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Register a new userThe geometrical nature and some properties of the capacitance coefficients based on Laplaces Equation
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Added by
William Javier Herrera
Publication date
2007
fields
Physics
and research's language is
English
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The fact that the capacitance coefficients for a set of conductors are geometrical factors is derived in most electricity and magnetism textbooks. We present an alternative derivation based on Laplaces equation that is accessible for an intermediate course on electricity and magnetism. The properties of Laplaces equation permits to prove many properties of the capacitance matrix. Some examples are given to illustrate the usefulness of such properties.
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In transformation optics, the space transformation is viewed as the deformation of a material. The permittivity and permeability tensors in the transformed space are found to correlate with the deformation field of the material. By solving the Laplaces equation, which describes how the material will deform during a transformation, we can design electromagnetic cloaks with arbitrary shapes if the boundary conditions of the cloak are considered. As examples, the material parameters of the spherical and elliptical cylindrical cloaks are derived based on the analytical solutions of the Laplaces equation. For cloaks with irregular shapes, the material parameters of the transformation medium are determined numerically by solving the Laplaces equation. Full-wave simulations based on the Maxwells equations validate the designed cloaks. The proposed method can be easily extended to design other transformation materials for electromagnetic and acoustic wave phenomena.
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In the first sections of this article, we discuss two variations on Maxwells equations that have been introduced in earlier work--a class of nonlinear Maxwell theories with well-defined Galilean limits (and correspondingly generalized Yang-Mills equations), and a linear modification motivated by the coupling of the electromagnetic potential with a certain nonlinear Schroedinger equation. In the final section, revisiting an old idea of Lorentz, we write Maxwells equations for a theory in which the electrostatic force of repulsion between like charges differs fundamentally in magnitude from the electrostatic force of attraction between unlike charges. We elaborate on Lorentz description by means of electric and magnetic field strengths, whose governing equations separate into two fully relativistic Maxwell systems--one describing ordinary electromagnetism, and the other describing a universally attractive or repulsive long-range force. If such a force cannot be ruled out {it a priori} by known physical principles, its magnitude should be determined or bounded experimentally. Were it to exist, interesting possibilities go beyond Lorentz early conjecture of a relation to (Newtonian) gravity.
I had the marvelous good fortune to be Ken Wilsons graduate student at the Physics Department, Cornell University, from 1972 to 1976. In this article, I present some recollections of how this came about, my interactions with Ken, and Cornell during this period; and acknowledge my debt to Ken, and to John Wilkins and Michael Fisher, who I was privileged to have as my main mentors at Cornell. I end with some thoughts on the challenges of reforming education, a subject that was one of Kens major preoccupations in the second half of his professional life.
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