Mie–Lorenz Field Components

This paper presents the results of a theoretical study of the electromagnetic field in the vicinity of a rod antenna transmitting over a conductive ground plane. In the course of studying compatibility problems one occasionally finds it necessary to predict accurately the field strength within a few wavelengths of a radiating element. This problem i s more difficult than one might expect, especially if the length of the radiating element is greater than one tenth of a wavelength. In fact, the equations involved are sufficiently complex to effectively prohibit their use as a practical engineering tool. What is needed is a simple and rapid method for applying these equations. This paper presents such a method. The three general equations which describe the three components of the field are presented and discussed. A digital computer is used to obtain solutions for a specific antenna height and several specific frequencies. The graphical computer results are presented and discussed. A method for normalizing the three general equations is developed. It i s shown that the data obtained from the computer (for specific values of the variables) can be transformed into a normalized presentation in which all distances are measured in terms of wavelength. It therefore becomes possible to construct normalized field strength graphs. These graphs can easily be applied to obtain theoretical field strength values for the three components of the field over a wide range of parameters. Examples are presented, illustrating the method of application. Figure 1 is a typical normalized field strength graph. It is constructed for the case in which antenna height (h ) is equal to 0.685 wavelength. The horizontal axis represents horizontal distance from the base of the antenna, measured in wavelengths. Each of the curves on the graph corresponds to certain height above the ground plane, again measured in wavelengths. The vertical axis is field strength expressed in decibels above one microvolt per wavelength per ampere of antenna current. This graph i s for the vertical component of electric field. Similar graphs are constructed for the horizontal component of electric field, and for the magnetic field. The complete family of graphs is made up of a set of three graphs for each ratio of antenna height-to-wavelength to be evaluated.

This paper shows that amorphous Finemet films can be used as a medium for visualization and topography of inhomogeneous magnetic fields. The intensity of magnetooptical images obtained in the geometry of the polar magneto-optical Kerr effect is proportional to the normal component of the inhomogeneous magnetic field. This allows us to construct two-dimensional topograms of the normal component of the magnetic field. The images observed in the geometry of the longitudinal magneto-optical Kerr effect carry information about the planar component of the field. The vector field of the plane component has singular points that are displayed by magneto-optical images. Applying an external homogeneous field leads to the appearance of new singular points and their motion. At special points, the plane component of the field is equal to the value of the external field. This allows you to display a planar component by recording the coordinates of specific points.

The linear complexity of an N-periodic sequence with components in a field of characteristic p, where N = np ν and gcd(n, p) = 1, is characterized in terms of the n th roots of unity and their multiplicities as zeroes of the polynomial whose cofficients are the first N digits of the sequence. Hasse derivatives are then introduced to quantify these multiplicities and to define a new generalized discrete Fourier transform that can be applied to sequences of arbitrary length N with components in a field of characteristic p, regardless of whether or not gcd(N, p) = 1. This generalized discrete Fourier transform is used to give a simple proof of the validity of the well-known Games-Chan algorithm for finding the linear complexity of an N-periodic binary sequence with N = 2ν and to generalize this algorithm to apply to N-periodic sequences with components in a finite field of characteristic p when N = p ν. It is also shown how to use this new transform to study the linear complexity of Hadamard (i.e., component-wise) products of sequences.Keywordsdiscrete Fourier transformDFTGames-Chan algorithmHadamard productHasse derivativehyperderivativelinear complexitystream ciphers

Magnetic reconnection provides an efficient conversion of the so-called ‘free magnetic energy’ to kinetic and thermal energies of cosmic plasmas, hard electromagnetic radiation, and accelerated particles. This phenomenon was found in laboratory and space, but it is especially well studied in the solar atmosphere where it manifests itself as flares and flare-like events. We review the works devoted to the tearing instability — the inalienable part of the reconnection process — in current sheets which have, inside of them, a transverse (perpendicular to the sheet ‘plain’) component of the magnetic field and a longitudinal (parallel to the electric current) component of the field. Such ‘non-neutral’ current sheets are well known as the energy sources for flare-like processes in the solar corona. In particular, quasi-steady high-temperature turbulent current sheets are the energy sources during the ‘main’ or ‘hot’ phase of solar flares. These sheets are stabilized with respect to the collisionless tearing instability by a small transverse component of magnetic fiel, normally existing in the reconnecting and reconnected magnetic fluxes. The collision tearing mode plays, however, an important and perhaps dominant role for non-neutral current sheets in solar flares. In the MHD approximation, the theory shows that the tearing instability can be completely stabilized by the transverse fieldB n if its value satisfies the conditionB n /B≫S−3/4B is the reconnecting component of the magnetic field just near the current sheet,S is the magnetic Reynolds number for the sheet. In this case, stable current sheets become sources of temporal spatial oscillations and usual MHD waves. The application of the theory to the solar atmosphere shows that the effect of the transverse field explains high stability of high-temperature turbulent current sheets in the solar corona. The stable current sheets can be sources of radiation in the radio band. If the sheet is destabilized (atB n /B≪S−3/4) the compressibility of plasma leads to the arizing of the tearing instability in a long wave region, in which for an incompressible plasma the instability is absent. When a longitudinal magnetic field exists in the current sheet, the compressibility-induces instability can be dumped by the longitudinal field. These effects are significant in destabilization of reconnecting current sheets in solar flares: in particular, the instability with respect to disturbances comparable with the width of the sheet is determined by the effect of compressibility.

The research area includes the White Sea and adjacent land located in the junction zone of the eastern part of the Fennoscandian Shield and the Russian Plate. The purpose of the study is to construct a model of the lithospheric structure of the region using decomposition of anomalous gravitational and magnetic fields and inverse problem solving for components of gravity and magnetic fields, respectively. The decompositions of the fields were provided by the singular spectral method in the software package "R 4.3.1". The inverse problems were solved using the programs of the "Integro" complex. The components of the fields help to identify and analyze buried geological structures. The rift system of the White Sea is most clearly represented by the fourth component of the gravitational and magnetic fields. The positions of density and magnetic inhomogeneities of the Earth’s crust corresponding the components of the fields have been determined. The component model is compared with the seismic density and magnetic models of the lithosphere along the 3-AР geotraverse (Kem – White Sea Throat).

Measurement and estimation of performance characteristics (i.e. precision, bias, performance range, interferences and sensitivity) are often neglected in the development and use of biological sampling methods. However, knowledge of this information is critical in enabling potential users to assess data quality and make comparisons among different sampling methods. In this study, the performance characteristics were evaluated for both the field and laboratory components of a new large river macroinvertebrate bioassessment protocol (mLR‐BP) for non‐wadeable streams. We sampled 19 sites across two depth classes, collecting three replicate samples at each site and sorting three 300‐organism subsamples from each sample. The replicate samples provided data for estimates of precision in the laboratory and field, and abiotic variables allowed for measurements of overall sensitivity. Precision and performance range differed between shallow and deep sites, particularly for the field component. As compared with precision measured in other studies of bioassessment methods, the field component of the mLR‐BP performed similarly, particularly in shallow sites. Based on the measures of combined field and laboratory sensitivity, this protocol should be able to detect differences of approximately 20–25% in the metrics evaluated in this study, if used for bioassessment in similar types of rivers. With all sites and the field and laboratory components combined, metrics were most responsive to a gradient of urban land cover but also showed some relationship with agricultural land cover. However, metric responsiveness does not necessarily correlate with precision, and metric selection can influence the performance characteristics of the method. Overall, the sampling protocol shows great utility for bioassessment and monitoring of non‐wadeable rivers, as well as for measuring the success of restoration efforts. In addition, the design of this study provides a template for estimating performance characteristics in other non‐wadeable systems. Copyright © 2008 John Wiley & Sons, Ltd.

Geomagnetic jerks are rapid time variations of the magnetic field at the Earth's surface that are thought to be of primarily internal origin. Jerks are relevant for studies of the Earth interior: they likely give information on core dynamics and possibly on mantle electrical conductivity. In such studies a precise determination of the jerk occurrence time and its error bar at each observatory is required. We analyze the most well-known global jerks (1969, 1978, and 1991) and a possible local jerk in 1999, considering all three components of the magnetic field (X, Y, and Z). Different data sets are investigated: annual means, 12 month running averages of observatory monthly means in rotated geomagnetic dipole coordinates, and data representing the core field contribution synthesized from the CM4 time-dependent field model. The secular variation in each component of the field around the time of a jerk was modeled by two straight line segments, using both least squares and 1-norm methods. The 1969, 1978, and 1991 jerks were globally detected, while the 1999 event was only locally identified. Using this simple method enables us to calculate error bars in the jerk occurrence times and to quantify their nonsimultaneous behavior. We find that our error bars are not, in general, symmetric about the mean occurrence time and that the mean errors on the X and Z components of 1.7 years and 1.5 years are larger than that of 1.1 years on the Y component. Generally, the error bars were found to be larger in the Southern Hemisphere observatories. Our results are necessary prerequisites for further studies of the inverse problem that attempt to determine mantle electrical conductivity from variations in jerk occurrence times.

ISEE 1 and 2 magnetic field measurements are used to examine the structure of the low beta, quasi‐perpendicular shock. A shock crossing database consisting of ISEE 1 satellite crossings from the beginning of the mission in 1977 to the end of 1980 is utilized to identify shock crossings for this study. A set of 20 low beta, quasi‐perpendicular shock crossings are drawn from the database for study. Analysis of the shock overshoots indicates that the strength of the overshoot of low beta, quasi‐perpendicular shocks increases as the ratio of the Mach number to the first critical Mach number (or ratio of criticality) increases. There are subcritical crossings which have nonnegligible overshoots and other subcritical crossings which exhibit no overshoot. Wave analysis shows that the power of the downstream waves also increases as a function of this ratio of criticality. Upstream of the shock, large‐amplitude, low‐frequency whistler mode and higher‐frequency (f ∼ 1 Hz) whistler waves are evident for subcritical and marginally critical shocks. The lower‐frequency whistlers are right‐hand elliptically polarized and phase stand upstream of the shock, propagating along the shock normal direction. The thickness of the shock is found to be within a factor of 1 and 2 times greater than the wavelength of this precursor wave. This result is inconsistent with the conjecture that the shock is merely the last amplified cycle of the precursor wave, for if this were true, the thickness of the shock from minimum to maximum would be one half of the precursor wavelength. The 1‐Hz waves are right‐hand elliptically polarized and propagate upstream obliquely to the magnetic field direction. Downstream of the marginally critical and supercritical shock, left‐hand elliptically polarized waves are found to propagate along the magnetic field direction and have frequencies of about 0.2‐0.8 fci. These ion cyclotron waves appear to result from the excitation of the Alfvén ion cyclotron (AIC) instability. The AIC instability is driven by the Ti⊥>Ti∥ temperature anisotropy created in front of the shock by the reflection of solar wind ions. Ion cyclotron waves act to pitch angle scatter the ions downstream of the shock and remove the temperature anisotropy. A transitional behavior in the noncoplanar component of the magnetic field occurs at or about the first critical Mach number. Below the critical Mach number, the noncoplanar component is associated with the upstream whistler train. When the ratio of criticality is approximately unity, the noncoplanar component is isolated from any upstream or downstream wave activity. In the supercritical regime, this component of the field is associated with the downstream ion cyclotron wave train. For all ranges of criticality, the noncoplanar component is seen to lie within the shock ramp, and the transitional behavior of this component of the field indicates that it is an inherent part of the shock.

Models of uniformly rotating stars in purely radiative equilibrium containing multipole magnetic fields are constructed by a perturbation technique. Initially calculations are performed for a mixture of P/sub 2/ and P/ sub 4/ fields. It is found that for a given surface P/sub 2/ field both the P/sub 2/ and P/sub 4/ components of magnetic field have to be very strongly concentrated in the interior as the rotation speed increases. For a given interior magnetic flux there is found to be a maximum rotation speed at which it is possible to find solutions. However, unlike the analogous problem with a dipole-type magnetic field, this does not correspond to the limit of zero surface field. The calculations are extended to include a P/sub 6/ component of field, and some evidence for the convergence of the expansion procedure for the field is found. (auth)

In the work formalism for the determination of electronic structure, the exchange-correlation energy and (local) potential of the electrons both arise via Coulomb's law from the same source, viz., the quantum-mechanical Fermi-Coulomb hole charge. The potential is the work ${\mathit{W}}_{\mathrm{xc}}$(r) done to move an electron in the field of its Fermi-Coulomb hole and the energy is the interaction energy between the electronic and hole charge densities. For nonsymmetrical electronic density systems for which the curl of the field may not vanish, a local effective exchange-correlation potential ${\mathit{W}}_{\mathrm{xc}}^{\mathrm{eff}}$(r) is determined from the irrotational component of the field, the solenoidal component being neglected. In this paper we investigate this approximation further to better understand its accuracy for nonspherical density systems by application to a degenerate ground state of the carbon atom within the Pauli-correlated approximation. The results of the non-self-consistent calculations indicate that the solenoidal component of the force field due to the Fermi hole is negligible and two orders of magnitude smaller than the irrotational component. Therefore, essentially all the effects of Pauli correlation are accounted for by the latter and the effective exchange potential ${\mathit{W}}_{\mathit{x}}^{\mathrm{eff}}$(r) is thereby an accurate representation of the local exchange potential in the atom. Further, the solenoidal component of the field vanishes at the nucleus, in the classically forbidden region and along certain axes of symmetry. Thus the work ${\mathit{W}}_{\mathit{x}}$(r) in the field of the Fermi hole for such nonspherical atoms is path independent over substantial regions of configuration space. Finally, we discuss the structure of the local many-body potential of nonspherical-density atoms when both Pauli and Coulomb correlations are present.

Chapter 2 - Wave-Equation Description of Nonlinear Optical Interactions

Mars Global Surveyor (MGS) measured the most strongly magnetized crust in the heavily cratered southern hemisphere of Mars. Our analysis concentrates on the magnetic lineations or patterns centered near latitude 40°S, longitude 180°W, with a range of values ±40°, using a rotated Cartesian coordinate system. We downward continued the magnetic field measured at ∼400 km elevation and very closely match the corresponding component measured during the aerobraking phase at altitudes extending down to ∼100 km. Using the vertical component of the magnetic field alone, we construct a unique scalar potential and independently obtain from the derivatives of this scalar potential the x and y components of the field. These derived components agree very well with the observed horizontal components. This demonstrates the validity and utility of the method and the Cartesian approximation, and also it confirms the consistency of the MGS magnetic data set. A model constructed with just 8 vertical dipoles accounts for 80% of the variance of the scalar potential at 400 km over the region analyzed, but 14 dipoles can account for only 64% of the variance at 100 km. We also construct the vector potential, the curl of which generates the three components of the magnetic field. This more complicated description may contain more physical meaning than the scalar potential. The vector potential shows abrupt changes in direction over the analyzed region, suggesting either different stages of magnetization or local demagnetization.

By using purely geometric forces on a noncommutative spacetime, noncommutative spectral geometry (NCSG) was proposed as a possible way to unify gravitation with the other known fundamental forces. The correction of the NCSG solution to Einstein's general relativity (GR) in the four-dimensional spacetime can be characterized by a parameter \(\beta\sim 1/\sqrt{f_{0}}\), where \( f_{0}\) denotes the coupling constants at the unification. The parameter \( \beta\) contributes a Yukawa-type correction \(\mathrm{exp}(-\beta r)/r\) to the Newtonian gravitational potential at the leading order, which can be interpreted as either the massive component of the gravitational field or the typical range of interactions carried by that component of the field. As an extension of previous works, we mainly focus on the Solar System and stellar tests of the theory, and the constraints on \(\beta\) obtained by the present work is independent of the previous ones. In the Solar System, we investigate the effects of the NCSG on the perihelion shift of a planet, deflection of light, time delay at superior conjunction (SC) and inferior conjunction (IC), and the Cassini experiment by modeling new observational results and adopting new datasets. In the binary pulsars system, based on the observational data sets of four systems of binary pulsars, PSR B1913+16, PSR B1534+12, PSR J0737-3039, and PSR B2127+11C, the secular periastron precessions are used to constrain this theory. These effects in the scale of the Solar System and binary pulsars were not considered in previous works. We find that the lower bounds given by these experiments are \(\beta \simeq 10^{-9} \sim 10^{-10}\) m-1, considerably smaller than those obtained in laboratory experiments. This confirms that experiments and observations at smaller scales are more favorable for testing the NCSG theory.

Large scale (many minutes to 10 hours) magnetic field structures consisting predominantly of nearly north‐south field directions have been discovered in Jupiter's magnetosheath from the data of Voyagers 1 and 2 and Pioneer 10 during their outbound encounter trajectories. The Voyager 2 data, and those of Voyager 1 to a lesser extent, show evidence of a quasi‐period of 10 hours (and occasionally 5 hours) for these structures. For all three spacecraft the changes in the field throughout these structures for many tens of hours are approximately restricted to a plane parallel to Jupiter's local magnetopause, according to a variance analysis of the field. Similar directional changes in the field occurred in the inbound magnetosheath for the Voyager spacecraft, but the occurrence was much less frequent, no quasiperiodicity was apparent, and the scale lengths were on average much shorter. The north‐south components of the field and plasma velocity are strongly correlated in the outbound magnetosheath as observed by Voyagers 1 and 2, and the components orthogonal to the north‐south direction show weak correlations. For both Voyager encounters the sense (positive or negative) of the north‐south correlations has been directly related to the direction of the ecliptic plane component of the interplanetary magnetic field (IMF) using the field and plasma measurements of the non‐encountering spacecraft. Some outbound magneto pause and bow shock crossings, on Voyager 2 especially, are phase locked in system III with some of the large scale magnetosheath field and plasma structures. These structures may be accounted for in terms of field line draping around the magnetopause of the convected IMF and solar wind, where the temporal properties are controlled by the motion and shape of a flattened magnetosphere which, in turn, depend on the rapid rotation of the current sheet within the magnetosphere.

A set of classical solutions of a singular type is found in a 5D SUSY bulk-boundary system. The “parallel” configuration, where the whole components of fields or branes are parallel in the iso-space, naturally appears. It has three free parameters related to the scale freedom in the choice of the brane-matter sources and the “free” wave property of the extra component of the bulk-vector field. The solutions describe brane, anti-brane and brane–anti-brane configurations depending on the parameter choice. Some solutions describe the localization behaviour even after the non-compact limit of the extra space. Stableness is assured. Their meaning in the brane world physics is examined in relation to the stableness, localization, non-singular (kink) solution and the bulk Higgs mechanism.