# Analysis of Electromagnetic Propulsion on a Two-Electric-Dipole System

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2 where H 0 is the permittivity of a vacuum, k is the wave number of the generated electromagnetic wave, and k Z/c (c: speed of light in vacuum). These electromagnetic fields can be divided into three components. They are the static electric field related to 1/r 3, the inductive electromagnetic field related to 1/r 2, and the radiated electromagnetic field related to 1/r. Of these components, the radiated electromagnetic field is the propagating component and is dominant in the region sufficiently away from the source. The remaining two components are nonpropagating components effective near the source. The electromagnetic field components E 2i and H 2i i x, y, z generated at point P by dipole P 2 can be obtained by replacing P 1 t, r, T, M, with P 2 t, rc, Tc, Mc. The electromagnetic fields E i and H i generated at point P by the two dipoles P 1 and P 2 can be expressed as the sums of individual components: From the geometric relationship in Fig. 1, the following relationships exist in each model between r, T, M and rc, Tc, Mc: (3) (4) (5) The first term on the right-hand side of Eq. (6) represents the electromagnetic momentum flowing into V through S per unit time, while the second term represents the increment of the electromagnetic momentum in V per unit time. Hence, the force F may be interpreted as the electromagnetic momentum that has vanished in V per unit time. From the spatial symmetry of the electromagnetic field in each analysis model, it is considered that a propulsion is generated along the z axis in the CA model and along the x axis in the PA model. Further, let us consider the time average F B i [i = z (CA model), x (PA model)] of the component of F in Eq. (6) along the direction of propulsion in order to derive only the effective propulsion component. In the steady state of each analysis model, the time average of the electromagnetic momentum in V is constant. Hence, the second term on the right-hand side of Eq. (6) is 0. Therefore, the propulsion to be derived is as follows: where the bar above the symbol indicates the time average Analysis of CA model (7) (8) where P 0 is the permeability of vacuum. From the condition of integration range r!! d and from Eq. (4), the following relationship is derived: The above and the condition for the integration range kr!! 1r!! O are applied to Eqs. (2) and (3), and then the results are substituted into Eq. (10). Further, by means of time averaging, the terms with perturbation in time are eliminated: From the above, the electromagnetic propulsion in the CA model can be derived as follows: 3.3. Analysis of PA model (10) (11) (12) (13) Let us consider the integration the same as that for the CA model. From Eq. (8) and the first expression in Eq. (7) together with Eq. (9), the following is obtained: The above and the condition kr!! 1r!! O are applied to Eqs. (2) and (3). The results are substituted into Eq. (14) and the time averaging is taken. Then, From the above, the electromagnetic propulsion of the PA model is derived as follows: 4. Discussion (16) (17) 4.1. Dependence of electromagnetic propulsion on the spacing between dipoles From expressions (13) and (17) for the propulsion, it is found that the magnitude of the propulsion is proportional to sin D and reaches a maximum when the phase difference D of the two dipoles is S/2 or 3S/2. When sin D z 0 and the wave number k is constant, the variation of the propulsion versus the dipole spacing d is as shown in Fig. 2. As d becomes larger, the propulsion decreases toward 0 with positive and negative undulations. The undulation is due to the portions related to the phase difference in Eqs. (13) and (17), sin(kd) and cos(kd). Also, the amplitude becomes smaller in inverse proportion to the power of d. As d becomes larger, the propulsion approaches 0. This is attrib- 3. Analysis 3.1. Fundamental equations for electromagnetic propulsion The force F exerted on a system placed in an electromagnetic field can be expressed as follows in terms of Maxwell s stress tensor T and the electromagnetic momentum density g: Let the spherical surface with a radius r centered at the origin (dipole P 1 ) be the integration range S. It is assumed that r is sufficiently larger than the dipole spacing d and the wavelength O of the generated electromagnetic field. The normal vector to this spherical surface is given by (9) (14) (6) Hence, if the first expressions in Eqs. (7) and (9) are applied to Eq. (8), When the condition r!! d for the integration range is applied to Eq. (5), the following relationships hold: where S is the surface area of V, and n is the normal vector of S. Here, the component T ij of T and g can be expressed as follows in terms of the electric field E, the magnetic field H, the electric flux density D, and the magnetic flux density B: (15) Fig. 2. Dependence of propulsion on d

5 AUTHORS (from left to right) Noriaki Obara (member) graduated from the Department of Applied Physics, Tohoku University, in 1990 and joined the Communications Research Laboratory, Ministry of Posts and Telecommunications. Since then, he has been engaged in research on antennas for satellite communication and on propagation. He also has an interest in the fundamental theory of electromagnetic fields. From 1993 to 1995, he was a member of the 35th Japanese Antarctic Research Expedition and was engaged in observation of upper atmosphere physics. In 1999, he completed a doctoral course at Iwate University. He holds a D.Eng. degree. Presently, he is with Satellite Communications Section, Kashima Space Research Center. He is a member of the Japan Society of Applied Physics. Mamoru Baba completed a doctoral course (majoring in electronic engineering) at Tohoku University in 1974 and became a researcher at Toshiba R&D Center. In 1976, he became a research associate in the Department of Electronic Engineering, Iwate University and is presently a professor in the Department of Electrical and Electronic Engineering. He holds a D.Eng. degree. He has been engaged in research on photonic properties of inorganic materials, semiconductor photoelectronic devices, and metal oxide thin film devices. He is a member of the Japan Society of Applied Physics and the Japan Physical Society. 39

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