Hertz potential

XIX. Connection between Spinors, Hertz Potential, and Beltrami Fields... 

In utilizing a complex three-vector (self-dual tensor) rather than a real antisymmetric tensor to describe the electromagnetic field, Hillion and Quinnez discussed the equivalence between the 2-spinor field and the complex electromagnetic field [63]. Using a Hertz potential [64] instead of the standard 4-vector potential in this model, they derived an energy momentum tensor out of which Beltrami-type field relations emerged. This development proceeded from the Maxwell equations in free homogeneous isotropic space... 

XIX. CONNECTION BETWEEN SPINORS, HERTZ POTENTIAL, AND BELTRAMI FIELDS... 

Similar to the Hillion-Quinnez model, Rodrigues and Vaz defined an EM field that is a function of a specific Hertz potential ... 

For their Hertz potential, Rodrigues and Vaz chose the factor < )(f,x) = 4>(x) exp (j fl t). Now, since II satisfies the wave equation, we conclude that the factor (<))x) in turn satisfies the Helmholtz equation ... 

Once again, using the Pauli algebra, we express the Hertz potential as a sum of its electric and magnetic parts ... 

It will prove useful in the sequel to use the hertz potentials (Born and Wolf, 1980, pp. 76-84) to describe the electromagnetic field. The hertz potentials also satisfy the homogeneous vector Helmholtz equation in free space. The advantage of the hertz potentials is that they display much higher symmetry than the conventional vector and scalar potentials. Furthermore, they may be written in a form that displays the paraxial approximation of quasioptics directly. 

Fig. 5 Variations of the interaction energy between particles versus the center-to-center distance, (a) Hertzian potentials for particles with bulk elasticity the dashed line represents the usual Hertz potential (1), and the solid line the generalized Hertzian potential (2). (b) Potentials for emulsions with surface elasticity the dashed line denotes the approximate solution for small compression ratios (3), and the solid line the general solution (4). (c) Ultrasoft potentials for star polymers (where a 1.3Rg, with Rq being the radius of gyration of the star, following [123]) (5) dashed line f = 256 solid line f = 128 dashed and dotted line f = 64. (d) Hard-sphere potential...

Let a CNT bundle contains N infinitely long metallic CNTs, closely packed together, with surface conductivity aQ. The bundle radius Rb is much less than the wavelength A. Since the incident field is almost homogeneous over the bundle cross-section, a symmetrical surface wave is excited in the bundle. In order to take into account the symmetrical local field distribution inside and outside the bundle we model one as a system of n coaxial thin-walled cylinders with the radii R, (l = l,2.,.n, Rb = Rn> Rn x> > Rx) and the surface conductivity cr, /(2 R,), where cr, is equal to the sum of linear conductivities of CNTs placed between the surfaces of cylinders with radii R, and R,, . Boundary conditions for electric Hertz potential on the surface of I -th cylinder in the cylindrical coordinate system (p, q>, z) is as follows [2] ... 

---