Current, density

An alternative approach to evaluating the magnetic properties is based on the concept of current density [15,24,25]. The electronic current density is an expectation value of the current operator, the product of the observables for the electron density and velocity 

The energy of interaction between the field A and the electrons of a molecule, flowing with current density J, is 

The M-electron wave function can be written in the form of a perturbation expansion 

At the same time the electronic current density can be expanded in powers of the magnetic induction 

For a field applied along the ath Cartesian axis, the induced current density can be rewritten in the form 

---

H0S(0)200S(0)20H. Dibasic acid formed as salts by electrolysis of sulphates at low temperatures and high current density. The acid and persulphates are strong oxidizing agents ( "[S20a] to S04 -t-2 01 volts in acid) but the reactions are often slow. Compare permonosulphuric acid. 

This correction function was calculated for different kinds of excitation coils, like circular coils without ferrite core, spiral coils, double-D coils and a sheet inducer. For this purpose the eddy current density was determined for frequencies between 10 and 1000 Hz and for depths between 0 and 30 mm. 

First, the eddy current density is damped while penetrating into the conductor (penetration effect). Here the frequency dependence of the penetration depth implies that for deep lying cracks low frequencies must be used for obtaining a sufficient current density in the vicinity of the crack. Secondly, due to the induction law the induced current density at the surface jco is diminished when using lower frequencies. Therefore, in total, there is a certain excitation frequency which results in a maximum response field from the crack. 

The simulation of the actual distortion of the eddy current flow caused by a crack turns out to be too time consuming with present means. We therefore have developed a simple model for calculating the optimum excitation frequencies for cracks in different depths of arbitrary test sarriples Using Equ. (2.5), we are able to calculate the decrease in eddy current density with increasing depth in the conductor for a given excitation method, taking into account the dependence of the penetration depth c on coil geometry and excitation frequency. 

Figure 3.3 shows the increasing attenuation for cracks in a depth between 5 and 30 mm, using the optimised excitation frequency for each depth. The coils (circular, double-D) have a current density of lOWm. In case of circular and double-D coil, this corresponds to an... 

We have perfomied some simulations of the eddy current distribution in a test object for a spiral coil and a circular one (see Fig. 4.1). Both coils had 9 turns and the excitation current was 6 mA. Figs. 4.1 show the cross section of the sample at the location of the crack and the amplitude of the eddy current density. One observes a 1.5 higher current density at the sides of the crack for the case of the circular coil. 

Left Fig. 4.1 Excitation with spiral (top) or circular (bottom) coil. While color corresponds to a high eddy current density in the sample (9turns, 6mA). 

This modelisation is based upon the computation of the eddy current density J z,r,0 ) when there are no flaw, then of the perturbation induced by the presence of a flaw. The magnetic field can be then computed in any point of the space. The modelisation is quite easier if the excitation coil is axi-symetric, as the computation of becomes a 2D problem. 3D computations are needed only for perturbation and resultant magnetic field. 

The first law of electrode kinetics, observed by Tafel in 1905 [197], is that overvoltage i) varies with current density i according to the equation... 

We then have, from A2.4.24 . ,where the charge on ions of type a is and the applied electric field is E. Given that the current density, J, in d Lis... 

At higher current densities, the primary electron transfer rate is usually no longer limiting instead, limitations arise tluough the slow transport of reactants from the solution to the electrode surface or, conversely, the slow transport of the product away from the electrode (diffusion overpotential) or tluough the inability of chemical reactions coupled to the electron transfer step to keep pace (reaction overpotential). 

For a more general reaction of the fonn Ox + ne Red, with differing concentrations of Ox and Red, the exchange current density is given by... 

In the absence of free charges and current densities, we have in cgs units ... 

Cyclic voltammetry provides a simple method for investigating the reversibility of an electrode reaction (table Bl.28.1). The reversibility of a reaction closely depends upon the rate of electron transfer being sufficiently high to maintain the surface concentrations close to those demanded by the electrode potential through the Nemst equation. Therefore, when the scan rate is increased, a reversible reaction may be transfomied to an irreversible one if the rate of electron transfer is slow. For a reversible reaction at a planar electrode, the peak current density, fp, is given by... 

The expression for the mass-transport-limiting current density may be employed together with the Nemst equation to deduce the complete current-potential response in a solution containing only oxidized or reduced species... 

The effects of ultrasound-enlianced mass transport have been investigated by several authors [73, 74, 75 and 76]. Empirically, it was found that, in the presence of ultrasound, the limiting current for a simple reversible electrode reaction exhibits quasi-steady-state characteristics with intensities considerably higher in magnitude compared to the peak current of the response obtained under silent conditions. The current density can be... 

---