A Motor driven by Electrostatic Forces Published in physic.philica.com Abstract Fundamental principle Reference [1] can be regarded as a preparation of the explanations presented here. There it is demonstrated, that every electrical charge permanently emanates energy carried by the electrical field produced by this charge. Therefore the finite speed of propagation of the electrostatic field has to be taken into account, and thus we see a close connection with retarded fields and retarded potentials known from electromagnetic fieldtheory (see [2], [3]). At the end of preceding article we will come back to the question of the origin of the energy driving the rotor. But now it shall be described how electrical fieldenergy emitted by an electrical charge can be converted into mechanical energy. The method of energyconversion developed here, consists in a special guidance of the electrical flux (which is illustrated in textbooks by drawing field strength lines) with the use of metallic surfaces, in such a way that mechanical forces will act onto the guiding metallic surfaces, so that these surfaces will feel a force and consequently they will begin to move. The electric flux can be defined in analogy to the magnetic flux through a closed area C as .
An imaginable setup for this energy conversion is shown in fig.1. There, the electrical charge q is constant and the rotorblades are electrically connected to ground. Of course it would be possible, to imagine many different types of constructions for the electrostatic motor. For instance if the pointcharge q would be replaced by a flat plate (which has the same diameter as the rotor or even more) parallel to the xyplane, the forces onto the metallic rotorblades would be remarkably larger than in our example of fig.1. Furthermore it would be possible to change the angle between the rotorblades and the xyplane as well as several other geometrical parameters in order to optimize the forces onto the metallic blades, but such an optimization would be subject to further development of the engine for technical applications. For the principle explanation of the concept of the engine it is advantageous to find a setup as easy to understand as possible. And just therefore the use of a pointcharge as fieldsource is very convenient, because it is easy to calculate its electric field and its electric potential using Coulomb’s law. This is the reason, why we decide to construct the assembly with a pointcharge q as fieldsource as shown in fig.1.
Fig.1: Possible setup of an electrostatic motor, consisting of a rotor with three metallic blades. An electrical charge q causes a permanent electrostatic force onto the rotor and so it permanently drives the rotor, as long as the practical setup guarantees, that the forces of friction are not stronger than the driving electrostatic forces. In the picture we see the charge q and the corresponding imagecharge q’ with regard to the rotorblade no.1, as it will be subject to the considerations following now.
In order to determine the Coulombforce acting onto the rotorblades, we will now apply the imagecharge method (see for instance [4]). For this purpose we begin with a consideration of the geometry of the apparatus. For the sake of simplicity, we arrange an angle of 45° between the blade no.1 and the xyplane. In the moment of our consideration, the middle line of the rotorblade shall be oriented along the xaxis. Consequently the blade no.1 defines a plane z:=z(x,y) following the functional equation z=–y. Thus the position vectors of the points of this plane are with two free parameters x and y. Because of the symmetry of the assembly, the considerations for the determination of the forces do not alter by principle, when the rotorblades rotate during time. Also because of the symmetry, the forces are analogously for all three rotorblades. Thus it is sufficient to calculate the force and the torque in the moment of consideration chosen here, and to do this calculation just for the blade no.1. In any case, the axis of rotation is the zaxis, so that all rotorblades move within the xyplane. The charge q is placed at the zaxis with the zcoordinate z_{o}. The position of the corresponding imagecharge q’ (with respect to the blade no.1) can be found as illustrated in fig.2. There we see the view from the direction of the xaxis onto the yzplane. In this view we see the cut of blade no.1 with the yzplane being the straight line z=–y (in agreement with the parametrisation of the above given function of the plane of the blade). Constructing the position of the imagecharge q’ will lead us to yaxis, and there to the point with the ycoordinate y=z_{o}. The xcoordinates of q and q’ are zero. This is the reason, why we could construct the position of the imagecharge just in a two dimensional cut as we did. Thus the position vectors of the charge q and of the imagecharge q’ are
Fig.2: Sketch for the determination of the position of the imagecharge. The charge q as well as the imagecharge q’ have the xcoordinates x=0. Thus it is sufficient, to construct the position of the imagecharge using a twodimensional cut of the xyplane with the assembly of the motor, especially with the rotorblade no.1, for which the position of the imagecharge is determined.
For now we know the positions of the fieldsource q and its image q’, we can easily calculate the forces onto the rotorblades, just by applying Coulomb’s law. Therefore we put the value of q’=–q for the imagecharge into the formula. So the Coulombforce between the charge and the imagecharge can be written as The crucial point is: The charge q as well as the rotorblade feels a component of the force in ydirection, which causes a rotation of the rotorblades around the zaxis. For illustration we can again have a look to fig.1. This force is attractive, because the imagecharge has the opposite algebraic sign as the charge itself. From this consideration we understand the direction of the rotation as indicated in fig.1. Example for a possible testsetup
Fig.3: Rotor with three blades and a diameter of 20 centimeters, rotating around the zaxis. In the picture we see the projection of the rotor onto the xyplane together with some values of the angles covered by the metalblades. The charge q is mounted on the zaxis at the position z_{o}=5cm, so that the imagecharge q’ turns out to be on the yaxis at the position y=z_{o}=5cm.
For a really existing setup, the electrical charge q has to be put onto some really existing matter. Let us chose an electrically conducting sphere with a diameter of 2R=1.0cm, and let us mount its centre at the position z_{o}=5cm. The capacity of such a spherical capacitor (against infinity) is C=4??_{o}R. If we put this sphere to an electrical voltage of U=10kV (which should be a good value in order to avoid electrical breakthrough), it will take an electrical charge of q=C·U=4??_{o}·R·U_{?}5.56·10^{9}C. The imagecharge thus will have a value of q’_{?} 5.56·10^{9}C. Putting these values into the formula for the force between the charge and the imagecharge, we come to
The zcomponent of this force, which is parallel to the direction of the axis of rotation will not be recognized (and not be important at all), but the ycomponent directly causes the rotation of blade no.1 around the zaxis (if this force is strong enough to overcome the force of friction). Because of the symmetry of the assembly, the forces onto the other blades are understood analogously. This means that the principle of functioning of the motor is already explained now.
It should be mentioned, that our calculation of up to now gives the total force between the charge q and the infinite plane z:=z(x,y)=–y. But the rotorblade of our assembly only covers a finite part of this plane. For the determination of the force actually working on the blade, we again want to turn our attention to fig.3 showing a projection of the assembly. And now we have to calculate which percentage of the electric flux through the whole plane z:=z(x,y)=–y will pass the finite blade. Therefore we have to calculate the electric flux, and we begin this calculation by writing the potential of the charge and the imagecharge, valid for the space between the plane z:=z(x,y)=–y and the charge q: Coulomb’s potential of the imagecharge q’ is
with and being the distances between the charge respectively the imagecharge and the space point, at which the potential has to be calculated. From there (and because of q’=–q), we come to the total potential within the space between the charge and the plane z:=z(x,y)=–y, and we find:
The electrostatic fieldstrength is calculated as usual: . From there we calculate the percentage of the electric flux through the plane of the rotorblade relatively to the electric flux through the total plane z:=z(x,y)=–y. This was done in numerical approximation, leading to a result of about (4±0.5)%.This means, that the force acting onto the finite rotorblade is about (4±0.5)% of the total force . Consequently, we get the ycomponent of the force acting on each single finite rotorblade as F_{y}_{?}3.93·10^{5}N·4%_{?}1.6·10^{6}N, and thus the force acting on all three rotorblades is 3·F_{y}_{?}4.7·10^{6}N.
At least we want to know the torque, with which the charge q turns our rotorblades. Therefore we have to take into account, that the force 3·F_{y} does not act onto one single point, but its action is distributed along several different radii of rotation. The calculation of the torque is a simple mechanical problem, which does not need a detailed demonstration here. Its result is a torque with an absolute value of about M_{tot}_{?}9·10^{8}Nm acting in sum on all three rotorblades. (The value is again given as “approximately”, because the values of the forces originate from a numerical approximation.)
Resumée and origination of the energy This conclusion again awakes the question about the origin of the energy driving the rotor. For the answer to this really crucial question, we have to come back to the article [1] again. There it is demonstrated, that the electrical charge as a source of electrostatic field permanently emits fieldenergy. But it is also demonstrated, that this fieldenergy is absorbed by the mere space, when the field propagates into the space. This means that the mere space (with an other word, the vacuum) is not only responsible for the propagation of the field, but also for some absorption and for a repropagation of the fieldenergy (doing the latter one without the repropagation of fieldstrength). According to this idea, the vacuum would absorb energy from the propagating field, would distribute this energy all over the space, and would provide this energy to fieldsources, which take this energy and convert it again back to fieldstrength. Fieldsources typically are called electrical charges. These considerations are new, same as the motor which was developed on the basis of these thoughts. The fact, that the mere empty space (the vacuum) really contains energy is well known from the cosmological constant ? of the theory of General Relativity [5], and it is also known from experimental investigations of astrophysics [6], [7] (with values being measured in the order of magnitude of about 10^{9}J/m^{3}), where the standard model of astrophysics comes to the conclusion, that about 65% of the universe consists of invisible vacuumenergy. And also quantumelectrodynamical considerations regarding the nontrivial structure of the vacuum [8] (see for instance vacuum polarisation) confirms the fact, that the vacuum contains energy. Up to now, there is no clarity about the real value of the energy density, but this open question does not affect the functioning principle of the electrostatic driven motor presented here. In this sense, the motor presented here does nothing else, than the conversion of vacuumenergy into mechanical energy.
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Adress of the Author Prof. Dr. Claus W. Turtur University of Applied Sciences BraunschweigWolfenbüttel Salzdahlumer Strasse 46 / 48 Germany – 38302 Wolfenbüttel Email: cw.turtur@fhwolfenbuettel.de Tel.: (++49) 5331 / 939 – 3412 Information about this Article Published on Monday 18th February, 2008 at 17:58:33.

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