Nonlinear analytical calculation of the equilibrium shape of a charged drop uniformly accelerated in an electrostatic field

An analytical asymptotic expression for the equilibrium shape of a charged drop of an ideal incompressible conducting liquid uniformly accelerated in collinear electrostatic and gravitational fields is derived in an approximation quadratic with respect to the deviation of the equilibrium shape of the drop from a sphere. It is found that the equilibrium shape of the drop is close to a prolate spheroid when its self-charge and the external electric field strength are far from their values critical in terms of instability against the self-charge and induced charge. This spheroid experiences an insignificant pear-shaped distortion even when the charge of the drop and the electrostatic field strength are high.     

Nonlinear analysis of the equilibrium shape of a charged conducting drop in an electrostatic suspension

An analytical asymptotic expression is derived that describes the equilibrium shape of a charged drop of an ideal incompressible conducting liquid suspended in superposed collinear uniform electrostatic and gravitational fields. The expression is obtained in an approximation quadratic in the small amplitude of deviation of the equilibrium drop from a sphere, with the electrostatic field dimensionless strength taken as a measure of the deviation amplitude. With allowance for the gravitational and electrostatic fields and interaction between the drop self-charge and external electrostatic field, the equilibrium shape of the drop is found to be very close to a spheroid when the charge and the electrostatic field strength are far from their critical values. The analysis is carried out with a refined procedure of calculation of the equilibrium shape of drops placed in external force fields.     

Nonlinear vibrations of a charged drop in an electrostatic suspension

The problem of nonlinear vibrations of a charged drop of an ideal incompressible conducting fluid in an electrostatic suspension is analytically solved in an approximation quadratic in two small parameters: vibration amplitude and equilibrium deformation of the shape of the drop in an electrostatic field. To solve the problem analytically, the desired quantities are expanded in semiinteger powers of the small parameters. It is shown that the charge of the drop and the gravitational field influence the shape of the drop, nonlinear corrections to the vibration frequencies, and critical conditions for instability of the drop against the surface charge. At near-critical values of the charge, the shape of the nonlinearly vibrating drop falls far short of being a sphere or a spheroid, which should be taken into account in treating experimental data.     

Nonlinear analysis of the equilibrium shape of a charged drop in the tornado funnel wall

From the pressure balance condition on the free surface of a conducting liquid charged drop, an expression is derived for the equilibrium shape of the drop placed in the field of centrifugal forces acting in the tornado wall. The analysis is carried out in an approximation quadratic in small parameter (the ratio of the deformation amplitude to the radius of the initially spherical drop). In the linear approximation, the drop is a spheroid oblate in the direction normal to the tornado axis. The eccentricity of the spheroid squared is proportional to the angular velocity squared and the radius of the drop cubed. In the quadratic approximation, the equilibrium shape of the drop is other than spheroidal.     

Modification of the boundary layer theory for calculating viscous drop oscillations in a uniform electrostatic field

Capillary oscillations on the free surface of a viscous conductive liquid drop placed in an electrostatic field are calculated. In an approximation linear in stationary deformation amplitude, the drop in this field has the shape of a spheroid extended along the field. The initial problem is modified and simplified in terms of the boundary layer theory by applying an approximation that is linear in the oscillation amplitude and quadratic in the eccentricity of the drop. The accuracy of the approximate solution relative to an exact one is estimated. It is shown that, with a rise in the electrostatic field strength (with an increase in the eccentricity of the drop) and in the viscosity of the liquid, the boundary layer at the free surface of the drop becomes thicker.     

Electrostatic potential of point charges inside dielectric oblate spheroids

As a sequel to a previous paper on electrostatic potential of point charges inside dielectric prolate spheroids [1], this note further presents the exact solution to the electrostatic problem of finding the electric potential of point charges inside a dielectric oblate spheroid that is embedded in a dissimilar dielectric medium. Numerical experiments have demonstrated the convergence of the proposed series solutions.     

Beugung elektromagnetischer Wellen in Braunbekscher Näherung. I

An approximation method proposed by Braunbek for the diffraction of scalar waves by plane obstacles, or apertures in a plane screen, is extended to the diffraction of electromagnetic waves. As an example, the diffraction of a linearly polarized plane electromagnetic wave normally incident on a circular aperture in a perfectly conducting plane screen and on a perfectly conducting plane circular disk is considered in detail. The results are compared with those obtained from the exact solution ofMeixner andAndrejewski and with the results ofKirchhoff's approximations. It is shown that, in contrast to theKirchhoff approximations,Braunbek's method leads in all cases to the correct ‘optical’ limits and represents a good approximation even for wavelengths comparable with the size of the obstacle. After a general formulation ofBraunbek's method in the electromagnetic diffraction problem, we calculate in the present paper the far field of the circular aperture and the transmission coefficient by means of the cross section theorem. The near field of the circular aperture and the circular disk will be considered in a subsequent paper.     

New technique to measure particle size using electrostatic sensor

A new technique is proposed to measure the particle mean size using an electrostatic sensor in frequency domain. This paper starts with a finite-element modeling simulator to model the induced electric charge of a ring electrode and to find the electrode sensitivity. The mathematical modeling was used to extract particle size information from the simulated signal in frequency domain. The method is applied in an experimental test where a low-noise signal conditioning was designed with a ring electrode as the electrostatic sensor. The method can be used to establish a cost effective size measurement system using electrostatic sensor.     

Effect of the size of macroparticles on their electrostatic interaction in a plasma

The electrostatic interaction of two spherical macroparticles in a plasma has been considered. Primary attention has been focused on investigating the electrostatic interaction at short distances where polarization effects of the surface charge of finite-size macroparticles begin to play a dominant role. The first part of this study is devoted to the interaction of a point charge with a charged conducting sphere in an equilibrium plasma. It has been shown that the presence of a plasma in the system leads to a decrease in the potential barrier when two like-charged macroparticles approach each other and that this decrease proves to be the most significant in the case where the macroparticle radius is comparable to the Debye screening length. The second part of this study is concerned with the interaction of two conducting spheres in the bispherical system of the coordinates under the assumption that the charges of the conducting spheres are constant and under the assumption that the surface potentials of the spheres are constant. The latter case is closer to the physics of electrostatic interaction of two macroparticles in a plasma medium where the electrostatic potential of their surface is determined by the floating potential of the plasma. It has been demonstrated that the interaction potentials in these two cases are substantially different from each other and that, at constant macroparticle charges, the energy of the electrostatic field is an interaction potential, but, in the case of macroparticles with constant surface potentials, which are independent of the interparticle distance, the energy of the electrostatic field is not an interaction potential. In the latter case, account must be taken of the work done by external sources on the macroparticle potentials to maintain them constant. The form of the interaction potential has been established in this case from the analysis of the interaction force in terms of the Maxwell tension tensor. In the third part of this study, the interaction of two macroparticles has been considered in the spherical system of coordinates and analytical expressions for the interaction potentials have been derived for both the case of constant macroparticle charges and the case of constant surface potentials of the macroparticles.     

光电反馈式静电悬浮及静电力机制研究

本文介绍了光电反馈式静电悬浮方法 ,给出了静电悬浮力的理论计算公式 ,讨论了静电力与电极电压、电极 -悬浮体间距、悬浮体材料等因素之间的关系。通过实验测定了静电力与上述因素之间的关系曲线 ,获得了理想的结果。理论分析与实验结果均表明 ,实现导体与常规非导体的静电悬浮是完全可行的     

氧化物正电荷影响下半导体空间电势的近似解析解研究

建立了半导体空间电势与界面氧化物正电荷之间联系的解析表达式。从一维情况下精确的泊松方程及其边界条件出发,对N(P)型硅半导体中的泊松方程作积累(耗尽)近似,根据德拜屏蔽效应对边界条件作截断近似,得到了氧化物正电荷影响下两种类型半导体内电势的近似解析解。另外,还进行了精确数值计算,并将它与近似解析解的结果进行比较,结果表明,当氧化物正电荷增加到使P型半导体发生强反型后,近似解不再成立。根据强反型的条件,给出了P型半导体中近似解的适用范围。     

Extended boundary condition method in combination with field expansions in terms of spheroidal functions

The problem of light scattering by nonspherical particles, which arises in many applications, is nowadays most frequently solved by the method of extended boundary conditions in combination with the expansion of the fields in terms of spherical wave functions. However, such an approach encounters difficulties if the shape of particles is far from spherically symmetric, even in the simplest case of spheroids with the semiaxis ratio a/b > 5?10. A new approach to solving this problem is proposed, which also applies the extended boundary condition method but involves the expansion of the fields in terms of spheroidal functions. In this case, to obtain effective solutions for strongly prolate and oblate particles, the fields are divided in two parts with known properties and specific scalar potentials are used for each part. The basic relations of the approach are presented and some results of calculations of the optical properties of spheroids and spheroidal Chebyshev particles that are performed using computer codes realizing this approach are given. The convergence of the results for different cases and the domain of applicability of the method are discussed.     

电偶极子位于均匀介质球中时球外电场的研究

采用分离变量法求解了电偶极子位于均匀介质球中时复连通域的拉普拉斯方程和泊松方程,求出了球内外两种不同介质的电势分布和球面上的极化电荷分布;通过求解二阶非线性微分方程得到了球外的电场线函数;利用计算软件Math-ematica 5.0,作出了相应的相互正交的等势线簇图形和电场线簇图形,并且进行了必要的讨论.     

Scattering of electromagnetic radiation by a two-layer nonconfocal spheroid that is small compared to the wavelength

We have considered the electrostatic problem for a two-layer nonconfocal spheroid. The approach is based on surface integral equations that are similar to equations in terms of the extended boundary condition method for wave problems. Electrostatic fields are related to scalar potentials, which are represented as expansions in terms of eigenfunctions of the Laplace equation in two spheroidal coordinate systems, while unknown expansion coefficients are determined from infinite systems of linear algebraic equations. The constructed rigorous solution to the problem coincides with the known solution in a particular case of a confocal two-layer spheroid. In addition, for the nonconfocal two-layer spheroid, we have constructed an explicit approximated solution assuming that the field in the particle core is constant. This solution coincides with the rigorous solution if the scatterer shells are confocal. The formula found for the polarizability of the two-layer nonconfocal spheroid has a very simple form compared to the previously proposed cumbersome algorithm (B. Posselt et al., Measur. Sci. Technol. 13, 256 (2002)) and is more efficient numerically.     

The theory of electrostatic potential in superconductors

The problem of electrostatic potential distribution in superconductors is studied based on the microscopic theory of superconductors. Local chemical potential of the superconductor is introduced, and an approximation is made to BG theory, which is similar to the Thomas-Fermi (TF) method used in quantum mechanics. The electrostatic potential and charge distribution around an isolated vortex in type-II superconductors is discussed within the approximation. A correction to GL theory considering the electrostatic potential distribution is suggested.Received: 9 February 2004, Published online: 9 April 2004PACS: 74.20.Fg BCS theory and its development - 74.20.De Phenomenological theories (two-fluid, Ginzburg-Landau, etc.) - 74.25.Jb Electronic structureXiao-Yue Jin: Current address: Physikalisches Institut III, Erlangen-Nurenberg University, Erwin-Rommel-Str. 1, 91058 Erlangen, Germany     

Equilibrium shape and stability of a charged drop in an air flow under an electric field

The pressure balance on the surface of a charged liquid drop moving along a uniform electrostatic field is analyzed. The liquid is assumed to be nonviscous and incompressible. In the approximation linear in deformation amplitude, the equilibrium shape of the drop as a function of the charge, field strength, and velocity of travel can be both a prolate and an oblate spheroid. Critical conditions for the surface instability of such a drop are obtained analytically in the form of a relationship between the charge, field strength, and velocity of travel. An instability criterion is found by extrapolating to large Reynolds numbers. This makes it possible to fit the earlier model of a corona-initiated lightning in the vicinity of large charged water drops or hailstones to the charges of the drops, field strengths, and velocities of travel (relative to the medium) typical of thunderclouds.     

Determination of volume, shape and refractive index of individual blood platelets

Light scattering patterns (LSP) of blood platelets were theoretically and experimentally analyzed. We used spicular spheroids as a model for the platelets with pseudopodia. The discrete dipole approximation was employed to simulate light scattering from an individual spicular spheroid constructed from a homogeneous oblate spheroid and 14 rectilinear parallelepipeds rising from the cell centre. These parallelepipeds have a weak effect on the LSP over the measured angular range. Therefore, a homogeneous oblate spheroid was taken as a simplified optical model for platelets. Using the T-matrix method, we computed the LSP over a range of volumes, aspect ratios and refractive indices. Measured LSPs of individual platelets were compared one by one with the theoretical set and the best fit was taken to characterize the measured platelets, resulting in distributions of volume, aspect ratio and refractive index.     

Nonlinear oscillations of an uncharged conducting drop in an external uniform electrostatic field

The problem of nonlinear oscillations of the finite amplitude of an uncharged drop of an ideal incompressible conducting liquid in an external uniform electrostatic field is solved for the first time by analytical asymptotic methods. The problem is solved in an approximation quadratic in amplitude of the initial deformation of the equilibrium shape of the drop and in eccentricity of its equilibrium spheroidal deformation. Compared with the case of nonlinear oscillations of charged drops in the absence of the field, the curvature of the vertices of uncharged drops nonlinearly oscillating in the field is noticeably higher, whereas the number of resonant situations (in the sense of internal resonant interaction of modes) is much smaller.     

Magnetic field effects in the presence of electrostatic interactions

Magnetic field effects on ion radical pair (IRP) recombination reactions are considered. In the diffusion approximation simple analytical expressions for the magnitudes of these effects are obtained. They are reduced to those obtained within the simple cage approximation in the case, when the intramolecular spin interactions are strong enough and to those obtained within the free radical approximation in the opposite case. The applicability of both simple approximations are shown to be closely connected with peculiarities of the IRP decay kinetics. Namely, the cage approximation is applicable for the strong spin interactions because the IRP decays exponentially at small times. The free radical approximation is valid in the opposite limit of weak interactions since at large times the IRP decays as a free radical pair.