09.01.2016

## What is the law of electrostatic attraction

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Coulomb's Law states that the magnitude of the electrostatic force (Fe) between two objects is equal to a constant, k, multiplied by each of the two charges, q1 and q2, and divided by the square of the distance between the charges (r2). Formally, a positive value for the electrostatic force indicates that the force is a repelling force, while a negative value for the electrostatic force indicates that the force is an attractive force.
Question: A beam of electrons is directed into the electric field between two oppositely charged parallel plates, as shown in the diagram below. Answer: (4) toward the top of the page because the electron beam is negative, and will be attracted by the positively charged upper plate and repelled by the negatively charged lower plate. Question: The diagram below shows a beam of electrons fired through the region between two oppositely charged parallel plates in a cathode ray tube. Ecology Law Currents is the online-only publication of Ecology Law Quarterly, one of the nation's most respected and widely read environmental law journals. To be notified when the latest Currents articles are published, subscribe to the Currents list serve by emailing here. Sulfuric acid mist emissions from coal-fired power plants, which creates tell-tale blue plumes (not pictured here), has increasingly been under scrutiny by the EPA over the past decade. The LIFAC (limestone injected into the furnace with activation of untreated calcium oxide) sorbent injection desulfurization process. On January 2, 2013, the US Environmental Protection Agency and the US Department of Justice announced the settlement of a sulfuric acid mist case with Kentucky Utilites Company. See John Amos Power Plant emitted more sulphuric acid than reported, Herald-Dispatch, (Feb.
See Nicole Cohen, A Disappearing Town in the Shadow of Big Coal, National Public Radio, (Apr. A large body of experimental data now exists for (e,2e) differential cross section (DCS) ionisation studies in which the scattered and ejected electrons are detected with the same energy and at the 'same' asymptotic scattering angles with respect to the incident electron direction. A detection plane spanned by the scattered and ejected electron momenta ka and kb is initially defined. The energy region from around 3eV to 100eV excess energy is defined here as the 'intermediate energy' region. Studies at lower energies are considered to lie in the threshold region where correlation effects between outgoing electrons dominate the reaction. Differential cross section studies at energies in excess of 100eV are predominantly governed by single binary collisions between the incident and ejected electrons, the core playing a decreasing role in the reaction as this energy increases. Common to all these models is the importance of correlations between the outgoing electrons brought about through electrostatic coupling as they emerge from the reaction zone (see figure 2).
As the excess energy above ionisation decreases, the outgoing electrons have more time to mutually interact and so the probability that they will emerge asymptotically at a mutual angle of p radians increases, whereas their 'memory' of the incident electron direction correspondingly decreases.
In this page a model-independent parameterisation of the ionisation differential cross section is discussed. The parameterisation is in terms of a complete orthogonal set of irreducible tensorial angular functions which define correlation between any three vectors in space. This parameterisation was first proposed for the (e,2e) process by Klar and Fehr (1992), and has a long and distinguished history in the field of nuclear particle interactions proceeding through sharply defined states, where the analysis gives information about the angular momenta carried into and out of the reaction. By contrast, in the (e,2e) reaction the intermediate state is not normally sharply defined, and so the parameters do not directly reveal the angular momenta associated with the reaction, but rather indicate the degree of correlation between ingoing and outgoing electrons. The (e,2e) differential cross section is parameterised as a coherent superposition of partial waves up to L = 2 defining scattering in the reaction zone which is modulated by a Gaussian function modelling correlations between the outgoing electrons.
There is no evidence of the complex structure discussed above for the intermediate energy regime. Comprehensive experimental data exists for ionisation of helium in the intermediate energy region over a very wide range of scattering geometries. In these results the DCS is assumed to be zero when x = 0 and x = p since the probability of detection of two electrons of equal energy emerging in the same direction is very small. At 1eV excess energy a single structure is observed due to strong outgoing electron correlation, as previously noted. At 3eV and 5eV above ionisation both a forward and a backward structure start evolving from the central peak at low incident electron angles y, whereas at higher angles y the DCS is still approximately Gaussian. Backward scattering, probably dominated by elastic scattering of the incoming electron from the atom into the backward direction in the detection plane followed by a binary collision with a valence electron, is the dominant contribution to the overall DCS structure. At the higher excess energy (30 - 50eV) forward scattering starts to dominate, reflecting the importance of single binary collisions. Only the central peak is attainable through a single collision process and there will additionally be significant contributions to this peak through outgoing electron correlations at the lower energies. Indistinguishability of the outgoing electrons requires the DCS at (y,x) to be equated to that at (y, 2mp-x) where m is an integer. Reflection symmetry is necessary in the detection plane since no preferred direction is defined with respect to this plane, and hence s(y,x) = s(-y,x).

When electron spin is not considered the (e,2e) differential cross section is a function only of the momenta of the ingoing and outgoing electrons. The choice of angular functions Ilalbl0 required to characterise the (e,2e) DCS is restricted by the symmetries inherent in the ionisation process. Here the set for which la = lb when l0 is even and (la,A l0)A < lb when l0 is odd is chosen. It is clear that plotting the absolute magnitude of the I110 function as a radial vector when scanning over all possible values of theta and phi produces a 3-D surface which represents the function.
Since the magnitude can be positive as well as negative, it is important to represent this facet when describing the function in this way. It is necessary to consider the magnitude and sign of the functions as the (e,2e) Differential Cross Section is parameterised as a linear sum of the angular functions (see equation 6), and so for various values of theta and phi the functions can cancel each other to allow zeroes in the cross section.
The complete set of 3-D images of the 44 Angular Functions Ila,lb,l0 required to parameterise the symmetric (e,2e) Differential Cross Section over the energy range from 1eV to 50eV above the Helium ionisation threshold have been generated.
The summation is over the experimental data points i each with an associated weight wi evaluated from the uncertainty in the data. This linear fitting was fast and reliable, yielding excellent starting parameters for the second non-linear fitting method.
A Simplex method was used to minimise the c2 function by adjusting the Blalbl0 parameters (see figure 7). Any trial for which the fitting function did not obey the assumptions was rejected and the Simplex routine was reset to explore alternative regions of the c2 surface. This method converges far more slowly than the linear fitting method but requires no inferred data.
The maximum significant values of la, lb and l0 were established using a statistical F-test. At 1eV excess energy only l0 = 0 to 4 contribute, and the la and lb terms are all of the same sign, reinforcing each other in each l0 manifold. The relative magnitudes of the terms in each manifold combine to produce the lobe structure at the mutual angle of p radians. As the energy increases this regular pattern quickly disappears as the forward and backward lobes evolve from the central 'Gaussian' profile.
Higher order l0 correlation terms become significant as the intermediate energy region is entered, indicating an increasing contribution of higher order partial waves to the DCS. Additionally the la and lb terms start to compete in magnitude and sign to produce the more complex structures observed. At excess energies above about 30eV the parameters tend to stabilise in sign and vary smoothly with energy. The maximum significant number of la and lb terms in each l0 manifold also decreases as the energy increases, indicating that the number of partial waves required in the outgoing channel decreases with increasing energy. Figure 9 shows 3-D representations of the (e,2e) differential cross sections at each energy from 1eV to 50eV excess energy as obtained from fitting the data to the parameterisation. The 3-D surface is generated from a vector whose length is given by the magnitude of the differential cross section as the surface is generated throughout space for all angles (q,f). The backward scattering region is towards the viewer, whereas the forward scattering region is away from the viewer. The results shown in figure 9 show 3 dimensional representations of the (e,2e) differential cross sections derived from the parameterisation at the energies where experiments were conducted.
It is useful to establish a technique which allows interpolation between these results at the eight discrete excess energies, to enable the differential cross section to be estimated as a function of the three angular and one energy variables over the complete range of energies from threshold to 50eV excess energy. The (e,2e) differential cross section s is then projected onto a four dimensional space, whose axes are defined by the independant variables (s, q,f & Eexc).
As this cannot be visualised as a single projection on these pages, it is instructive to map one of these variables onto the time co-ordinate. In this case, we choose the excess energy to be varied in time, whereas the other 3 variables are mapped onto conventional 3-dimensional co-ordinate space, as has been done in the previous figures.a€?Hence, once the DCS is parameterised as a function of energy as well as a function of the scattering angle, this parameterisation can be used to derive 3-D images at discrete energy intervals.
These images can then be projected as a moving film, where the time from the start to the end of the film depicts the energy change from 1eV to 50eV excess energy.
A cubic spline is interpolated through the parameters as a function of the ratio of excess energy to the ionisation energy of 24.6eV.
These are fitted using a ninth order polynomial fit which closely emulates the cubic spline through the data. Details of these cubic spline fits to the Blalbl0 parameters can be found by accessing the appropriate interlink page splinefit.htm. The b-parameters which define the normalised DCS as a function of energy and angle can be downloaded here as a text file.
Although the b-parameters do not have a direct physical meaning, they allow the normalised DCS surfaces to be generated at any energy from 1eV to 50eV as has been discussed.
100 of these images have been generated at 0.5eV intervals, and compressed into a Quicktime movie. Note that these files are around 3MB in size, so may take some time to download should internet access be busy.
Additionally the evolution of the 44.6eV DCS as successive angular functions are added together can be seen!
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Both Newton's Law of Universal Gravitation and Coulomb's Law follow the inverse-square relationship, a pattern that repeats many times over in physics. Currents features short-form commentary and analysis on timely environmental law and policy issues.
This was the first case resolved by the DOJ that directly addresses sulfuric acid mist in a stand-alone enforcement case. These experiments have recently been carried out principally by two groups at Kaiserslautern group and at Manchester (see references for examples).
This region is now successfully modelled to a large degree using the Born approximation and its associated derivatives.
Although theoretical interest in this region is increasing, no theory adequately explains the experimental data accumulated so far. The Paris coplanar and Manchester Perpendicular plane results at an excess energy of 1eV ionising Helium as the target. In the experimentally inaccessible regions centred at x = 0A° and x = p the DCS is constrained to be positive and to have no points of inflection, and at the angles x = 0A° and p themselves the DCS is put equal to zero. The choice of la, lb and l0 used to define the subset is arbitrary within the constraint that the angular functions must be unrelated. These images are established by plotting the magnitude of the angular function against the angles theta and phi that are defined in the function. These artificially generated points were given low weighting to minimise the constraints placed upon the fit. Consideration of the data sets at all excess energies established that 44 angular functions up to la, lb = 7 and l0 = 6 gave the best overall fit to the data. The fitting parameters can be downloaded as a TAB deliminated text file by linking to the file Blalbl0.dat.
Further information can be obtained by linking from the figures to the appropriate WWW page.
3-D representations of the DCS calculated from the fitted parameters Blalbl0 for the 8 data sets from 1eV to 50eV. The 3D Quicktime movie can be downloaded at the end of this page by linking to the appropriate file. Similar to the manner in which the force of attraction between two masses is determined by the amount of mass and the distance between the masses, as described by Newton's Law of Universal Gravitation, the force of attraction or repulsion is determined by the amount of charge and the distance between the charges. To determine the direction of the force vector, once you have calculated its magnitude, use common sense to tell you the direction on each charged object.
The Wannier model parameterisation transformed to the (y,x) geometry is shown as a fit to the data. The function can either be positive or negative depending upon the value of theta and phi, and the sign is therefore represented by two colours, PURPLE indicating a positive magnitude and RED representing a negative magnitude. The example here shows the method for finding the minimum of a 3-D Gaussian function of the form z = - exp(-x^2 + y^2).
Higher resolution images together with a downloadable EPS file for each of the data sets can be obtained by linking to either of the above low resolution images.
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If you were to double the distance from a charge, you would quarter the electrostatic force on a charge. If the objects have opposite charges, they are being attracted, and if they have like charges, they must be repelling each other. Jantzen photographed various locales in Cheshire, Ohio once a year between 2002 and 2004, when American Electric Power bought up much of the land in the the town in response to complaints by residents of severe sulfuric acid mist pollution. This electron, and the incident electron that has lost almost all of its energy in the reaction have similar energies and therefore as they emerge from the reaction zone feel the Coulombic influence of each other and the ion core following ionisation. The scattering is more complex than in the threshold region, since the incident electron can penetrate deeper into the neutral target electron cloud, and therefore there is more complex interactions between the electrons and the core. The Simplex, in this case a 2-D triangle whose vertices are defined by points on the surface, proceeds by reflection, contraction, expansion or shrinking along the surface as shown. There are elements of work here, such as a subset of the images in the archive from WikiPedia, that originated as GNU General Public License works, so take care to follow the unique stipulations of that license in printed reproductions. The low relative velocity of the electrons allows them to mutually interact for sufficient time to tightly correlate their respective asymptotic directions.
There is now a finite probability of exchange, capture, ingoing and outgoing correlations between all charged particles in the reaction. This leads to both forward and backward scattering possibilities, depending upon the complexity of these interactions.
Double continuum wave functions and threshold law for electron atom ionisation, Physics Review A.