Showing papers on "Scalar potential published in 1970"

Eigenfunctions of the Curl Operator, Rotationally Invariant Helmholtz Theorem, and Applications to Electromagnetic Theory and Fluid Mechanics

Abstract: In the present paper we introduce eigenfunctions of the curl operator. The expansion of vector fields in terms of these eigenfunctions leads to a decomposition of such fields into three modes, one of which corresponds to an irrotational vector field and two of which correspond to rotational circularly polarized vector fields of opposite signs of polarization. Under a rotation of coordinates, the three modes which are introduced in this fashion remain invariant. Hence we have introduced the Helmholtz decomposition of vector fields in an irreducible, rotationally invariant form.These expansions enable one to handle the curl and divergence operators simply. As illustrations of the use of the curl eigenfunctions, we solve four problems. The first problem that is solved is the initial value problem of electromagnetic theory with given time- and space-dependent sources and currents and we show that the radiation and longitudinal modes uncouple in a very simple way. In the second problem we show how fluid motion...

Potential functions in electromagnetic field problems

Abstract: In two dimensional problems with the current flow in only one direction, the magnetic field can be solved by computing a scalar potential or one component of the vector potential. The general formulation for three dimensional solutions, including nonlinearities, is more complex and requires all three components of the vector potential as well as a scalar potential for the description of the fields. It is the purpose of this paper to present a general systematic formulation at low frequencies in terms of potential functions for three dimensional numerical solutions of the nonlinear electro-magnetic field problems that include either nonlinear magnetic materials or nonlinear electric materials. Special cases of the magnetostatic as well as eddy-current problems are discussed.

Use of Scalar Theory in Optics

Abstract: Justification of the use of scalar theory in optics is discussed. Although it is difficult to relate the scalar quantity V(t) commonly used in the scalar theory with the components of the electric field vector E(t), we show that there exists a simple relationship between the corresponding quantities quadratic in the field variables. Thus a scalar theory based on this identification can be used to obtain correct answers for the measurement of irradiance when the description of polarization is not of primary interest.

Cherenkov Radiation in Anisotropic Media

Abstract: The problem of Cherenkov radiation in anisotropic media is studied in a Lorentz frame in which the charged particle is at rest and the medium is moving with a uniform velocity Both electric and magnetic anisotropy are assumed to be present, but the axes of the permittivity and permeability ellipsoids are taken to be parallel to one another The electromagnetic field generated by the charge is described by two scalar potentials Each of these satisfies a partial differential equation of the fourth order when the velocity vector lies in a principal plane of the ellipsoids The two equations closely resemble one another, and passage from one to the other is possible by means of certain simple symmetry operations The equation for the scalar potential of the electric field is solved by the standard technique of Fourier transformation In evaluating the Fourier integrals, however, it is found necessary to assume that two of the ratios $\epsilon \_{i}$/$\mu \_{i}$ of the principal permittivities and permeabilities are equal With this additional restriction the integrals are evaluated easily by the residue theorem and expressions for the field and the radiated energy are obtained in closed forms

Dynamic Potentials in Gyrotropic Plasma

Abstract: Wave equations are derived for the scalar and vector potentials of guided electromagnetic waves propagating in a laterally bounded magnetized plasma along the axis of the static magnetic field. Solutions are given in the Coulomb and the Lorentz gauge. These solutions are used in a critical examination of the conceptual basis of the quasistatic theory because they allow convenient partitioning of the high‐frequency electric field into a lamellar and a rotational part. In the quasistatic approximation the electric field is considered to be entirely lamellar, i.e., the gradient of a scalar potential that is not consistent with Maxwell's equations. To avoid inconsistencies the problem is formulated so that the dynamic scalar potential replaces the quasistatic potential. The rotational part of the electric field is readily obtained as the time derivative of the vector potential. Exact quantitative comparison of two parts of the electric field is made possible in this way and the relative importance of each part is evaluated for a few typical cases of interest. There exist some waves with dispersion characteristics that can not be obtained with the quasistatic formula although the condition that the phase velocity be smaller than the velocity of light presumed to be necessary for the applicability of the quasistatic approximation is satisfied. In these waves the rotational part of the electric field is comparable or greater than the lamellar part. Conversely, in modes with dispersion characteristics that can be well approximated with the quasistatic formula even when the phase velocity exceeds the velocity of light the lamellar part of the electric field is much larger than the rotational part. It can be inferred that the essential element of the quasistatic theory is the assumption that the high‐frequency electric field is predominantly lamellar rather than the condition that the phase velocity be smaller than the velocity of light and related criteria.

Electric Field Generated by a Transient Current Distribution

Abstract: The retarded Hertz vector, obtained earlier [J. Math. Phys. 11, 9 (1970)], is differentiated to give the scalar potential and the electric‐field vector associated with a general transient source‐current distribution. The scalar and vector quantities are expanded in terms of scalar and vector spherical harmonics. For each mode, characterized by specific values for n and m in the expansions, the scalar potential and electric‐field vector are expressed in the (r, t) plane as two‐dimensional integrals over the causally accessible portion of the source‐current distribution.

Some Remarks on Classical Scalar Radiation from Point Sources

Abstract: The radiation from a point source with proper time‐dependent coupling strength to a real classical scalar field is worked out. The adiabatic switching limit reproduces the results of the physical case of constant coupling for the radiation. The problem of defining a ``radiated'' field is discussed.