The Gauss law

A vector flux through an area and vector circulation along a closed contour is the basic characteristic of a vector field in vector algebra. The apphcation of these concepts to an electrostatic field appears extremely fruitful. 

We apply this concept, which we first met in Section 2.8.3, to a flux and density of flux. Accordingly, the elementary flux d5 of the electrostatic field strength E through an elementary area dS is 

Having written down an expression of scalar product and then attributing cos a first to E and then to dS, we obtain  

Similar to the way in which the total flat angle is equal to I dUr = Inrlr = 2n rad, the total solid angle is equal to  

Values Q can be determined if distribntion of the charge in a given problem is known. If the field is prodnced by a sum of N individual charges, then according to the superposition principle 

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By careful proceeding of measurements random variations can be minimized, but fundamentally not eliminated. The appearance of random errors follow a natural law (often called the Gauss law ). Therefore, random variations may be characterized by mathematical statistics, namely, by the laws of probability and error propagation. 

The Gauss and Ampere laws of magnetism are obtained mathematically, and somewhat artificially, from the fact that using a Cartesian basis gives Eq. (143) (the Gauss law) and from the fact that there is no current and no B, so we have... 

Using the Gauss law and die Green theorem, the electrostatic contribution can be rewritten as... 

If there is no charge accumulation, the Gauss law provides the equation... 

Another expression of the Coulomb law is the Gauss law, which states that the electric field associated with a charge distribution p (r) satisfies the relationship... 

S. The Gauss law (1.212) relates the surface-integrated field on the boundary of a volume to the total net charge inside the volume. Using the divergence theorem... 

By such a procedure, Feldberg (19) treated the relaxation of the diffuse double layer following a coulostatic injection of charge. When a charge density is injected into the electrode, assumed to be initially at the field at any time is given by the Gauss law ... 

The Gauss law of electrostatics is used to calculate the electric field produced by the total charge distribution obtained in the previous step. The new function V(r) generally differs from the estimation made in the first step. The procedure is repeated until two consecutive values of V(r) become essentially the same. Then the obtained self-consistent solution describes the electrons in the ground state of the multi-electron atom. 

In Eq. (15) q is the vacuum dielectric constant. The condition for electroneutrality of the solution as a whole is equivalent to the Gauss law, which determines the surface charge density, q. In the lubrication approximation it reads (Kralchevsky et al. 1999) ... 

To find the exact value of Ilei applying Eq. (26) the adsorption isotherms of ions are needed. Combining the isotherms and the Gauss law, Eqs. (15) and (16) written using the surface potential ips in the form... 

Experimentally, the fraction of free counterions in salt-free polyelectrolyte solutions is believed to give the main contribution to the system osmotic pressure. In the framework of the two-state models, the osmotic pressure is equal to the osmotic pressure of the free counterions. The more accurate analysis of the counterion effect on the solution osmotic pressure can be done in the framework of the Poisson-Boltzmann approach and its two-zone model simplification. In order to obtain an expression for the osmotic pressure in the framework of the two-zone model, one has to know the counterion concentration at the outer boundary of the spherical region. This requires knowledge of the electrostatic potential within the spherical zone. However, one can avoid solving the nonlinear Poisson-Boltzmann equation and use the relation between the local pressure P(r) and the electric field (r). To obtain this relation, one has to combine the differential form of the Gauss law ... 

In view of space limitations we here consider only the uncertainty of the Gibbs surface excess mass adsorbed according to Eq. (2.8). The Gauss law of error propagation immediately leads to the expressions... 

Uncertainties of Gibbs excess masses (3.52) as represented by their mean square deviations (MSD) (a ) can be calculated by applying the Gauss Law of propagation of uncertainty of error to eqs. (3.51) and (3.52) respectively ... 

The characterization is based on the assumption that selected orientations for the major axes of magnetic resonance tensors exist. Another assumption is the random scattering of major axes with respect to this direction according to the Gauss law. The orientation and disordering parameters are optimised by simulation of theoretical ESR spectra and comparison with experimental spectra. The mathematical algorithm developed for the simulation program is also discussed. Results of this approach were applied to experimental data for films of copper phthalocyanine and dipivaloil methanate, obtained by sedimentation on quartz plates. 

Poisson equation, i.e., the electroneutrality equation is basically the Gauss law, that is to say the Maxwell s equation giving the dependence between the electric induction vector and charge density... 

Since the injected hole concentration is much higher than the electron concentration the resulting charge concentration can be assumed to be consisting entirely of hole. The Poisson s equation in integral form transforms into the Gauss law ... 

In the case of large effects Cio 1 C20 the simplification on the right is obtained, which naturally must correspond to Eq. (5.238). On the basis of the Gauss law it is also possible to interpret the condition (5.252) as continuity of the dielectric displacement (D = dielectric constant times electric field sEq = SsEg). The electrical potential varies linearly (zero charge density) with x, and the difference < (s) — (0) is given by the product of the (constant) field strength and s. 

If the source of field is a charged line, the Gauss law can be given as... 

It is worth mentioning here that the Gauss law is valid for aU fields whose strength falls as 1/r. Obviously, only in this case the terms in the nominators and denominators of the expression (4.1.12) are cancelled. We know two such fields electrostatic and gravitation in these two cases the Gauss theorem is valid. 

Let us start with those equations that describe stationary phenomena. One of the equations is the Gauss law (Section 4.1.3). Its physical sense concerns the statement the sources... 

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