Tangential field surfaces

Templating by electric fields equipotential and tangential field surfaces... 

This correspondence between equipotential and tangential field surfaces leads to a simple construction for the likely diffusion trajectories of ions within fast-ion conductors, also called "solid electrolytes". These solids (typically binary salts) are electrically highly conducting (the electrical conductivity of... 

The surface current leads to a build up of charge over one portion of the double layer and a depletion over the remainder. This in turn leads to a back field, and this alters the average tangential field around the particle. 

Figure 4.7 (a) Magnitude of the tangential field spectrum on the particle for different values of the surface conductance parameter X (b) effect of surface conductance on the tangential field argument (°). 

The physical fact that the tangential fields are continuous across the intersecting surface is the key to the structure of the null-field equations, and it is not used explicitly latter on. To simplify the null-field equations (2.158) we use the Stratton-Chu representation theorem for the incident field in the interior and exterior of the closed surface 2c... 

Figure 40 shows a scatterer embedded in a hypothetical closed surface. The surface currents are defined on the closed surface. The equations that connect the surface currents to the tangential fields and govern the radiation from the currents can be found in the popular FDTD texts. 

Equation (1.36) is the surface analogy of the first equation of the attraction field, and it shows that the tangential component of g is a continuous function at the boundary between media with different densities. Next, imagine an elementary cylinder around some point q of this surface, Fig. 1.6d. Then, applying Equation (1.26), we have... 

Equation (1.84) form Dirichlet s boundary value problem, which can be either exterior or internal one. Fig. 1.8a, and it has several important applications in the theory of the gravitational field of the earth. It is worth to notice that in accordance with Equation (1.83) we can say that along any direction tangential to the boundary surface, the component of the field is also known, since = dU/dt. Consequently, the boundary value problem can be written in terms of the field as... 

We see that the gradient of the density and that of the gravitational field are parallel to each other. This means that at each point the field g has a direction along which the maximal rate of a change of density occurs. The same result can be formulated differently. Inasmuch as the gradient of the density is normal to the surfaces where 5 is constant, we conclude that the level surfaces U = constant and 5 — constant have the same shape. For instance, if the density remains constant on the spheroidal surfaces, then the level surfaces of the potential of the gravitational field are also spheroidal. It is obvious that the surface of the fluid Earth is equip-otential otherwise there will be tangential component of the field g, which has to cause a motion of the fluid. But this contradicts the condition of the hydrostatic equilibrium. 

Taking into account the fact that on the surface of the ellipsoid e = eq and d — dg we arrive at the obvious result, namely, at the level surface the tangential component of the gravitational field vanishes, — 0. Next, consider the component Again differentiating Equation (2.153) we obtain... 

All four processes have the same origin, since they are all based on the phenomenon of slip of the hquid along the surface of the other phase when a tangential electric field is present, or conversely, on the phenomenon that an electric field will arise during slip of the liquid. 

---