APPLIED PHYSICS LETTERS

VOLUME 80, NUMBER 15

15 APRIL 2002

Coulomb motor by rotation of spherical conductors via the electrostatic force Anders O. Wistrom and Armik V. M. Khachatourian Department of Chemical and Environmental Engineering, University of California, Riverside, California 92521

共Received 22 October 2001; accepted for publication 17 February 2002兲 Three spherical conductors fixed in space and held at constant potential produces a rotational force that causes the conductors to rotate about their axis. The motor is described by an expression for the moment of force given by Coulomb’s law complemented by Gauss’ law of the electric potential. The observed rotation is likely to be general and apply to machines of all size scales where the electrostatic force is the dominant operative force. This would include systems ranging in size from molecular to macroscopic and be useful for devices that require rotational motion. © 2002 American Institute of Physics. 关DOI: 10.1063/1.1470259兴

In a series of experiments, each comprising three supported metal spheres held at constant potential, we observed a rotation for two spheres with application of voltage to the third sphere, which was held stationary. The rotation proceeded until standstill where the induced torque was counterbalanced by the restoring torque of the suspending wire. Only when the external power supply was disconnected did the spheres return to their original position. This observation opens the possibility of a Coulomb motor where a moment of force is obtained directly via the Coulomb interaction between charged spherical conductors. Because the laws of electrostatics are invariant with size, the phenomena reported here might occur at all length scales where the electrostatic force is the dominant operative force. The experiment comprises three metal spheres of equal size arranged such that a plane passing through the sphere centers is the horizontal 共x,z兲 plane 共Fig. 1兲. The center of the spheres form the vertices of a triangle where the connecting angle between sphere pairs 1 and 2 and sphere pairs 3 and 2 is denoted ␭ 13 . Spheres of different materials have been used, including spheres with graphite and nickel coatings as well as stainless steel spheres. In these experiments, three stainless steel spheres, diameter 270 mm and weight 780 g, were used. Two of the spheres were suspended from the top by thin steel wires, diameter 127 ␮m, to allow rotation about the y axis passing through the center of each sphere, while the third sphere was rigidly supported. The use of metal spheres affords a convenient macroscopic model for constant potential surfaces in combination with an external power 共dc兲 supply. In these experiments, the applied voltage was between 400 and 5000 V provided to the stationary sphere via a connecting wire from the power supply. In all experiments, the surface-to-surface separation distance between spheres exceeded that for sparking in dry air by at least one order of magnitude. Surface-to-surface separation distances were typically larger than 5 mm. Also, the experiments were conducted in isolation from the surroundings by carefully insulating all fasteners and connectors, by utilizing a large open space, or by installing the experimental assembly in a Faraday box. Hence, current flow was negligible. We have found that our experimental model offers a

particularly powerful approach to investigating electrostatic phenomena, and we have previously used similar setups to successfully calibrate the electrostatic force.1,2 The absence of electrical current and magnetic materials, natural or induced, leads us to conclude that the experimental assembly was electrostatic. Experimentally, we measure a net angular displacement of the spheres at an initial, uncharged, configuration and at a final, charged, configuration. Rotation is defined as either up

FIG. 1. Schematic of experimental assembly of three metal spheres where the plane passing through the sphere centers is the horizontal 共x,z兲 plane. A—Sphere alignment in the horizontal plane is given by the connecting angle ␭ 13 . B—Sphere center-to-center separation is denoted by h i j between spheres i and j, respectively. The vector associated with the rotation for each sphere passes through respective sphere center along the y axis. Observed direction of rotation is given by the arrow.

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Appl. Phys. Lett., Vol. 80, No. 15, 15 April 2002

A. O. Wistrom and A. V. M. Khachatourian

or down 共y axis兲 taken perpendicular to the horizontal 共x,z兲 plane. The spheres are stationary in both configurations, held in place by a combination of electrostatic, gravitational, and tension forces. Evidence for moment of force is based on experimental observations after 10, 20, and 200 h after voltage is applied to the center sphere at which times the spheres were deemed to be in static equilibrium. The experimental evidence for an electrostatic moment of force was as follows: 共1兲 translational movement is not observed, 共2兲 rotation about the vertical axis of the sphere proceeds until standstill where the induced moment of force is offset by the restoring torque of the suspending wire 共only after the external power supply is disconnected does the sphere return to its starting position兲, 共3兲 the direction of the rotation remains invariant between replicate experiments, and 共4兲 the magnitude of the net angular displacement increases with the length of the suspending wire, We propose that rotation is the result of the continuum of static charges residing on the surface of the metal spheres. Theoretical evidence for rotation about the geometrical center of each sphere is obtained from the classical definition of the static moment of force, which, in the spirit of Cavendish and Coulomb, is determined from an action-at-a-distance perspective. The electrostatic potential on the surface of the spheres is given by Gauss’ law

兺冕 j⫽1 N

V i ⫽K

dQ j , Rij

共1兲

which relates the surface potential of sphere i to their corresponding charges dQ i ’s located at position Ri j on spheres j weighted with the separation distance R i j . Here N is the number of the spheres, and K is 1/4␲ ⑀ 0 , where ⑀ 0 is the permittivity constant for a vacuum. Theoretical evidence for rotation for sphere i about the geometrical center of each sphere is obtained from the classical definition of the static moment of force:

兺冕 j⫽1 N

M i ⫽K

j⫽i

dQ i dQ j R 3i j

a i ⫻R i j

共2兲

given by the integration of cross product of the position vector of the points a on the surface of sphere i and the Coulomb

2801

force due to sphere j. Note, that the term i⫽ j in the moment would result in the self-force which is not observed experimentally. The Coulomb force is obtained by integrating the effect of the known charge distribution via the Fredholm integral equation.2– 4 Asymptotic analysis of equation for potential, Eq. 共1兲, gives the surface charge distribution on each sphere, which once substituted for moment of force, Eq. 共2兲, reveals an interaction where the leading term is proportional to the inverse of the sixth power of the separation distance M ⬀1/KV 2 a 7 h ⫺6 where V is the applied potential, a is the sphere radius, h is the separation distance, and K as defined previously. Preliminary evaluation of the first few terms yields a sin ␭13 dependence for angles 共␭ 13 in Fig. 1兲 greater than ␲/3. The correspondence between the theoretical prediction and the experimental observations lends considerable support to the notion that the rotation is an electrostatic entity. It is important to note that the direction of the rotation is explicit in the equation for moment of force. Rotation is either up or down, taken perpendicular to a plane passing through the sphere centers. We have demonstrated a Coulomb motor where the moment of force is induced by an assembly of three spherical conductors held at constant potential. The rotation of the spheres about the axis perpendicular to the plane passing through the center of the spheres is shown to be a natural consequence of the electrostatic force. We find that when the charged spheres are stationary the only degree of motion that remains is rotation. Hence, the electrostatic coupling between the three charged spheres is converted to a net rotation beyond the observation that the rotation is likely to be general. The Coulomb motor appears to be feasible in systems ranging in size from molecular to macroscopic, and would be a useful device in situations that require angular motion.

1

A. O. Wistrom and A. V. Khachatourian Meas. Sci. Technol. 10, 1296 共1999兲. 2 A. V. Khachatourian and A. O. Wistrom J. Phys. A 33, 307 共2000兲. 3 R. P. Kanwal, Linear Integral Equations 共Academic, New York, 1971兲. 4 I. N. Sneddon, The Use of Integral Transforms 共McGraw–Hill, New York, 1972兲.

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