energy into thermal energy.

Key words: Modified theories of gravity, Gravitational lenses, Solar instruments.

PACS: 04.50.Kd, 98.62.Sb, 95.55.Ev.

1. Introduction

2. Theory

It is known that Gravitational fields can bend light. This effect was confirmed in 1919 during a solar eclipse, when Arthur Eddington observed the light from stars passing close to the sun was slightly bent, so that stars appeared slightly out of position [1]. Einstein realized that a massive astronomical object can bend light making what is called a gravitational lens. The gravitational lensing is one of the predictions of Einstein's general theory of relativity.

Although this phenomenon was first mentioned in 1924 by Orest Chwolson [2], the effect is more commonly associated with Einstein, who published a more famous article on the subject in 1936 [3, 4].

Here we show that it is possible to produce gravitational lenses at laboratory scale, by means of a toroidal device which strongly intensifies the radial gravitational acceleration at its nucleus, and can make the acceleration repulsive besides attractive * [5].

This means that a light flux through the toroid can becomes convergent or divergent from its central axis. These lenses are similar to optical lenses and can be very useful for telescopes, microscopes, and for the concentration of solar light in order to convert solar energy into thermal energy.

From the quantization of gravity it follows that the gravitational mass mg and the inertial mass mi are correlated by means of the following factor [5]:

*

De Aquino, F. (2008) Process and Device for Controlling the Locally the Gravitational Mass and the Gravity Acceleration, BR Patent Number: PI0805046-5, July 31, 2008.

2 ⎧ ⎡ ⎤⎫ ⎛ Δp ⎞ ⎪ ⎪ ⎢ ⎟⎟ − 1⎥⎬ χ= = ⎨1 − 2 1 + ⎜⎜ ⎢ ⎥⎪ mi 0 ⎪ ⎝ mi 0 c ⎠ ⎣ ⎦⎭ ⎩

mg

(1)

where mi 0 is the rest inertial mass of the particle and Δp is the variation in the particle’s kinetic momentum; c is the speed of light.

When Δp is produced by the absorption of a photon with wavelength λ , it is expressed by Δp = h λ . In this case, Eq. (1) becomes 2 ⎡ ⎤⎫ m g ⎧⎪ ⎛ h mi 0 c ⎞ ⎪ = ⎨1 − 2⎢ 1 + ⎜ ⎟ − 1⎥ ⎬ ⎢ ⎥⎪ mi 0 ⎪ ⎝ λ ⎠ ⎣ ⎦⎭ ⎩ 2 ⎧ ⎡ ⎤⎫ ⎛λ ⎞ ⎪ ⎪ (2) = ⎨1 − 2⎢ 1 + ⎜ 0 ⎟ − 1⎥ ⎬ ⎢ ⎥ λ ⎝ ⎠ ⎪⎩ ⎣ ⎦ ⎪⎭ where λ0 = h mi0 c is the De Broglie wavelength for the particle with rest inertial mass mi 0 .

It has been shown that there is an additional effect - Gravitational Shielding effect - produced by a substance whose gravitational mass was reduced or made negative [6]. The effect extends beyond

2 substance (gravitational shielding) , up to a certain distance from it (along the central axis of gravitational shielding). This effect shows that in this region the gravity acceleration, g 1 , is reduced at the same proportion, i.e., g1 = χ 1 g where χ 1 = m g mi 0

and

g

is

the

v=c

v = c/nr

nr

λ = c/f

λmod = v/f = c/nr f

gravity

acceleration before the gravitational shielding). Consequently, after a second gravitational shielding, the gravity will be given by g 2 = χ 2 g 1 = χ 1 χ 2 g , where χ 2 is

Fig. 1 – Modified Electromagnetic Wave. The wavelength of the electromagnetic wave can be strongly reduced, but its frequency remains the same.

the value of the ratio m g mi 0 for the second gravitational shielding. In a generalized way, we can write that after the nth gravitational shielding the gravity, g n , will be given by

(3)

g n = χ 1 χ 2 χ 3 ...χ n g

This possibility shows that, by means of a battery of gravitational shieldings, we can strongly intensify the gravitational acceleration.

From Electrodynamics we know that when an electromagnetic wave with frequency f and velocity c incides on a material with relative permittivity ε r , relative magnetic permeability μ r and electrical conductivity σ , its velocity is reduced to v = c nr where nr is the index of refraction of the material, given by [7]

ε μ c 2 nr = = r r ⎛⎜ 1 + (σ ωε ) + 1⎞⎟ ⎠ v 2 ⎝

(4)

v c f λ = = = f nr nr

(5)

4π μfσ

n=

N0 ρ A

(7)

where N 0 = 6.02 × 10 26 atoms / kmole is the Avogadro’s number; ρ is the matter density of the lamina (in kg/m3) and A is the molar mass(kg/kmole).

When an electromagnetic wave incides on the lamina, it strikes N f front atoms, where Nf ≅ n Sf φm , φ m is the “diameter” of

Thus, the wavelength of the incident radiation (See Fig. 1) becomes

λmod =

of atoms reached by the radiation is N = nS f ξ . The number of atoms per unit of volume, n , is given by

( )

If σ >> ωε , ω = 2πf , Eq. (4) reduces to

μrσ nr = 4πε0 f

If a lamina with thickness equal to ξ contains n atoms/m3, then the number of atoms per area unit is nξ . Thus, if the electromagnetic radiation with frequency f incides on an area S of the lamina it reaches nSξ atoms. If it incides on the total area of the lamina, S f , then the total number

(6)

the atom. Thus, the electromagnetic wave incides effectively on an area S = Nf Sm , where Sm = 14 πφm2 is the cross section area of one atom. After these collisions, it carries out ncollisions with the other atoms (See Fig.2).

3 Substitution of Eq. (12) into Eq. (11) yields 2 ⎧ ⎡ ⎤⎫ ⎡⎛ P ⎞ λ0 ⎤ ⎪ ⎪ ⎢ = ⎨1 − 2 1 + ⎢⎜⎜ 2 ⎟⎟(nl Sξ ) ⎥ −1⎥⎬ (13) ⎢ ⎥ mi0(l ) ⎪ λ ⎦⎥ ⎣⎢⎝ hf ⎠ ⎥⎦⎪⎭ ⎣⎢ ⎩

mg (l )

atom Sm Wave

Substitution of P given by Eq. (9) into Eq. (13) gives

Fig. 2 – Collisions inside the lamina.

Thus, the total number of collisions in the volume Sξ is

Ncollisions= Nf +ncollisions=nl Sφm +(nl Sξ −nmSφm) = =nmSξ

(8)

The power density, D , of the radiation on the lamina can be expressed by D=

P P = S N f Sm

(9 )

We can express the total mean number of collisions in each atom, n1 , by means of the following equation n1 =

ntotal

photons N collisions

N

2 ⎧ ⎡ ⎤⎫ ⎡⎛ N f SmD⎞⎛ n Sξ ⎞ 1 ⎤ ⎥⎪ ⎪ ⎢ ⎟⎜ l ⎟ ⎥ −1 ⎬ (14) = ⎨1− 2 1+ ⎢⎜⎜ 2 ⎟⎜ ⎥ ⎟ mi0(l ) ⎪ ⎢ f ⎢⎣⎝ ⎠⎝ mi0(l )c ⎠ λ ⎥⎦ ⎥⎪ ⎢ ⎦⎭ ⎩ ⎣

mg(l )

Substitution of N f ≅ (nl S f )φm and into Eq. (14) results

2 ⎧ ⎡ ⎤⎫ ⎡⎛ nl3S 2f Sm2φm2ξD ⎞ 1 ⎤ ⎪ ⎢ ⎟ ⎥ −1⎥⎪⎬ (15) = ⎨1− 2⎢ 1+ ⎢⎜ 2 ⎥ mi0(l ) ⎪ ⎢⎣⎜⎝ mi0(l )cf ⎟⎠ λ ⎥⎦ ⎢⎣ ⎥⎦⎪ ⎩ ⎭ where mi 0(l ) = ρ (l )V(l ) .

Now, considering that the lamina is inside an ELF electromagnetic field with E and B , then we can write that [8]

mg(l )

(10 )

Since in each collision a momentum h λ is transferred to the atom, then the total momentum transferred to the lamina will be Δp = (n1 N ) h λ . Therefore, in accordance with Eq. (1), we can write that 2 ⎧ ⎡ ⎤⎫ λ ⎤ ⎡ ⎪ ⎪ = ⎨1− 2⎢ 1+ ⎢(n1 N) 0 ⎥ −1⎥⎬ = ⎥⎪ mi0(l ) ⎪ ⎢ λ⎦ ⎣ ⎦⎭ ⎩ ⎣ 2 ⎧ ⎡ ⎤⎫ λ0 ⎤ ⎡ ⎪ ⎪ ⎢ = ⎨1− 2 1+ ⎢ntotal photonsNcollisions ⎥ −1⎥⎬ (11) ⎥⎪ λ⎦ ⎣ ⎪⎩ ⎢⎣ ⎦⎭

S = N f Sm

D=

n r (l ) E 2 2μ 0 c

(16 )

Substitution of Eq. (16) into Eq. (15) gives 2 ⎧ ⎡ ⎤⎫ ⎡⎛ nr(l )nl3S2f Sm2φm2ξE2 ⎞ 1 ⎤ ⎥⎪ ⎪ ⎢ ⎟ ⎥ −1 (17) = ⎨1− 2 1+ ⎢⎜ ⎜ 2μ0mi0(l )c2 f 2 ⎟ λ ⎥ ⎥⎬ mi0(l ) ⎪ ⎢ ⎢ ⎠ ⎦ ⎥⎪ ⎣⎝ ⎢ ⎦⎭ ⎩ ⎣

mg(l )

mg(l )

Since Eq. (8) gives N collisions = nl Sξ , we get

ntotal

photons N collisions

⎛ P ⎞ = ⎜⎜ 2 ⎟⎟(nl Sξ ) ⎝ hf ⎠

(12)

In the case in which the area S f is just the area of the cross-section of the lamina (Sα ) , we obtain from Eq. (17), considering that mi0(l) = ρ(l)Sαξ , the following expression

2 ⎧ ⎡ ⎤⎫ ⎡⎛ nr(l )nl3Sα Sm2φm2 E2 ⎞ 1 ⎤ ⎥⎪ ⎪ ⎢ ⎟ ⎥ −1 (18) = ⎨1− 2 1+ ⎢⎜ ⎜ 2μ0ρ(l )c2 f 2 ⎟ λ ⎥ ⎥⎬ mi0(l ) ⎪ ⎢ ⎢ ⎠ ⎦ ⎥⎪ ⎣⎝ ⎢ ⎦⎭ ⎩ ⎣

mg(l )

4 According to Eq. (6) we have v c λ = λ mod = = f nr (l ) f

(19 )

Substitution of Eq. (19) into Eq. (18) gives ⎧ ⎡ 4φ4 E4 ⎤⎫ nr4(l )nl6Sα2 Sm m ⎪ ⎢ ⎥⎪ 1 = ⎨1− 2⎢ 1+ − ⎥⎬ (20) 2ρ 2 c6 f 2 mi0(l ) ⎪ μ 4 ⎢ ⎥⎪ 0 (l ) ⎦⎭ ⎩ ⎣ mg(l )

Note that E = E m sin ωt .The average value for E 2 is equal to

1

sinusoidaly ( E m

is the maximum value

2

E m2 because E varies

for E ). On the other hand, E rms = E m 4

2.

4 rms

Consequently we can change E by E , and the equation above can be rewritten as follows χ=

m g (l )

= mi0(l ) ⎧ ⎡ ⎤⎫ 4φ 4 E4 nr4(l ) nl6 Sα2 S m m rms ⎪ ⎢ ⎥⎪ = ⎨1 − 2⎢ 1 + − 1⎥ ⎬ (21) 2 2 6 2 4μ 0 ρ (l )c f ⎪ ⎢ ⎥⎪ ⎣ ⎦⎭ ⎩

Now consider the Artificial Gravitational Lenses shown in Fig.3.

Basically they are rectangular toroids.

Inside them there are two dielectric rings with ε r ≅ 1 and an Aluminum ring with mass density ρ = 2700kg .m −3 (See Fig.3). The rectangular toroid is filled with air at ambient temperature and 1atm. Thus, inside the tube, the air density is

ρ air ≅ 1 . 2 kg .m − 3

(22 )

The number of atoms of air (Nitrogen) per unit of volume, n air , according to Eq.(7), is given by

nair =

N 0 ρ air = 5.16×1025 atoms/ m3 AN

(23)

Here, the area Sα refers to the area of the ring inside the air toroid, with average radius r = re + ri 2 and height α , i.e.,

S α = 2πα (ri + re ) 2 = πα (ri + re ) where ri is the inner radius and re the outer radius of the rectangular toroid. For ri = 400mm , re = 650mm and α = 60mm , we get

S α = πα (ri + re ) = 0 .198 m 2

(24 )

The parallel metallic plates (p), shown in Fig.3 are subjected to different drop voltages. The two sets of plates (D), placed on the extremes of the toroid, are subjected to V( D )rms = 3.065V at f = 2.5Hz, while the central set of plates (A) is subjected to V( A)rms = 51.623V at f = 2.5Hz.Since d = 98mm , then the intensity of the electric field, which passes through the 36 cylindrical air laminas (each one with 5mm thickness) of the two sets (D), is E( D)rms = V( D)rms d = 31.28V / m and the intensity of the electric field, which passes through the 9 cylindrical air laminas of the two sets (A), is given by E ( A )rms = V ( A )rms d = 526 .77V / m Note that the metallic rings (5mm thickness) are positioned in such way to block the electric field out of the cylindrical air laminas (also 5mm thickness). The objective is to turn each one of these laminas into a Gravity Control Cell (GCC) [6]. Thus, the system shown in Fig. 3 has 4 sets of GCC. Two with 18 GCC each and two with 9 GCC each. The two sets with 18 GCC each are positioned at the extremes of the tube (D).

They work as gravitational decelerator while the other two set with 9 GCC (A) each works as a gravitational accelerator, intensifying the gravity acceleration produced by the Aluminum ring. According to Eq. (3), this gravity after the 9 th GCC becomes g 9 = χ 9 g 0 , where χ = mg (l ) mi (l ) given by Eq. (21), and g 0 can be calculated starting from the expression of the gravitational mass of the half-toroid of Aluminum, M g ( 1 toroid ) , which is 2

given by

5

∫

Mg

0

πri

dM g (1 toroid) = ρα(re − ri )∫ dz 0

2

whence

M g ( 1 toroid) = πρα(re − ri )ri

(25)

2

On the other hand, we have that

∫

g

0

dg = −

G Mg Gαρ (re − ri ) πri dM g = − 2 ∫0 ∫0 dz r r2

whence we get

g=−

Gπαρ (re − ri )ri r2

(26)

which gives the value of g produced by the half-toroid at a point inside the nucleus of the toroid, distant r from the center of the crosssection of the rectangular toroid. Thus, the value of g 0′ (r = r0 ) , due to the first halftoroid is

⎛r −r g 0′ ≅ −Gπρα ⎜⎜ e 2 i ⎝ r0

⎞ ⎟⎟ri ⎠

The value of g 0′′ , due to the opposite halftoroid is ⎧ ⎫ ⎪ ⎪ ( re − ri )ri ⎪ ⎪ g 0′′ ≅ −Gπρα⎨ 2⎬ ⎛ re − ri ⎞ ⎛ re − ri ⎞⎤ ⎪ ⎪⎡ − + 2 r r ⎜ ⎟−⎜ ⎟⎥ ⎪ e 0 ⎢ ⎪ 2 ⎝ ⎠ ⎝ 2 ⎠⎦ ⎭ ⎩⎣

Consequently, the resultant is

⎛r −r ⎞ g 0 ≅ −Gπρα⎜⎜ e 2 i ⎟⎟ri − ⎝ r0 ⎠ ⎧ ⎫ ⎪ ⎪ (re − ri )ri ⎪ ⎪ − Gπρα⎨ = 2⎬ ⎛ re − ri ⎞ ⎛ re − ri ⎞⎤ ⎪ ⎪⎡ ⎪ ⎢2re − ⎜ 2 + r0 ⎟ − ⎜ 2 ⎟⎥ ⎪ ⎝ ⎠ ⎝ ⎠⎦ ⎭ ⎩⎣ ⎡ (r − r )r ⎤ − Gπρα⎢ e i 2i ⎥ ⎣⎢ (ri + r0 ) ⎦⎥ In the case of ri >> r0 , the equation above reduces to ⎛r −r ⎞ (27) g 0 ≅ −Gπρα ⎜⎜ e 2 i ⎟⎟ri ⎝ r0 ⎠ where ri is the inner radius of the toroid; r0 is the distance between the center of the cross-section of the Aluminum ring and the surface of the first GCC of the set (A); α is the thickness of the Aluminum ring. Here, r0 = 35mm and α = 60mm (See Fig. 3 (a)).

The objective of the sets (D), with 18 GCC each, is to reduce strongly the value of the external gravity along the rectangular toroid of air in D region. In this case, the value of the external gravity, gext , is reduced by the factor χ d18 gext , where χ d = 10−2 . For example, if gext = 9.81m / s 2 then this value is reduced to χ d18 gext and, after the set A, it is increased by χ 9 . Since the system is designed for χ = −627 .1 , then the gravity acceleration on the Aluminum ring becomes χ 9 χ d18 g ext = 1 .47 × 10 −10 m / s 2 , this value is smaller than 2 −8 −2 g 0 ≅ −Gπρα [(re − ri )ri r0 ] = 9.9 × 10 m.s .

The values of χ and χ d , according to Eq. (21) are given by

6 ⎧ ⎡ ⎤⎫ 6 S 2 S 4φ 4 E4 nr4(air) nair α m m ( A)rms ⎥⎪ ⎪ ⎢ −1⎥⎬ = χ = ⎨1− 2⎢ 1+ 2 ρ 2 c6 f 2 4 μ ⎪ ⎢ ⎥⎪ 0 air ⎣ ⎦⎭ ⎩ ⎧ ⎫ = ⎨1− 2⎡⎢ 1+1.289×10−6 E(4A)rms −1⎤⎥⎬ ⎣ ⎦⎭ ⎩

(24)

⎧ ⎡ ⎤⎫ 6 S 2 S 4φ 4 E4 nr4(air) nair α m m ( D)rms ⎥⎪ ⎪ ⎢ −1⎥⎬ = χ d = ⎨1− 2⎢ 1 + 2 ρ 2 c6 f 2 μ 4 ⎪ ⎢ ⎥⎪ 0 air ⎣ ⎦⎭ ⎩ ⎧ ⎫ = ⎨1 − 2⎡⎢ 1 +1.289×10−6 E(4D)rms −1⎤⎥⎬ ⎣ ⎦⎭ ⎩ where n r ( air ) =

δ =−

−20 2

4 = 1.88×10 m

f = 2.5Hz. Since E( A)rms = 526.77V / m , E( D)rms = 31.28V /m , we get

δ =

(30)

δ =

χ d ≅ 10

⎡ (r − r )r ⎤ g 9 = χ 9 g 0 = − χ 9 Gπρα ⎢ e 2 i i ⎥ ⎣ r0 ⎦

(32)

It is known that gravitational fields can bend light, and that due to this effect, a light ray that passes very close to a body with gravitational mass M g is deviated of an angle δ (deflection angle) given by [3]

†

The electrical conductivity of air, inside the dielectric tube, is equal to the electrical conductivity of Earth’s atmosphere near the land, whose average value is

σ air ≅ 1 × 10 −14 S / m [9].

4 g 0 r02 c2d

4 g 0 r02 4 g 0 r02 − 2 δ= 2 c d′ c d ′′

(31)

Then the gravitational acceleration after the 9th gravitational shielding is

c2d

(34 )

(35 )

However, considering the symmetry of the gravitational lenses shown in Fig.3, it is easy to see that Eq. (35) must be rewritten as follows

and −2

4 gr 2

For r = r0 we have g = g 0 and equation above can be rewritten as follows

and

χ = −627.1

(33)

this expression into Eq. (33) leads to the following equation

nair = 5.16 ×1025 atoms/ m3 , φm = 1.55×10−10 m , Sm = πφm2

c2d

where d is the distance of closest approach.

Here, we can obtain the expression of δ as follows: by comparing Eq. (26) with Eq. (25) we obtain GM g = gr 2 . Substitution of

(25)

ε r μ r ≅ 1 , since (σ << ωε ) † ;

4GM g

(36 )

where d ′ and d ′′ are respectively, the distances of closest approach of the light ray with respect to the two sides of the Aluminum ring (See Fig 3 (b)).

When the gravitational lenses are activated the value of g 0 is amplified to

χ 9 g 0 , then Eq. (36) becomes δ=

4 χ 9 g 0 r02 4χ 9 g 0 r02 − c2d ′ c 2 d ′′

(37)

Note that, for d ′ = d ′′ (light ray at the center of the Gravitational lens) Eq. (37) gives δ = 0 ( null deflection). On the other hand, if d ′ < d ′′ we have δ > 0 (the light ray is gravitationally attracted to the inner edge of rectangular toroid). Under these conditions, when a light flux crosses the gravitational lens (nucleus of the rectangular toroid), it becomes divergent in respect to the central axis of the toroid (See Fig. 3(c)). If

d ′ > d ′′ then Eq. (37) shows that δ < 0 (the light ray is gravitationally repelled from the inner edge of rectangular toroid). In this case, when a light flux crosses the gravitational lens, it becomes convergent in respect to the central axis of the toroid (See Fig. 3(b)).

Substitution of the known values into Eq. (37) yields 1⎞ ⎛1 − ⎟ ⎝ d ′ d ′′ ⎠

δ ≅ 0.1⎜

(38)

Note that the values of δ can be easily controlled simply by controlling of the value of χ . Also note that the curvatures of the light rays are proportional to the distances d ′ and d ′′ , similarly to the curvature of the light rays in the optical lenses. Then it is easy to see that these gravitational lenses can be very useful in building of telescopes, microscopes, and in concentrating solar light in order to convert solar energy into thermal energy.

I would like to thank Physicist André Luis Martins (RJ, Brazil) who came up with the original idea to build Artificial Gravitational Lens using sets of Gravitational Shieldings, as shown in my previous papers.

7

8 Metallic ring (5 mm thickness)

Air 1 atm 300K

Dielectric rings εr ≅ 1

Rectangular Toroid

EA(rms)

D

A

d = 98 mm α=60mm

ED(rms)

1

2 Air17

ξ = 5 mm

18

88

Mg

1

Aluminum ring

r0=35mm

2 Air 6

Gravitational Lens

9

Parallel plate capacitor (p) (Metallic ring)

ri

re (a)

Light

d ′′

d′

D

Mg

A

g’

g’

Gravitational Lens

D and A Sets of Gravitational Shieldings

A

Mg

D

Rectangular Toroid

(b)

Rectangular Toroid Light

D

Mg

A

g’

Gravitational Lens

g’

A

Mg

D

(c) Fig. 3 – Artificial Gravitational Lens. (a) Cross-section of the Artificial Gravitational Lens. (b) Cross-section of a Convergent Gravitational Lens. The light rays are gravitationally repelled from the inner edge of toroid (c) Cross-section of a Divergent Gravitational Lens. The light rays are gravitationally attracted to the inner edge of toroid.

9 References [1] Dyson, F.W., Eddington, A.S., and Davidson, C.R. (1920). A Determination of the Deflection of Light by the Sun's Gravitational Field, from Observations Made at the Solar eclipse of May 29, 1919. Phil.

Trans. Roy. Soc. A 220 (571-581): 291–333.

[2] Chwolson, O (1924). Über eine mögliche Form fiktiver Doppelsterne. Astronomische Nachrichten 221 (20): 329.

[3] Einstein, A. (1936). Lens-like Action of a Star by the Deviation of Light in the Gravitational Field. Science 84 (2188): 506–7.

[4] Renn, J., Tilman S., and Stachel, J., (1997). The Origin of Gravitational Lensing: A Postscript to Einstein's 1936 Science paper. Science 275 (5297): 184–6.

[5] De Aquino, F. (2010) Mathematical Foundations of the Relativistic Theory of Quantum Gravity, Pacific Journal of Science and Technology, 11 (1), pp. 173232.

[6] De Aquino, F. (2010) Gravity Control by means of Electromagnetic Field through Gas at Ultra-Low Pressure, Pacific Journal of Science and Technology, 11(2) November 2010, pp.178-247, Physics/0701091.

[7] Quevedo, C. P. (1977) Eletromagnetismo, McGrawHill, p. 270.

[8] Halliday, D. and Resnick, R. (1968) Physics, J. Willey & Sons, Portuguese Version, Ed. USP, p.1124.

[9] Chalmers, J.A., (1967) Atmospheric Electricity, Pergamon press, Oxford, London; Kamsali, N. et al., (2011) Advanced Air Pollution, chapter 20, DOI: 10.5772/17163, edited by Farhad Nejadkoorki, Publisher: InTech, ISBN 978-953-307-511-2, under CC BY-NCSA 3.0 license.