Green function for the Grad-Shafranov operator

The Grad-Shafranov equation, often written in cylindrical coordinates, is an elliptic partial differential equation in two dimensions. It describes magnetohydrodynamic equilibria in axisymmetric toroidal plasmas, such as tokamaks, and yields the poloidal magnetic flux function, which is related to the azimuthal component of the vector potential for the magnetic field produced by a circular (toroidal) current density. The Green function for the differential operator can be obtained from the vector potential for the magnetic field of a circular current loop, which is a typical problem in magnetostatics. The purpose of the paper is to collect results scattered in electrodynamics and plasma physics textbooks for the benefit of students in the field, as well as attracting the attention of a wider audience, in the context of electrodynamics and partial differential equations.

An Existence Result for a Class of Magnetic Problems in Exterior Domains

In this paper we deal with the existence of solutions for the following class of magnetic semilinear Schrödinger equation $$\begin{aligned} (P) \qquad \qquad \left\{ \begin{aligned}&(-i\nabla + A(x))^2u +u = |u|^{p-2}u,\;\;\text{ in }\;\;\Omega ,\\&u=0\;\;\text{ on }\;\;\partial \Omega , \end{aligned} \right. \end{aligned}$$ ( P ) ( - i ∇ + A ( x ) ) 2 u + u = | u | p - 2 u , in Ω , u = 0 on ∂ Ω , where $$N \ge 3$$ N ≥ 3 , $$\Omega \subset {\mathbb {R}}^N$$ Ω ⊂ R N is an exterior domain, $$p\in (2, 2^*)$$ p ∈ ( 2 , 2 ∗ ) with $$2^*=\frac{2N}{N-2}$$ 2 ∗ = 2 N N - 2 , and $$A: {\mathbb {R}}^N\rightarrow {\mathbb {R}}^N$$ A : R N → R N is a continuous vector potential verifying $$A(x) \rightarrow 0\;\;\text{ as }\;\;|x|\rightarrow \infty .$$ A ( x ) → 0 as | x | → ∞ .

Calculation of the Eddy Current Losses in a Laminated Open-Type Transformer Core Based on the A→,T→−A→ Formulation

Losses due to eddy currents in an open-type transformer core are significantly reduced by the lamination of the transformer core. In order to further reduce the eddy current losses, the open-type core often has a multi-part structure, i.e., it is composed of several more slender cores. The complete homogenization of such a core is not possible when an A→,V−A→ formulation is used, where A and V represent the magnetic vector potential and electric scalar potential, respectively. On the other hand, an A→,T→−A→ formulation, where T represents the electric vector potential, enables the complete homogenization of the general open-type core, but the simulation converges poorly due to the large number of degrees of freedom. By eliminating the redundant degrees of freedom, the convergence rate is significantly improved, and is at least twice as good as the convergence rate of the simulation based on the A→,V−A→ formulation. In this paper, a method for the calculation of the eddy current losses in an open-type core based on the A→,T→−A→ formulation with the elimination of redundant degrees of freedom is presented. The method is validated by comparison with a brute force simulation based on the A→,V−A→ formulation, and the efficiency of the method is determined by comparison with the standard homogenization method based on the A→,V−A→ formulation.

Vector Potential, Magnetic Field, Mutual Inductance, Magnetic Force, Torque and Stiffness Calculation between Current-Carrying Arc Segments with Inclined Axes in Air

In this paper, the improved and the new analytical and semi-analytical expressions for calculating the magnetic vector potential, magnetic field, magnetic force, mutual inductance, torque, and stiffness between two inclined current-carrying arc segments in air are given. The expressions are obtained either in the analytical form over the incomplete elliptic integrals of the first and the second kind or by the single numerical integration of some elliptical integrals of the first and the second kind. The validity of the presented formulas is proved from the particular cases when the inclined circular loops are addressed. We mention that all formulas are obtained by the integral approach, except the stiffness, which is found by the derivative of the magnetic force. The novelty of this paper is the treatment of the inclined circular carting-current arc segments for which the calculations of the previously mentioned electromagnetic quantities are given.

Quantum Vacuum Gravitation Matter-Antimatter Antigravity

Without stating any assumptions or making postulates we show that the electromagnetic quantum vacuum plays a primary role in quantum electrodynamics, particle physics, gravitation and cosmology. Photons are local oscillations of the electromagnetic quantum vacuum field guided by a non-local vector potential wave function. The electron-positron elementary charge emerges naturally from the vacuum field and is related to the photon vector potential. We establish the masse-charge equivalence relation showing that the masses of all particles (leptons, mesons, baryons) and antiparticles have electromagnetic origin. In addition, we deduce that the gravitational constant G is an intrinsic property of the electromagnetic quantum vacuum putting in evidence the electromagnetic nature of gravity. We show that Newton’s gravitational law is equivalent to Coulomb’s electrostatic law. Furthermore, we draw that G is the same for matter and antimatter but gravitational forces could be repulsive between particles and antiparticles because their masses bear naturally opposite signs. The electromagnetic quantum vacuum field may be the natural link between particle physics, quantum electrodynamics, gravitation and cosmology constituting a basic step towards a unified field theory.

A Theory for Electromagnetic Radiation and Coupling

A theory for analyzing the radiative and reactive electromagnetic energies of a radiator in vacuum is presented. In vacuum, the radiative electromagnetic energies will depart from their sources and travel to infinity, generating a power flux in the space. However, the reactive electromagnetic energies are bounded to their sources. They appear and disappear almost in the same time with their sources, and their fluctuation also causes a power flux in the space. In the proposed theory, the reactive electromagnetic energies of a radiator are defined by postulating that they have properties similar to the self-energies in the charged particle theory. More importantly, in addition to a main term of source-potential products, the reactive energies contain a special energy term which will last to exist a short time after the sources disappear. This oscillating energy is related to the electric displacement and the vector potential, and seems to be responsible for energy exchanging between the reactive energy and the radiative energy in the radiation process, performing like the Schott energy term. As the Poynting vector describes the total power flux density related to the total electromagnetic energy, it should include the contributions of the propagation of the radiative energies and the fluctuation of the reactive energies. The mutual electromagnetic couplings between two radiators are also defined in a similar way in which the vector potential plays a central role. The reactive electromagnetic energies can be evaluated with explicit expressions in time domain and frequency domain. The theory is verified with the Hertzian dipole and numerical examples.

A Theory for Electromagnetic Radiation and Coupling

A theory for analyzing the radiative and reactive electromagnetic energies of a radiator in vacuum is presented. In vacuum, the radiative electromagnetic energies will depart from their sources and travel to infinity, generating a power flux in the space. However, the reactive electromagnetic energies are bounded to their sources. They appear and disappear almost in the same time with their sources, and their fluctuation also causes a power flux in the space. In the proposed theory, the reactive electromagnetic energies of a radiator are defined by postulating that they have properties similar to the self-energies in the charged particle theory. More importantly, in addition to a main term of source-potential products, the reactive energies contain a special energy term which will last to exist a short time after the sources disappear. This oscillating energy is related to the electric displacement and the vector potential, and seems to be responsible for energy exchanging between the reactive energy and the radiative energy in the radiation process, performing like the Schott energy term. As the Poynting vector describes the total power flux density related to the total electromagnetic energy, it should include the contributions of the propagation of the radiative energies and the fluctuation of the reactive energies. The mutual electromagnetic couplings between two radiators are also defined in a similar way in which the vector potential plays a central role. The reactive electromagnetic energies can be evaluated with explicit expressions in time domain and frequency domain. The theory is verified with the Hertzian dipole and numerical examples.

A Theory for Electromagnetic Radiation and Coupling

A theory for analyzing the radiative and reactive electromagnetic energies of a radiator in vacuum is presented. In vacuum, the radiative electromagnetic energies will depart from their sources and travel to infinity, generating a power flux in the space. However, the reactive electromagnetic energies are bounded to their sources. They appear and disappear almost in the same time with their sources, and their fluctuation also causes a power flux in the space. In the proposed theory, the reactive electromagnetic energies of a radiator are defined by postulating that they have properties similar to the self-energies in the charged particle theory. More importantly, in addition to a main term of source-potential products, the reactive energies contain a special energy term which will last to exist a short time after the sources disappear. This oscillating energy is related to the electric displacement and the vector potential, and seems to be responsible for energy exchanging between the reactive energy and the radiative energy in the radiation process, performing like the Schott energy term. As the Poynting vector describes the total power flux density related to the total electromagnetic energy, it should include the contributions of the propagation of the radiative energies and the fluctuation of the reactive energies. The mutual electromagnetic couplings between two radiators are also defined in a similar way in which the vector potential plays a central role. The reactive electromagnetic energies can be evaluated with explicit expressions in time domain and frequency domain. The theory is verified with the Hertzian dipole and numerical examples.

Vector Potential, Magnetic Field, Mutual Inductance, Magnetic Force, Torque and Stiffness Calculation between Current-Carrying Arc Segments with Inclined Axes in Air

In this paper we give the improved and new analytical and semi-analytical expression for calcu-lating the magnetic vector potential, magnetic field, magnetic force, mutual inductance, torque, and stiffness between two inclined current-carrying arc segments in air. The expressions are ob-tained either in the analytical form over the incomplete elliptic integrals of the first and the sec-ond time or by the single numerical integration of some elliptical integrals of the first and the second kind. The validity of the presented formulas is proved from the special cases when the inclined circular loops are treated. We mention that all formulas are obtain by the integral ap-proach except the stiffness which is found by the derivative of the magnetic force.