Electrostatic Field Theory Research Articles

On the theory of electrostatic fields under the conditions of the ionosphere

On the theory of electrostatic fields under the conditions of the ionosphere

Approximate solution of axially symmetric problems

A variety of physical problems in such diverse fields as electrostatic field theory, heat and ideal fluid flow, and stress concentration theory reduce, under the assumption of axial symmetry, to the study of the elliptic partial differential equation ∂ 2 u/∂x 2 + ∂ 2 u/∂y 2 + k/y(∂u/∂y) = 0. Dirichlet-type problems associated with this equation are studied on regions whose boundaries include a nondegenerate portion of the x-axis and exceedingly accurate numerical methods are given for approximating solutions.

Molecular orbital theory of the ligand field in hydrated V3+salts

The theory of magnetic anisotropy and susceptibility of V3+ salts has been worked out on the basis of the molecular orbital method of Stevens and others, since the usual electrostatic field theory is not quite adequate to explain the magnetic behaviour. It is found that the anisotropic part of the crystal field changes with temperature possibly owing to the thermal expansion and relaxation of the crystal lattice. The spin-orbit coupling coefficient has to be decreased anisotropically by 35 to 40% from its free ion value which indicates asymmetric overlap between the 3d V3+ and s and p ligand charge clouds. The agreement of the theoretical values with the experimental is good.

The electronic structure and spectra of octacyanide complexes

The optical and electron spin resonance spectra of molybdenum and tungsten octacyanides have been measured and the results are interpreted by electrostatic crystal field theory using a point charge model of D 2d symmetry.

Discussion of paper by John R. Spreiter and Benjamin R. Briggs on “Theory of electrostatic fields in the ionosphere at polar and middle geomagnetic latitudes”

In their paper Spreiter and Briggs [1961] presented new computations of the electrostatic coupling between the E and F regions, based on ionospheric data much more extensive and reliable than that used in my earlier calculations [Farley, 1959; 1960]. The numerical results obtained were similar to mine but indicated an even stronger degree of coupling. In discussing the boundary condition at infinity to be used in the calculation, Spreiter and Briggs made some points that I believe could be debated, but since any differences of opinion that we may have are rather academic and do not affect the final answer, there seems to be little point in discussing them here. I certainly agree with their final results, which undoubtedly will be of interest in connection with various ionospheric problems.

Theory of electrostatic fields in the ionosphere at equatorial latitudes

The properties of the elongated electrostatic fields that are required to provide the coupling mechanism in the dynamo-motor concept of the E and F regions of the ionosphere are examined theoretically for the conditions that prevail in equatorial latitudes. The analysis is developed for an electrostatic field of arbitrary horizontal scale in a horizontally stratified partly ionized gas subject to an imposed magnetic field having the form of a parabolic arch over the equator. The anisotropic character and continuous variation with height of the conductivity are retained throughout , and numerical solutions are determined for the attenuation of the electric field with distance along the field line. The results are similar qualitatively to those found previously upon analysis of the corresponding problem for middle latitudes, but the attenuation of the electrostatic field with height is considerably greater. It develops, in particular, that the coupling between E and F regions is very small for fields having horizontal wavelengths of a few kilometers. It follows that the dynamo-motor concept could not be used to account for the presence of irregularities of this scale. On the other hand, it is found that almost all the attenuation occurs at heights near that of the E region. If an electrostatic field having a horizontal wavelength of a few kilometers could be produced at heights of, say 200 km or greater, it follows that it would be very elongated and extend with little change in amplitude from hemisphere to hemisphere. The results also indicate that effective coupling could be achieved for fields having horizontal wavelengths of the order of tens of kilometers, but the assumptions introduced to simplify the analysis may impair the quantitative reliability of the results for fields of this scale.

Theory of electrostatic fields in the ionosphere at polar and middle geomagnetic latitudes

An analysis is presented of the properties of the elongated electrostatic fields that provide the coupling mechanism required in the dynamo-motor concept of the E and F regions of the ionosphere. This concept has been proposed in recent years to account for a variety of observed phenomena ranging from large-scale effects associated with tidal flows in the F region to small-scale effects associated with the scintillation of radio signals from radio stars or from space vehicles. The analysis is developed for an electrostatic field of arbitrary horizontal scale in a horizontally stratified partially ionized gas subject to an imposed magnetic field selected to be representative of conditions in polar and middle geomagnetic latitudes. The anisotropic character and continuous variation with height of the conductivity are retained throughout, and numerical solutions are determined for the attenuation of the electric field with height above the dynamo region. The results show that the attenuation is much less than that which would be experienced in a homogeneous medium, and that it is possible, for instance, for electrostatic fields having horizontal scale as small as about 3 or 4 km to extend from 130 km to 400 km with a loss in magnitude of only about 25 per cent. The results also show that relatively small changes in the ionospheric conditions, well within the range of possibilities, can result in very substantial changes in the attenuation of the electric field with height.

The nature of the crystalline fields in Ti3+alum

A theory of susceptibility for titanium caesium alum is given. The crystalline field in this alum is treated on the molecular orbital method of Stevens and others, as the usual electrostatic field theory is found to fail to explain the magnetic behaviour. Experimental susceptibility data between 300 and 100°K as well as the paramagnetic resonance data at 2·5°K can all be accounted for satisfactorily by assuming that the trigonal field splitting changes from 800 to ≈ 170 cm-1, with temperature, which is also indicated by the large observed increase in the spin-lattice relaxation time from 300 to 1·2°K.

A theory of electrostatic fields in the ionosphere at nonpolar geomagnetic latitudes

The theoretical electrostatic coupling between the dynamo region and the F region of the ionosphere is examined at non-polar geomagnetic latitudes. It is found that, under certain conditions, significant coupling between these two regions can occur at all latitudes, even for electrostatic fields with horizontal scale sizes as small as a few kilometers. The coupling is strongest at the poles and weakest at the equator. Strong coupling will also occur between magnetically conjugate portions of the F region, while weaker but significant coupling will exist between conjugate portions of the dynamo region. The strength of the electric source field which would be produced by an irregular, horizontally stratified wind in the dynamo region is then computed, both for polar and for nonpolar latitudes. The results indicate that, to a large extent, the polarization charge that the local winds attempt to build up leaks away vertically and forms closed current loops. In other words, the ‘internal impedance’ of the thin source layer is greater than the ‘load impedance’ presented by the rest of the ionosphere. Finally, the possibility that electrostatic fields may cause significant electron-density variations in the F region is examined briefly. It is concluded that they will not.

A theory of electrostatic fields in a horizontally stratified ionosphere subject to a vertical magnetic field

A theory is developed to describe quantitatively the idea that in an ionized gas subject to an imposed magnetic field, such as the ionosphere, the lines of magnetic flux are approximately equipotential lines. The ionosphere is assumed to be horizontally stratified, and the case in which the earth's magnetic field is vertical is considered. Small-scale electrostatic fields are studied with a view towards elucidating the phenomena of spread F and radio star scintillation. The analysis indicates that in the ionosphere the results are strongly affected by the variation of conductivity with height, as well as by the anisotropy. For a reasonable model of the ionosphere it is shown that it is possible, under certain conditions, for a horizontal field three kilometers or larger in extent, at a height of about 120 or more kilometers, to produce a similar, localized electric field in the F region, not appreciably reduced in strength. The height of the source is the most important factor, but the temperature and ionization-density profiles are also significant. The fact that the strength of the small-scale fields in the F region could vary by one or two powers of 10 for plausible diurnal variations of the ionospheric parameters suggests that these fields could perhaps be responsible for the puzzling diurnal behavior of spread F and radio star scintillation.