Electromagnetism and differential geometry

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Differential Geometry ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Exterior Derivative ◽  
Exterior Product ◽  
Exterior Calculus ◽  
Definition Of

This chapter begins by examining p-forms and the exterior product, as well as the dual of a p-form. Meanwhile, the exterior derivative is an operator, denoted d, which acts on a p-form to give a (p + 1)-form. It possesses the following defining properties: if f is a 0-form, df(t) = t f (where t is a vector of Eₙ), which coincides with the definition of differential 1-forms. Moreover, d(α‎ + β‎) = dα‎ + dβ‎, where α‎ and β‎ are forms of the same degree. Moreover, the exterior calculus can be used to obtain a compact and elegant formulation of Maxwell’s equations.

scholarly journals Hydrodynamic construction of the electromagnetic field

2005 ◽  
Vol 461 (2063) ◽  
pp. 3659-3679 ◽  
Author(s):  
Peter Holland
Keyword(s):  
Electromagnetic Field ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Classical Model ◽  
Canonical Quantization ◽  
Flat Space ◽  
Single Orbit ◽  
State Construction ◽  
Lagrangian Coordinate ◽  
Definition Of

We present an alternative Eulerian hydrodynamic model for the electromagnetic field in which the discrete vector indices in Maxwell's equations are replaced by continuous angular freedoms, and develop the corresponding Lagrangian picture in which the fluid particles have rotational and translational freedoms. This enables us to extend to the electromagnetic field the exact method of state construction proposed previously for spin 0 systems, in which the time-dependent wavefunction is computed from a single-valued continuum of deterministic trajectories where two spacetime points are linked by at most a single orbit. The deduction of Maxwell's equations from continuum mechanics is achieved by generalizing the spin 0 theory to a general Riemannian manifold from which the electromagnetic construction is extracted as a special case. In particular, the flat-space Maxwell equations are represented as a curved-space Schrödinger equation for a massive system. The Lorentz covariance of the Eulerian field theory is obtained from the non-covariant Lagrangian-coordinate model as a kind of collective effect. The method makes manifest the electromagnetic analogue of the quantum potential that is tacit in Maxwell's equations. This implies a novel definition of the ‘classical limit’ of Maxwell's equations that differs from geometrical optics. It is shown that Maxwell's equations may be obtained by canonical quantization of the classical model. Using the classical trajectories a novel expression is derived for the propagator of the electromagnetic field in the Eulerian picture. The trajectory and propagator methods of solution are illustrated for the case of a light wave.

scholarly journals Updating Maxwell with Electrons and Charge Aug 2-1.pdf (Version: 1)

2019 ◽  
Author(s):  
Robert Eisenberg
Keyword(s):  
Dielectric Constant ◽  
Chemical Kinetics ◽  
Electromotive Force ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Total Current ◽  
Electric Force ◽  
Field Independent ◽  
Definition Of ◽  
Permanent Field

Maxwell's equations describe the relation of charge and electric force almost perfectly even though electrons and permanent charge were not in his equations, as he wrote them. For Maxwell, all charge depended on the electric field. Charge was induced and polarization was described by a single dielectric constant. Electrons, permanent charge, and polarization are important when matter is involved. Polarization of matter cannot be described by a single dielectric constant ?_(r )with reasonable realism today when applications involve 10^(-10) sec. Only vacuum is well described by a single dielectric constant ?_(0 ). Here, Maxwell's equations are rewritten to include permanent charge and any type of polarization. Rewriting is in one sense petty, and in another sense profound, in either case presumptuous. Either petty or profound, rewriting confirms the legitimacy of electrodynamics that includes permanent charge and realistic polarization. One cannot be sure ahead of time that a theory of electrodynamics without electrons or (permanent, field independent) charge (like Maxwell's equations as he wrote them) would be legitimate or not. After all a theory cannot calculate the fields produced by charges (for example electrons) that are not in the theory at all!After updating,Maxwell's equations seem universal and exact.Polarization must be described explicitly to use Maxwell's equations in applications. Conservation of total current (including ?_0 ?E??t) becomes exact, independent of matter, allowing precise definition of electromotive force EMF in circuits.Kirchhoff's current law becomes as exact as Maxwell's equations themselves.Classical chemical kinetics is seen to need revision to conserve current.

Sumudu Applications to Maxwell's Equations

PIERS Online ◽  
2009 ◽  
Vol 5 (4) ◽  
pp. 355-360 ◽  
Author(s):  
Fethi Bin Muhammad Belgacem
Keyword(s):  
Maxwell’S Equations ◽  
Maxwell's Equations

Maxwell’s Equations versus Newton’s Third Law

2018 ◽  
Author(s):  
Glyn Kennell ◽  
Richard Evitts
Keyword(s):  
Electric Fields ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Simulated Data ◽  
Numerical Data ◽  
Transport Equations ◽  
Concentration Gradients ◽  
Third Law

The presented simulated data compares concentration gradients and electric fields with experimental and numerical data of others. This data is simulated for cases involving liquid junctions and electrolytic transport. The objective of presenting this data is to support a model and theory. This theory demonstrates the incompatibility between conventional electrostatics inherent in Maxwell's equations with conventional transport equations. <br>

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