scholarly journals Simple Solutions of Lattice Sums for Electric Fields Due to Infinitely Many Parallel Line Charges

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 1040
Author(s):  
Erik Vigren ◽  
Andreas Dieckmann
Keyword(s):  
Electric Fields ◽  
Elliptic Functions ◽  
Parallel Line ◽  
Closed Form Solutions ◽  
Lattice Sums ◽  
Duality Symmetry ◽  
Long Line ◽  
Arbitrary Position ◽  
Electric Potentials ◽  
Lemniscate Functions

We present surprisingly simple closed-form solutions for electric fields and electric potentials at arbitrary position ( x ,   y ) within a plane crossed by infinitely long line charges at regularly repeating positions using angular or elliptic functions with complex arguments. The lattice sums for the electric-field components and the electric potentials could be exactly solved, and the duality symmetry of trigonometric and lemniscate functions occurred in some solutions. The results may have relevance in calculating field configurations with rectangular boundary conditions. Several series related to Gauss’s constant are presented, established either as corollary results or via parallel investigations conducted in the spirit of experimental mathematics.

scholarly journals Buckling analysis of nano-scale magneto-electro-elastic plates using the nonlocal elasticity theory

2018 ◽  
Vol 10 (8) ◽  
pp. 168781401879333 ◽  
Author(s):  
Weon-Tae Park ◽  
Sung-Cheon Han
Keyword(s):  
Electric Fields ◽  
Constitutive Relations ◽  
Shear Deformation Theory ◽  
Nonlocal Elasticity Theory ◽  
Nonlocal Theory ◽  
Buckling Analysis ◽  
Numerical Results ◽  
Elastic Plates ◽  
Electric Boundary Condition ◽  
Electric Potentials

Buckling analysis of nonlocal magneto-electro-elastic nano-plate is investigated based on the higher-order shear deformation theory. The in-plane magnetic and electric fields can be ignored for magneto-electro-elastic nano-plates. According to magneto-electric boundary condition and Maxwell equation, the variation of magnetic and electric potentials along the thickness direction of the magneto-electro-elastic plate is determined. To reformulate the elastic theory of magneto-electro-elastic nano-plate, the nonlocal differential constitutive relations of Eringen is applied. Using the variational principle, the governing equations of the nonlocal theory are derived. The relations between local and nonlocal theories are studied by numerical results. Also, the effects of nonlocal parameters, in-plane load directions, and aspect ratio on buckling response are investigated. Numerical results show the effects of the electric and magnetic potentials. These numerical results can be useful in the design and analysis of advanced structures constructed from magneto-electro-elastic materials.

scholarly journals Penetration of the electric fields of the geomagnetic sudden commencement over the globe as observed with the HF Doppler sounders and magnetometers

2021 ◽  
Vol 73 (1) ◽  
Author(s):  
Takashi Kikuchi ◽  
Jaroslav Chum ◽  
Ichiro Tomizawa ◽  
Kumiko K. Hashimoto ◽  
Keisuke Hosokawa ◽  
...  
Keyword(s):  
Electric Field ◽  
Electric Fields ◽  
Sudden Commencement ◽  
Ionospheric Currents ◽  
Republic Of China ◽  
F Region ◽  
Electric And Magnetic Fields ◽  
Magnetometer Data ◽  
Electric Potentials ◽  
Geomagnetic Sudden Commencement

AbstractUsing the HF Doppler sounders at middle and low latitudes (Prague, Czech Republic; Tucuman, Argentina; Zhongli, Republic of China, and Sugadaira, Japan), we observed the electric fields of the geomagnetic sudden commencement (SC) propagating near-instantaneously (within 10 s) over the globe. We found that the electric fields of the preliminary impulse (PI) and main impulse (MI) of the SC are in opposite direction to each other and that the PI and MI electric fields are directed from the dusk to dawn and dawn to dusk, respectively, manifesting the nature of the curl-free potential electric field. We further found that the onset and peak of the PI electric field are simultaneous on the day and nightsides (0545, 1250, 1345 MLT) within the resolution of 10 s. With the magnetometer data, we confirmed the near-instantaneous development of the ionospheric currents from high latitudes to the equator and estimated the location of the field-aligned currents that supply the ionospheric currents. The global simultaneity of the electric and magnetic fields does not require the contribution of the magnetohydrodynamic waves in the magnetosphere nor in the F-region ionosphere. The global simultaneity and day–night asymmetry of the electric fields are explained with the ionospheric electric potentials transmitted at the speed of light by the TM0 mode waves in the Earth-ionosphere waveguide.

The use of dipole lattice sums to estimate electric fields and dipole moment enhancement in molecular crystals

2007 ◽  
Vol 443 (1-3) ◽  
pp. 87-91 ◽  
Author(s):  
Mark A. Spackman ◽  
Parthapratim Munshi ◽  
Dylan Jayatilaka
Keyword(s):  
Dipole Moment ◽  
Electric Fields ◽  
Molecular Crystals ◽  
Lattice Sums

Exact solutions for the electron distribution in plane magnetrons

1965 ◽  
Vol 287 (1409) ◽  
pp. 183-200 ◽  
Keyword(s):  
Space Charge ◽  
Electric Fields ◽  
Stream Flow ◽  
Electron Distribution ◽  
Practical Interest ◽  
Space Charge Density ◽  
Complete Analysis ◽  
Field Variation ◽  
Self Consistent ◽  
Electric Potentials

The old problem of electron distribution in crossed electric and magnetic fields, such as exist in magnetrons, has in the past proved full of pitfalls, owing to the decisive influence which even very small initial electron velocities can have on the character of the solution. A complete analysis of the plane magnetron is presented, with a thermal emitter, i.e. with Maxwellian distribution of the initial velocities. Instead of looking for self-consistent solutions, which vary strongly with the space charge, the solution is given for three simple types of prescribed electric potentials, zero, linear and parabolic. The first two are mainly for orientation, the third is of practical interest as it is approximately self-consistent. For zero or weak electric fields the distribution is ‘triangular’, i.e. the function decreases monotonically as we move away from the cathode. For strong electric fields, the distribution has a peak away from the cathode and strongly resembles that obtained in the so called ‘double-stream’ flow. Finally, for a parabolic potential distribution (linear field variation) the space charge density exhibits a pronounced plateau which is highly reminiscent of the conditions in a Brillouin or ‘single-stream’ flow, although the electron motion is anything but rectilinear.

Elliptic functions and lattice sums for effective properties of heterogeneous materials

2021 ◽  
Author(s):  
Y. Espinosa-Almeyda ◽  
R. Rodríguez-Ramos ◽  
H. Camacho-Montes ◽  
R. Guinovart-Díaz ◽  
F. J. Sabina
Keyword(s):  
Effective Properties ◽  
Heterogeneous Materials ◽  
Elliptic Functions ◽  
Lattice Sums

scholarly journals Применение формулы Донкина в теории отражающих и поворотных устройств

2019 ◽  
Vol 89 (12) ◽  
pp. 1947
Author(s):  
Ю.К. Голиков ◽  
А.С. Бердников ◽  
А.С. Антонов ◽  
Н.К. Краснова ◽  
К.В. Соловьёв
Keyword(s):  
Electric Fields ◽  
Symmetry Plane ◽  
Optical Elements ◽  
Small Deviations ◽  
Similarity Principle ◽  
Beam Stability ◽  
Analytic Expressions ◽  
Electric Potentials ◽  
Direction Of Movement ◽  
Trajectory Similarity

Abstract Electrostatic turning devices are electron- and ion-optical elements changing the direction of movement of a parallel monochromatic beam of charged particles by a given angle without affecting a beam’s parallelism. The trajectory similarity principle for electric fields homogeneous in Euler terms ensures the fulfillment of this property for the fields with a homogeneity of a zero power. The Donkin formula for 3D homogeneous harmonic functions produces extremely wide class of analytic expressions describing homogeneous electric potentials of a zero power. This paper considers the algorithm of synthesis of electrostatic turning devices that transform input parallel monochromatic beams into output parallel monochromatic beams. The algorithm is based on the Donkin formula and ensures beam stability for small deviations from the electric field’s symmetry plane.

Electroelastic Fields Induced by Two Collinear and Energetically Consistent Cracks in a Piezoelectric Layer

2014 ◽  
Vol 30 (4) ◽  
pp. 361-372 ◽  
Author(s):  
X.-C. Zhong ◽  
K.-Y. Lee
Keyword(s):  
Stress Intensity ◽  
Boundary Conditions ◽  
Electric Fields ◽  
Stress Intensity Factors ◽  
Piezoelectric Layer ◽  
Intensity Factors ◽  
Crack Face ◽  
Closed Form Solutions ◽  
Electroelastic Fields ◽  
Crack Tips

AbstractWithin the framework of linear piezoelectricity, the problem of two collinear electrically dielectric cracks in a piezoelectric layer is investigated under inplane electro-mechanical loadings. The energetically consistent crack-face boundary conditions are utilized to address the effects of a dielectric inside the cracks on the crack growth. The Fourier transform technique is applied to solve the boundary-value problem. Under the consideration of two-case electromechanical loadings, the electroelastic fields near the inner and outer crack tips are obtained through the Lobatto-Chebyshev collocation method. The special case of two collinear energetically consistent cracks in an infinite piezoelectric solid is analyzed and the closed-form solutions of the crack-tip electroelastic fields are further determined. Numerical results show the variations of stress intensity factors and energy release rates near the inner and outer crack tips on the applied electric fields, the geometry of cracks and the width of the piezoelectric layer in graphics. The observations reveal that the stress intensity factors are dependent not only on the adopted crack-face boundary conditions, but also on the applied mechanical loading.

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