Dunkl–Maxwell equation and Dunkl-electrostatics in a spherical coordinate

This paper deals with Maxwell equations with Dunkl derivatives. Dunkl-deformed gauge transform is investigated. Dunkl-electrostatics in spherical coordinates is also studied. The multi-pole expansion of potential is obtained for even and odd potential for parity in z-direction. The conducting sphere in a uniform electric field in Dunkl-electrostatics is also discussed.

Negativity of the Casimir Self-Entropy in Spherical Geometries

It has been recognized for some time that, even for perfect conductors, the interaction Casimir entropy, due to quantum/thermal fluctuations, can be negative. This result was not considered problematic because it was thought that the self-entropies of the bodies would cancel this negative interaction entropy, yielding a total entropy that was positive. In fact, this cancellation seems not to occur. The positive self-entropy of a perfectly conducting sphere does indeed just cancel the negative interaction entropy of a system consisting of a perfectly conducting sphere and plate, but a model with weaker coupling in general possesses a regime where negative self-entropy appears. The physical meaning of this surprising result remains obscure. In this paper, we re-examine these issues, using improved physical and mathematical techniques, partly based on the Abel–Plana formula, and present numerical results for arbitrary temperatures and couplings, which exhibit the same remarkable features.

Negativity of the Casimir Self-Entropy in Spherical Geometries

It has been recognized for some time that even for perfect conductors, the interaction Casimir entropy, due to quantum/thermal fluctuations, can be negative. This result was not considered problematic because it was thought that the self-entropies of the bodies would cancel this negative interaction entropy, yielding a total entropy that was positive. In fact, this cancellation seems not to occur. The positive self entropy of a perfectly conducting sphere does indeed just cancel the negative interaction entropy of a system consisting of a perfectly conducting sphere and plate, but a model with weaker coupling in general possesses a regime where negative self-entropy appears. The physical meaning of this surprising result remains obscure. In this paper we re-examine these issues, using improved physical and mathematical techniques, partly based on the Abel-Plana formula, and present numerical results for arbitrary temperatures and couplings, which exhibit the same remarkable features.

Complementary spherical surface screens

Ideally conducting spherical screens (SS) are investigated in this work. Boundary conditions on the surface of the field inside the sphere and outside it give relations connecting the elements of the scattering matrix (SM) of SS. These relations make it possible to express 4 submatrices of the SS SM in terms of one submatrix. The problem of the relationship between the characteristics of additional SS is considered. This problem is an analogue of Babinet's problem on additional flat screens. Combining additional flat screens forms a perfectly conducting plane, and combining additional SS gives a perfectly conducting sphere. When establishing the relationship of the characteristics of additional SS, they are considered, on the one hand, as a system of two bodies. On the other hand, this system forms a perfectly conducting sphere, the SM of which is diagonal. Equating the SM of the system with the SM of an ideally conducting sphere gives the sought relationship between the SMs of additional SS. The results of numerical verification of the relationships for the SM of additional SS are presented. In numerical calculations, the dimensions of the SS SM, which are infinite in the general case, have to be limited, this leads to an error. It is shown that with an increase in the SM dimension, the error decreases, and the elements of the resulting matrix approach those of the SM of an ideally conducting sphere. This confirms the validity of the relations for the SM of additional SS. The interrelation of characteristics of additional SS obtained in this work is analogous to the Babinet principle for additional flat screens. We believe that it will turn out to be just as popular and useful in calculating spherical screens and antennas, as Babinet's principle for flat screens and flat antennas.