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Relativistic Engine Based on a Permanent Magnet

Miron Tuvala and Asher Yahalomb

a Atidron Ltd., San Martin 15, Ramat Gan 52237, Israelb Ariel University, Kiriat Hamata POB 3, Ariel 40700, Israel

e-mails: [email protected]; [email protected]

July 13, 2015

Abstract

Newton’s third law states that any action is countered by a reactionof equal magnitude but opposite direction. The total force in a systemnot affected by external forces is thus zero. However, according to theprinciples of relativity a signal can not propagate at speeds exceedingthe speed of light. Hence the action cannot be generated at the sametime with the reaction due to the relativity of simultaneity, thus thetotal force cannot be null at a given time. The following is a contin-uation of a previous paper [1] in which we analyzed the relativisticeffects in a system of two current conducting loops. Here the analysisis repeated but one of the loops is replaced by a permanent magnet.

It should be emphasized that although momentum can be created inthe material part of the system as described in the following workmomentum can not be created in the physical system, hence for anymomentum that is acquired by matter an opposite momentum is at-tributed to the electromagnetic field.

PACS: 03.30.+p, 03.50.De

Keywords: Newton’s Third Law, Electromagnetism, Relativity

1 Introduction

Among the major achievements of Sir Isaac Newton is the formulation of Newton’s third law stating that any action is countered by a reaction of equal magnitude but opposite direction [2, 3]. The total force in a system notaffected by external forces is thus zero. This law has numerous experimental

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verifications and seems to be one of the corner stones of physics. However,

by the middle of the nineteenth century Maxwell has formulated the lawsof electromagnetism in his famous four partial differential equations [4, 5, 7]which were formulated in their current form by Oliver Heaviside [8]. One of the consequences of these equations is that an electromagnetic signal cannottravel at speeds exceeding that of light. This was later used by AlbertEinstein [5, 7, 9] (among other things) to formulate his special theory of relativity which postulates that the speed of light is the maximal allowedvelocity in nature. According to the principles of relativity no signal (evenif not electromagnetic) can propagate at superluminal velocities. Hence anaction and its reaction cannot be generated at the same time because of therelativity of simultaneity. Thus the total force cannot be null at a given

time. In consequence, by not holding rigorously the simultaneity of actionand reaction Newton’s third law cannot hold in exact form but only as anapproximation. Moreover, the total force within a system that is not actedupon by an external force would not be rigorously null since the action andreactions are not able to balance each other and the total force on a systemwhich is not affected by an external force in not null in an exact sense.

Most locomotive systems of today are based on open systems. A rocketsheds exhaust gas to propel itself, a speeding bullet generates recoil. A carpushes the road with the same force that is used to accelerate it, the sameis true regarding the interaction of a plane with air and of a ship with wa-ter. However, the above relativistic considerations suggest’s a new type of motor in which the open system is not composed of two material bodies

but of a material b ody and field. Ignoring the field a naive observer willsee the material body gaining momentum created out of nothing, however,a knowledgeable observer will understand that the opposite amount of mo-mentum is obtained by the field. Indeed Noether’s theorem dictates thatany system possessing translational symmetry will conserve momentum andthe total physical system containing matter and field is indeed symmetricalunder translations, while every sub-system (either matter or field) is not.

In this paper we will use Jefimenkos equation [6] discuss the force be-tween a current loop and a permanent magnet. In this respect the currentpaper differs from a previous one [1] which discussed the case of two cur-rent carrying coils. The current configuration may seem attractive since a

permanent magnet does not require a power source.

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Figure 1: A cylindrical magnet (blue) and a current loop (red) above it.(three different sections)

the magnetization can be written as:

M z = M 0 (u(z + h1)− u(z + h)) u(R2 − r). (8)

The magnet and a current loop above it are depicted in figure 1.We can now calculate the magnetization current by evaluating equation

(4) using cylindrical coordinates:

J M = 1

r[∂ θ(M z) − ∂ z(rM θ)]r + [∂ z(M r)− ∂ r(M z)]θ

+ 1

r[∂ r(rM θ)− ∂ θ(M r)]z =

1

r∂ θ(M z)r − ∂ r(M z)θ. (9)

Since M z = M z(r, z) we obtain:

J M = −∂ rM zθ. (10)

Moreover, since the Dirac delta function δ (x) is related to the step functionby the formulae:

∂ xu(x) = δ (x), u(x) =

ˆ x−∞

δ (x′)dx′. (11)

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We obtain:

∂ ru(R2 − r) = ∂u(R2 − r)

∂ (R2 − r)

∂ (R2 − r)

∂r = −δ (R2 − r) (12)

Using equation (12) and equation (8) in equation (10) we obtain:

J M = M 0(u(z + h1) − u(z + h))δ (R2 − r)θ (13)

The above can also be written in the following form:

J M = M 0(

ˆ z+h1−∞

δ (z′)dz′ −ˆ z+h−∞

δ (z′)dz′)δ (R2 − r)θ

= M 0ˆ z+h1z+h

δ (z′)dz′δ (R2 − r)θ. (14)

Now consider a change of variables:

z′′ = z ′ − z, dz′′ = dz ′ (15)

This will lead to:

J M (r, z) = M 0

ˆ h1h

δ (z′′ + z)dz′′δ (r − R2)θ. (16)

Making an additional variable change:

z′′′ = −z′′, dz′′′ = −dz′′ (17)

Results in:

J M (r, z) = M 0

ˆ −h−h1

δ (z − z′′′)dz′′′δ (r −R2)θ. (18)

Hence we see that the magnetization current density is equivalent to a con-tinuum of loop currents each with a current density:

J (1)M (r,z,z′) ≡ M 0δ (z − z′)δ (r − R2)θ. (19)

We notice that the magnetization M 0 replaces the current I appearing in

the analog expression for charge current density. Hence we can write:

J M (r, z) =

ˆ −h−h1

J (1)M (r,z,z′)dz′. (20)

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4 Force Calculations

Since in a previous paper [1] we have calculated the relativistic total forceacting on a system of two coils (see equation (44) of [1]) we will take ad-vantage of this calculation taking into account that the magnetic currentdensity is static. For a single loop we have:

F (1)t =

µ0

8π(

h

c)2 K

(1)12,2I 2 I 1 =

µ0

8π(

h

c)2 K

(1)12,2M 0 I 1. (21)

In which K (1)(12,2) is defined in equation (39) of [1] and we have replaced I 2

with M 0. The total relativistic force is an integration over the contributionof a continuum of current loops:

F t =

ˆ −h−h1

F 1t dz ′ = µ0

8π(

h

c)2M 0 I 1

ˆ −h−h1

K (1)12,2dz′ ≡ µ0

8π(

h

c)2M 0 I 1 K 12,2. (22)

In which K 12,2 is an integration the K factor of all magnetization current

loops. Now we wish to calculate K (1)(12,2) for this we use equation (39) of [1]

which is: K (1)12,2(z′) = − 1

h2

˛ ˛ Rd l1 · d l2. (23)

Notice that R = X 1− X 2 is a difference vector between the location vectorsof current elements on the charge current and magnetization current loops.

Hence using suitable variables: X 1 = (R1 cos θ1, R1 sin θ1, 0), X 2 = (R2 cos θ2, R2 sin θ2, z′). (24)

We obtain:

R = X 1 − X 2 = (R1 cos θ1 − R2 cos θ2, R1 sin θ1 −R2 sin θ2,−z′). (25)

And thus R is:

R =

(R1 cos θ1 −R2 cos θ2)2 + (R1 sin θ1 − R2 sin θ2)2 + z′2

= R21 + R2

2

−2R1R2 cos (θ1

−θ2) + z′2. (26)

Now we can calculate the vector line elements:

d l1 = R1dθ1 θ1 = R1dθ1(− sin θ1, cos θ1, 0),

d l2 = R2dθ2 θ2 = R2dθ2(− sin θ2, cos θ2, 0). (27)

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The scalar product of those line elements is:

d l1 · d l2 = R1R2dθ1dθ2 cos(θ1 − θ2). (28)

Combining the above results we can write the K integral of equation ( 23)as:

K (1)12,2(z′) = −R1R2

h2

ˆ 2π0

dθ2

ˆ 2π0

dθ1 cos (θ1 − θ2) R(θ1, θ2, z′). (29)

We now make a change in variables:

θ′ = θ1 − θ2, dθ′ = dθ1. (30)

The second above equation is correct since θ2 is constant for the θ1 integral.Hence:

K (1)12,2(z′) = −R1R2

h2

ˆ 2π0

dθ2

ˆ 2π−θ2−θ2

dθ′ cos(θ′) R(θ′ + θ2, θ2, z′). (31)

In the above:

R = R

R =

(R1 cos (θ′ + θ2)− R2 cos θ2, R1 sin (θ′ + θ2) −R2 sin θ2,−z′) R2

1 + R22 − 2R1R2 cos θ′ + z′2

.

(32)Now we notice that all the function contained in the integrand of equation

(31) are periodic in θ ′ with a period of 2π hence we can replace´ 2π−θ2−θ2 dθ′ →´ 2π

0 dθ′. The following step would to change the order of integrals performingthe θ2 integral first and noticing that all the functions which are periodic inθ2 have a null contribution to the integral. Hence we obtain:

K (1)12,2(z′) =

2πR1R2

h2

ˆ 2π0

dθ′ cos θ′z′ R2

1 + R22 − 2R1R2 cos θ′ + z′2

z. (33)

Now we sum up contributions to the K factor from all loops:

K 12,2 =

ˆ −h−h

1

dz′ K (1)12,2(z′) =

2πR1R2

h2

ˆ 2π0

dθ′ cos θ′ˆ −h−h

1

dz′ z′√ α2

+ z′2

.

(34)in which we define: α2 ≡ R2

1 + R22−2R1R2 cos θ′. A simple integration leads

to:

K 12,2 = 2πR1R2

h2

ˆ 2π0

dθ′ cos θ′(

α2 + h2 −

α2 + h21)z. (35)

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Now we can write: α2 + h2 =

h2 + R2

1 + R22 − 2R1R2 cos θ′

=

R1R2

h2

R1R2+

R1

R2+

R2

R1− 2cos θ′. (36)

And define:

b ≡ h2

R1R2+

R1

R2+

R2

R1, b1 ≡ h2

1

R1R2+

R2

R1+

R1

R2. (37)

Hence:

α2 + h2 =

R1R2√ b− 2cos θ′,

α2 + h21 =

R1R2

b1 − 2cos θ′.

(38)In terms of the above definitions

K 12,2 = 2π(R1R2)1.5

h2 (g(b) − g(b1))z. (39)

In which:

g(b) ≡ˆ 2π0

dθ′ cos θ′√

b− 2cos θ′. (40)

It is obvious that when b ≫ 2 the respective part of the integral vanishes(the same is true for b1) this is evident since the integral is performed over aconstant time a cosine function of period 2π. It also clear that b is a sum of afactor dependent on the magnet vertical dimensions and a factor dependentthe ratio between R1 and R2 which we denote s ≡ R2

R1. The second factor

is given by:

f (s) = s + 1

s. (41)

It is obvious that f (∞) = ∞ and f (0) = ∞. Moreover, f ′(s) = 1 − 1s2

andfor f ′(s) = 0 we obtain the minimum value s = 1 which indicates an equal

radii to the magnet and the current loop. For this case b = 2 + h2

R2

1

and for

the case that the current loop is put on the magnet b = 2 and g(2) = −83 .

For a ”thick” magnet h1 → ∞

and thus also b1 → ∞

so that g(b1

)→−

π

√ b1which is unfortunately a rather slow decrease. Usually the magnet does nothave to be too thick only enough to make |g(b1)| < |g(b)|. The functiong(b) can be written explicitly in terms of the elliptic functions Ee(m) and

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4 6 8 10b

3.0

2.5

2.0

1.5

1.0

0.5

gb

Figure 2: The function g(b).

Ke(m) as:

g(b) = 1

3{√

b − 2[−bEe( 4

2 − b) + (b + 2)Ke(

4

2 − b)]

+√

b + 2[−bEe( 4

2 + b) + (b − 2)Ke(

4

2 + b)]}

Ee(m) ≡ˆ π

2

0dθ

1 −m sin2 θ

Ke(m) ≡ˆ π

2

0dθ 1

1− m sin2 θ(42)

A graph of g(b) is given in figure 2.

5 Specific Examples

Let us look at the case in which the radii of the magnet and the currentloop are equal and the current loop is placed on top of a thick magnet. Inthis case:

K 12,2 =

−16π

3

R31

h2

z. (43)

Inserting equation (43) into equation (22) will result in:

F t = −2

3µ0M 0 I 1

R31

c2 z. (44)

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However, since the residual magnetic flux density is related to the magneti-

zation as M 0 = Br

µ0 we may write:

F t = −2

3Br

I 1R3

1

c2 z. (45)

We notice that for strong magnets of the NdFeB type the residual magneticflux density Br ≈ 1− 1.4 Tesla. Hence the pre factor 2

3Br is of order 1.Let us now look at the more general case of a thick magnet with a current

loop on top but now the radii of the cylindrical magnet and the current loopare not equal hence h = 0, s = 1 and the force formula takes the form:

F t = −1

4Br

I 1R3

1

c2 s1.5g(s +

1

s)z. (46)

Since:

lims−>∞

−s1.5g(s + 1

s) = πs (47)

We arrive at the approximate force equation in the case that the radius of the magnet cylinder is much bigger than the radius of the loop current:

F t = π

4Br

I 1R2

1R2

c2 z. (48)

If we take the second derivative I 1 to be of the order I 1 ≈ I pτ 2r

we obtain:

F t ≈ π

4 BrI p R2

1

c2

τ 2r

R2z. (49)

we see that the decisive factor is the ration of the current rising time andthe time it will take a light signal to travel across the current loop.

Finally we would like to address the question of the possibility of thedevice to lift from the ground for this the force generated by the deviceshould be larger or equal to the gravitational force

F g = gm = gρV = gρLπR22 (50)

In the above g is the gravitational acceleration on earth (∼ 9.8ms2

not tobe confused with the function g(b)), m is the mass of the magnet, ρ is the

mass density of the magnet and V is the volume of the magnet. The ratioof relativistic and gravitational forces is given by:

F t

F g=− 1

4Br I 1

R3

1

c2 s1.5g(s + 1

s)

gρLπs2R21

=Br

I 1R2

c2

4gρLπ

−g(s + 1

s )√ s3

. (51)

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If the magnet is large with respect to the current loop we can take the limit:

lims−>∞

−g(s + 1

s)√

s3

=

π

s2 (52)

Which leads to the force ratio:

F t

F g≃ Br

I 1R2

c2

4gρLs2 . (53)

Now since the current loops are small we may consider to put N loops on

the magnet the maximum number in a single layer would be: N = R2

2

R2

1

= s2

hence the total force ratio is:

N F t

F g≃ Br

I 1R2

c2

4gρL . (54)

For a magnet of radius of 1 meter and thickens of 1 meter we obtain for aNdFeB magnet the density is ρ = 7500 kg

m3 hence we need at least a I 1 ∼1.9 ∗ 1022 A

s2 for this flying saucer to fly. This type of acceleration it typical

to microwave currents of frequency 10 GHz and current amplitude of about4.8 Ampere.

6 Conclusion

We have shown in this paper that in general Newton’s third law is notcompatible with the principles of special relativity and the total force on asystem of a current loop and a permanent magnet system is not zero. Stillmomentum is conserved if one takes the field momentum into account.

We have developed simple formulae for the total force in various casesincluding the case of equal radii of magnet and current loop and the caseof a small current loop. This was extrapolated to the case of multiple loopsand a specific example was done that the force of the system is equals itsweight.

References

[1] Miron Tuval & Asher Yahalom ”Newton’s Third Law in the Frameworkof Special Relativity” Eur. Phys. J. Plus (11 Nov 2014) 129: 240 DOI:10.1140/epjp/i2014-14240-x. (arXiv:1302.2537 [physics.gen-ph])

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[2] I. Newton, Philosophi Naturalis Principia Mathematica (1687).

[3] H. Goldstein , C. P. Poole Jr. & J. L. Safko, Classical Mechanics, Pear-son; 3 edition (2001).

[4] J.C. Maxwell, ”A dynamical theory of the electromagnetic field” Philo-sophical Transactions of the Royal Society of London 155: 459512(1865).

[5] J. D. Jackson, Classical Electrodynamics, Third Edition. Wiley: NewYork, (1999).

[6] Jefimenko, O. D., Electricity and Magnetism, Appleton-Century Crofts,New York (1966); 2nd edition, Electret Scientific, Star City, WV (1989).

[7] R. P. Feynman, R. B. Leighton & M. L. Sands, Feynman Lectures onPhysics, Basic Books; revised 50th anniversary edition (2011).

[8] O. Heaviside, ”On the Electromagnetic Effects due to the Motion of Electrification through a Dielectric” Philosophical Magazine, (1889).

[9] A. Einstein, ”On the Electrodynamics of Moving Bodies”, Annalen derPhysik 17 (10): 891921, (1905).

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