Casimir Force Between the Two Moving Conductive Plates. Jaroslav Hynecek 1 Isetex, Inc., 95 Pama Drive, Allen, TX 751 ABSTRACT This article resents the derivation of the Casimir force for the two moving conductive lates. The derivation is based on the validity of the Secial Relativity Theory, in articular on the Lorent length contraction and on the analogy derived from the Lorent force. The ossible alications of the derived formula are for the study of the Z inch effect and for the controlled nuclear fusion. Key words: Casimir force, Lorent force, Maxwell's equation, Casimir force of moving conductive lates, Secial Relativity Theory, Z inch effect, Controlled nuclear fusion. I. INTRODUCTION The Casimir force is a very interesting quantum field theory (FT) henomenon. ith the increasing activity and rogress in the nanotechnology it is clear that this force is gaining imortance when designing very small structures. The formula for the Casimir force between the two conductive lates that do not have any current flowing through them has been derived and verified by exeriments many times in the ast [1]. In this aer the new formula is derived for the force between the two conductive lates that are moving or that carry a current based on the analogy derived from the Maxwell Lorent equation of force existing between the two conductors. II. THE EM FORCE BETEEN THE TO CHARGED MOVING PLATES The attractive force observed between the two uniformly charged nonconductive moving lates can be calculated by considering that in addition to the electrostatic force attracting the lates there is also a force based on the Biot Savart law acting between the currents, which the moving lates now also 1 jhynecek@netscae.net, Isetex, Inc. 1 1
reresent. To calculate this force it is useful to first find the magnetic field H existing in the sace between the lates. For the selected configuration the simlest way is to use the integral form of Maxwell s equation: H ds I, (1) where the integration ath and the current flow are illustrated in a drawing in Fig.1. The magnetic field intensity between the lates u to their internal surfaces is thus equal to: H v L(. () Similarly for the electric field from the Gauss law it is: E ds, () where the integrating surface S encloses one of the lates. For the field magnitude then follows that: E L(. (4) Both, the magnetic field as well as the electric field intensities are, of course, ero on the external surfaces of the lates, but through the late s thickness increase linearly from ero to the full value found between the lates. This is the consequence of the assumtion that the embedded charge distribution within the lates volume is uniform. The formula for the force is obtained from the Lorent force equation: F q E v B, (5) which must be integrated over the late s thickness F q. v 1 L( c d. (6) After comletion of integration in Eq.6 where the substitutions for the arameters: 1/ c were also made, the result becomes: B H, and v F q 1. (7) L( c The otential energy of this arrangement is then calculated by integrating the force over the distance between the lates and including the Lorent length contraction factor: result is: L( L v c 1 /. The
q d m v. (8) 1 v / c A 1 v / c In this formula the subscrits ero indicate the laboratory stationary values. To determine the total energy the velocity deendent integration constant has also been added to this exression to make the energy consistent with the formula: q mc 1 v / / c. The imortant oint to note, however, is that the energy does not change when this arrangement is moving in the direction erendicular to the lates. The current does not flow for the erendicular motion and the late's area A is not contracted, but the sacing between the lates is shortened by the Lorent contraction factor. Therefore, it follows that for the motion arallel to the lates and erendicular to the lates it is: (9) q q Finally it is necessary to comment also on the force direction and thus on the sign of the energy. hen the current flows in the same direction for the insulating lates charged by the same charge, the electrostatic force between the lates is reulsive, but the Lorent comonent is attractive. hen the lates are conductive and charged it will be assumed that the result is the same as for the insulating lates, excet for the fact that charge will reside entirely on the inner surfaces of the lates. s +mq I v d v I E H -mq L( Fig.1. Orientation of current I generated by the moving charged lates and orientation of the resulting electric and magnetic fields. Charge mq. The magnetic field integration ath s used in Eq.1 is also indicated.
II. THE CASIMIR FORCE AND THE CASIMIR POTENTIAL The Casimir force [1,] is an interesting FT henomenon. The FT rovides the formula for the force between the two stationary conductive lates and thus the corresonding otential as follows: c hc A. (1) 144d For the moving lates oriented as in Fig.1 it is reasonable to anticiate the following modification: c hc A 1 v / c f ( 144d c m v 1 v / /, (11) where a velocity deended factor f ( and the velocity deendent integration constant were included similarly as in the revious case of the total energy aearing in Eq.(8). It will be further assumed that the energy will again be indeendent of the motion direction as in Eq. (9) and, therefore, for the total energy in the erendicular motion direction it should hold that: c hc A mv 144d / c 1 v / c / 1 v /. (1) By comaring these two formulas the velocity deendent factor f ( can be determined. 1 f (. (1) 1 v / c From this result then follows that the Casimir force for the moving conductive lates is: F c hc A 4 48d c 1 v / /. (14) This is an interesting result that could ossibly be exerimentally verified. This result also resents a considerable challenge to the MT theoreticians, since the theory should rovide its fundamental justification [,4,5,6]. The result is interesting also from the oint of view of exerimental confirmation of the Lorent length contraction. The very close arallel sheets of moving conductors or a very dense ionied lasma current sheets should thus attract each other by a much stronger force than the usual Lorent force suggests. This may have a significant consequences for the studies of the Z inch where the attemts are being made to achieve a controlled nuclear fusion. 4
III. CONCLUSIONS In this article the Casimir force between the two moving conductive lates and therefore between the two current sheets was derived. It is suggested that this force is stronger than the usual Lorent force thus having significant consequences for the Z inch and the study of the controlled nuclear fusion. The study of the motion deendence of the Casimir force could thus rovide additional insights into the nuclear fusion roblems. REFERENCES 1. htt://en.wikiedia.org/wiki/casimir_effect. htt://rd.as.org/abstract/prd/v65/i1/e155. T.G. Philbin, U. Leonhardt, "Casimir Lifshit force between moving lates at different temeratures", arxiv:94.148v [quant h] 11 May 9. 4. M. Tajmar, "Finite Element Simulation of Casimir Force in Arbitrary Geometries", htt://arxiv.org/ft/quant h/aers/45/4515.df 5. Anushree Roy, U. Mohideen, "A verification of uantum field theory measurement of Casimir force", Paramana journal of hysics, Indian Academy of Sciences Vol. 56, Nos & Feb. & Mar. 1. 9 4. 6. M. Bordag, U. Mohideen, V.M. Mosteanenko, "New Develoments in the Casimir Effect", Elsevier, Physics Reorts 5 Amsterdam, 1. 1 5. 5