38 research outputs found
Generalized second-order partial derivatives of 1/r
The generalized second-order partial derivatives of 1/r, where r is the
radial distance in 3D, are obtained using a result of the potential theory of
classical analysis. Some non-spherical regularization alternatives to the
standard spherical-regularization expression for the derivatives are derived.
The utility of a spheroidal-regularization expression is illustrated on an
example from classical electrodynamics.Comment: 12 pages; as accepted for publication by European Journal of Physic
On the electrodynamics of moving bodies at low velocities
We discuss the seminal article in which Le Bellac and Levy-Leblond have
identified two Galilean limits of electromagnetism, and its modern
implications. We use their results to point out some confusion in the
literature and in the teaching of special relativity and electromagnetism. For
instance, it is not widely recognized that there exist two well defined
non-relativistic limits, so that researchers and teachers are likely to utilize
an incoherent mixture of both. Recent works have shed a new light on the choice
of gauge conditions in classical electromagnetism. We retrieve Le
Bellac-Levy-Leblond's results by examining orders of magnitudes, and then with
a Lorentz-like manifestly covariant approach to Galilean covariance based on a
5-dimensional Minkowski manifold. We emphasize the Riemann-Lorenz approach
based on the vector and scalar potentials as opposed to the Heaviside-Hertz
formulation in terms of electromagnetic fields. We discuss various applications
and experiments, such as in magnetohydrodynamics and electrohydrodynamics,
quantum mechanics, superconductivity, continuous media, etc. Much of the
current technology where waves are not taken into account, is actually based on
Galilean electromagnetism
On the exact electric and magnetic fields of an electric dipole
We derive from Jefimenko's equations a multipole expansion in order to obtain
the exact expressions for the electric and magnetic fields of an electric
dipole with an arbitrary time dependence. A few comments are also made about
the usual expositions found in most common undergraduate and graduate textbooks
as well as in the literature on this topic
How to be causal: time, spacetime, and spectra
I explain a simple definition of causality in widespread use, and indicate
how it links to the Kramers Kronig relations. The specification of causality in
terms of temporal differential eqations then shows us the way to write down
dynamical models so that their causal nature /in the sense used here/ should be
obvious to all. To extend existing treatments of causality that work only in
the frequency domain, I derive a reformulation of the long-standing Kramers
Kronig relations applicable not only to just temporal causality, but also to
spacetime "light-cone" causality based on signals carried by waves. I also
apply this causal reasoning to Maxwell's equations, which is an instructive
example since their casual properties are sometimes debated.Comment: v4 - add Appdx A, "discrete" picture (not in EJP); v5 - add Appdx B,
cause classification/frames (not in EJP); v7 - unusual model case; v8 add
reference
Axiomatic geometric formulation of electromagnetism with only one axiom: the field equation for the bivector field F with an explanation of the Trouton-Noble experiment
In this paper we present an axiomatic, geometric, formulation of
electromagnetism with only one axiom: the field equation for the Faraday
bivector field F. This formulation with F field is a self-contained, complete
and consistent formulation that dispenses with either electric and magnetic
fields or the electromagnetic potentials. All physical quantities are defined
without reference frames, the absolute quantities, i.e., they are geometric
four dimensional (4D) quantities or, when some basis is introduced, every
quantity is represented as a 4D coordinate-based geometric quantity comprising
both components and a basis. The new observer independent expressions for the
stress-energy vector T(n)(1-vector), the energy density U (scalar), the
Poynting vector S and the momentum density g (1-vectors), the angular momentum
density M (bivector) and the Lorentz force K (1-vector) are directly derived
from the field equation for F. The local conservation laws are also directly
derived from that field equation. The 1-vector Lagrangian with the F field as a
4D absolute quantity is presented; the interaction term is written in terms of
F and not, as usual, in terms of A. It is shown that this geometric formulation
is in a full agreement with the Trouton-Noble experiment.Comment: 32 pages, LaTex, this changed version will be published in Found.
Phys. Let
Charges and fields in a current-carrying wire
Charges and fields in a straight, infinite, cylindrical wire carrying a
steady current are determined in the rest frames of ions and electrons,
starting from the standard assumption that the net charge per unit length is
zero in the lattice frame and taking into account a self-induced pinch effect.
The analysis presented illustrates the mutual consistency of classical
electromagnetism and Special Relativity. Some consequences of the assumption
that the net charge per unit length is zero in the electrons frame are also
briefly discussed
On the gravitodynamics of moving bodies
In the present work we propose a generalization of Newton's gravitational
theory from the original works of Heaviside and Sciama, that takes into account
both approaches, and accomplishes the same result in a simpler way than the
standard cosmological approach. The established formulation describes the local
gravitational field related to the observables and effectively implements the
Mach's principle in a quantitative form that retakes Dirac's large number
hypothesis. As a consequence of the equivalence principle and the application
of this formulation to the observable universe, we obtain, as an immediate
result, a value of Omega = 2. We construct a dynamic model for a galaxy without
dark matter, which fits well with recent observational data, in terms of a
variable effective inertial mass that reflects the present dynamic state of the
universe and that replicates from first principles, the phenomenology proposed
in MOND. The remarkable aspect of these results is the connection of the effect
dubbed dark matter with the dark energy field, which makes it possible for us
to interpret it as longitudinal gravitational waves.Comment: 18 pages, 4 figures. Final version: almost identical to the reference
journal; Cent. Eur. J. Phys. 201