Current density vector

By defining a four-vector with the components of the current-density vector J and charge density, i.e. 

Finally, recall that subtraction of (5.2.1b) from (5.2.1a) yields, taking into account (5.2.1c) and (5.2.5a), ix = 0 (this corresponds to solenoidality of the total current density vector in more than one dimension). 

The third-order interaction energy, equation 4, contains contributions, which can be expressed in terms of the second-order election current density vector JBE induced by the fields,... 

To solve the electrical problem at the electrodes, two variables need to be found (i.e., the electrical potential (scalar) and the current density (vector)). By analogy, two variables are also needed to find the ionic flux and the potential distribution. The vector relationship is given by Ohm s law (3.10), while the scalar relation is provided by expression (3.5), which can be re-written as ... 

Step 1. We represent the current density vector at the point r and at the instant t as a sum of the contributions produced by multiple moving point... 

In the above, E, D, H, B, Jt represent the electric field, electric displacement, magnetic field, magnetic induction, and free current density vectors respectively c is the velocity of light. Form the scalar product of (1.7.1a) with H and of (1.7.1b) with E and subtract to find... 

What current density vector y (units Am ) is produced by the application of an electric field of 10 V/m along the [1 1 1] direction of the crystal What is the angle between y and El What is the magnitude of a-along the [111] ... 

The blocks, or elementary volumes, form the grid. The blocks are numbered with two indexes, i and j, in the horizontal and vertical direction respectively. The horizontal axes is denoted as x and the vertical one - as z. The sizes of the block along these axes are denoted as Ax and Az. The values of cp, Ci, c2 and a are located in the centre of the block and are supposed to be constant in it. The current density vector j has the horizontal and vertical components j=(jx, jz) these components are assigned to the centres of the block sides see Fig.l. Ohm s law in vector notation gives us the relation between (p and j ... 

On the other hand, the selectivity of the electrochemical deposition of the metal on the substrate must be 100% of the current efficiency, with no interference from the other metal deposition processes. Therefore, the potential distribution needs to be presented for any serious electrochemical reactor study and the electrocatalyst selection problem. The major problem of current distribution depends on the type of the process that controls the entire reaction rate, such as charge transfer, ohmic contributions, or mass transport to or from the electrode. Many parameters have to be evaluated in the course of an electrochemical process to obtain the desired uniform potential and current distributions. One of the conditions that has to be fulfilled is the continuity equation for the current density vector, j ... 

To study the magnetic properties of matter one would often like to be able to obtain information on the currents in the system and their coupling to possible external magnetic fields. Important classes of experiments for which this information is relevant are nuclear magnetic resonance and the quantum Hall effects. SDFT does not provide explicit information on the currents. RDFT in principle does, but standard implementations of it are formulated in a spin-only version, which prohibits extraction of information on the currents. Furthermore, the formalism of RDFT is considerably more complicated than that of SDFT. In this situation the formulation of nonrelativistic current-DFT (CDFT), accomplished by Vignale and Rasolt [140, 141], was a major step forward. CDFT is formulated explicitly in terms of the (spin) density and the nonrelativistic paramagnetic current density vector jp(r). Some recent applications of CDFT are Refs. [142, 143, 144, 145]. E. K. U. Gross and the author have shown that the existence of spin currents implies the existence of a link between the xc functionals of SDFT and those of CDFT [146], Conceptually, this link is similar to the one of Eq. (99) between functionals of DFT and SDFT, but the details are quite different. Some approximations for xc functionals of CDFT are discussed in Refs. [146, 147, 148]. 

Ohm s law A linear relation between the electric field and current density vectors. 

The ACID ATj can be plotted as an isosurface similar to the total electron density. However, in contrast to the total electron density, only nonlocal (delocalized) electrons contribute to AT, . For a more detailed analysis of molecular magnetic properties, the current density vectors can be plotted onto the ACID hypersurface. It is important to note that the ACID, as defined earlier, is invariant with respect to the relative orientation of the molecule and the magnetic field. Therefore, the ACID hypersurface is unambiguous even in nonplanar and unsymmetric systems. 

Figure 13.13 Mobius [28]hexapor-phin(1.1.1.1.1.1). Current density vectors are plotted on top of the ACID isosurface. The ACID isosurface is not shown to provide a clearer picture of the electron flow within the Mobius twisted % system. For technical details see Figure 13.8. The delocalized system in the periphery includes 28 % electrons...

Figure 13.28 Coarctate Mobius transition state of the coarctate biscyclization (Figure 13.27). (a) calculated geometry (B3LYP/6-31G ), (b) ACID plot, isosurface value 0.05, current density vectors plotted...

Now let us introduce passive ion channels in the membrane of a polarized cell (Figure 5.7). The channels are normally closed, but now suddenly opened. Due to the potential difference, cations will immediately start to migrate into the negative cell interior. A current density field is suddenly created both intra- and extracellularly. The extracellular current density vector field J and the potential field are related by Eq. 2.1 V = — J/a. The current is generated by the ionic flow, and it terminates on the membrane capacitor in a discharge/charge process. 

Vi2- If the value of the potential is known everywhere in space can be interpreted as a potential field (x,y,z), V is simply the voltage difference between two defined points in a wired electric circuit. is a scalar quantity [energy per charge = volt], the current density J is a space vector and has a current density vector field J(x,y,z). 

Figure 6.16 Four-electrode system with current density and reciprocal current density lines. Sensitivity is illustrated at one point, the dot product therefore, the sensitivity there is small because the two current density vectors are almost perpendicular to each other.

---