The electromagnetic potentials

2018 ◽  
Author(s):  
J. Pierrus
Keyword(s):  
Electromagnetic Radiation ◽  
Gauge Transformation ◽  
Vector Potential ◽  
Point Charge ◽  
Wave Equations ◽  
Multipole Expansion ◽  
Inhomogeneous Wave ◽  
Electromagnetic Potentials ◽  
Starting Point

This chapter expresses the fields E ( r, t) and B ( r, t) in terms of the electromagnetic potentials Ф( r, t) and A ( r, t), and shows that these potentials are defined only up to a gauge transformation. This leads the reader naturally to the Coulomb and Lorenz gauges which are usually encountered in textbooks. The inhomogeneous wave equations whose solutions are the retarded electromagnetic potentials are also considered, as well as the Lienard–Wiechert potentials for an arbitrarily moving point charge. A few questions are included in which the Lagrangian and Hamiltonian of a point charge are expressed in terms of Ф and A. The chapter concludes by deriving a multipole expansion for the dynamic vector potential which will provide the starting point in our treatment of electromagnetic radiation later on in Chapter 11

Electromagnetic radiation

2018 ◽  
Author(s):  
J. Pierrus
Keyword(s):  
Electromagnetic Radiation ◽  
Magnetic Dipole ◽  
Antenna Arrays ◽  
Electric Dipole ◽  
Free Electron ◽  
Vector Potential ◽  
Electric Quadrupole ◽  
Multipole Expansion ◽  
Multipole Moments

This chapter begins by expressing the multipole expansion of the dynamic vector potential A ( r, t) in terms of electric and magnetic multipole moments. Differentiation of A ( r, t) leads directly to the fields E ( r, t) and B ( r, t), which have a component transporting energy away from the sources to infinity. This component is called electromagnetic radiation and it arises only when electric charges experience an acceleration. A range of questions deal with the various types of radiation, including electric dipole and magnetic dipole–electric quadrupole. Larmor’s formula is applied in both its non-relativistic and relativistic forms. Also considered are some applications involving antennas, antenna arrays and the scattering of radiation by a free electron.

Static magnetic fields in vacuum

2018 ◽  
Author(s):  
J. Pierrus
Keyword(s):  
Magnetic Field ◽  
Problem Solving ◽  
Magnetic Fields ◽  
Vector Potential ◽  
Scalar Potential ◽  
Multipole Expansion ◽  
Magnetic Vector Potential ◽  
Magnetic Dipoles ◽  
Static Magnetic Fields ◽  
The Magnetic Field

Wherever possible, an attempt has been made to structure this chapter along similar lines to Chapter 2 (its electrostatic counterpart). Maxwell’s magnetostatic equations are derived from Ampere’s experimental law of force. These results, along with the Biot–Savart law, are then used to determine the magnetic field B arising from various stationary current distributions. The magnetic vector potential A emerges naturally during our discussion, and it features prominently in questions throughout the remainder of this book. Also mentioned is the magnetic scalar potential. Although of lesser theoretical significance than the vector potential, the magnetic scalar potential can sometimes be an effective problem-solving device. Some examples of this are provided. This chapter concludes by making a multipole expansion of A and introducing the magnetic multipole moments of a bounded distribution of stationary currents. Several applications involving magnetic dipoles and magnetic quadrupoles are given.

WAVE AND MAXWELL'S EQUATIONS IN CARNOT GROUPS

2012 ◽  
Vol 14 (05) ◽  
pp. 1250032 ◽  
Author(s):  
BRUNO FRANCHI ◽  
MARIA CARLA TESI
Keyword(s):  
Vector Potential ◽  
Maxwell’S Equations ◽  
Wave Equations ◽  
Maxwell's Equations ◽  
Evolution Equations ◽  
Differential Forms ◽  
Higher Order ◽  
Carnot Groups ◽  
New Class ◽  
Euclidean Setting

In this paper we define Maxwell's equations in the setting of the intrinsic complex of differential forms in Carnot groups introduced by M. Rumin. It turns out that these equations are higher-order equations in the horizontal derivatives. In addition, when looking for a vector potential, we have to deal with a new class of higher-order evolution equations that replace usual wave equations of the Euclidean setting and that are no more hyperbolic. We prove equivalence of these equations with the "geometric equations" defined in the intrinsic complex, as well as existence and properties of solutions.

Time‐domain behavior of wide‐angle one‐way wave equations

Geophysics ◽  
1991 ◽  
Vol 56 (3) ◽  
pp. 382-384
Author(s):  
A. H. Kamel
Keyword(s):  
Wave Equation ◽  
Wave Equations ◽  
Source Function ◽  
Plane Waves ◽  
Wide Angle ◽  
Wave Processes ◽  
Inhomogeneous Wave ◽  
Dirac Delta ◽  
Inhomogeneous Wave Equation ◽  
Standard Tool

The constant‐coefficient inhomogeneous wave equation reads [Formula: see text], Eq. (1) where t is the time; x, z are Cartesian coordinates; c is the sound speed; and δ(.) is the Dirac delta source function located at the origin. The solution to the wave equation could be synthesized in terms of plane waves traveling in all directions. In several applications it is desirable to replace equation (1) by a one‐way wave equation, an equation that allows wave processes in a 180‐degree range of angles only. This idea has become a standard tool in geophysics (Berkhout, 1981; Claerbout, 1985). A “wide‐angle” one‐way wave equation is designed to be accurate over nearly the whole 180‐degree range of permitted angles. Such formulas can be systematically constructed by drawing upon the connection with the mathematical field of approximation theory (Halpern and Trefethen, 1988).

scholarly journals Local Fractional Function Decomposition Method for Solving Inhomogeneous Wave Equations with Local Fractional Derivative

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Shun-Qin Wang ◽  
Yong-Ju Yang ◽  
Hassan Kamil Jassim
Keyword(s):  
Fourier Series ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Decomposition Method ◽  
Wave Equations ◽  
Inhomogeneous Wave ◽  
Function Decomposition ◽  
Fractional Differential ◽  
Fractional Function

We propose the local fractional function decomposition method, which is derived from the coupling method of local fractional Fourier series and Yang-Laplace transform. The forms of solutions for local fractional differential equations are established. Some examples for inhomogeneous wave equations are given to show the accuracy and efficiency of the presented technique.

scholarly journals Gallilei covariant quantum mechanics in electromagnetic fields

1985 ◽  
Vol 8 (3) ◽  
pp. 589-597 ◽  
Author(s):  
H. E. Wilhelm
Keyword(s):  
Quantum Mechanics ◽  
Electromagnetic Fields ◽  
Electromagnetic Waves ◽  
Wave Equations ◽  
Reference Frames ◽  
Schroedinger Equation ◽  
Relativity Principle ◽  
Covariant Quantum ◽  
Electromagnetic Potentials ◽  
Galilean Relativity

A formulation of the quantum mechanics of charged particles in time-dependent electromagnetic fields is presented, in which both the Schroedinger equation and wave equations for the electromagnetic potentials are Galilei covariant, it is shown that the Galilean relativity principle leads to the introduction of the electromagnetic substratum in which the matter and electromagnetic waves propagate. The electromagnetic substratum effects are quantitatively significant for quantum mechanics in reference frames, in which the substratum velocitywis in magnitude comparable with the velocity of lightc. The electromagnetic substratum velocitywoccurs explicitly in the wave equations for the electromagnetic potentials but not in the Schroedinger equation.

scholarly journals The Field Of A Step–Like Accelerated Point Charge: Liénard–Wiechert Potentials And Čerenkov Shock Waves Revisited

2015 ◽  
Vol 66 (6) ◽  
pp. 317-322
Author(s):  
L’ubomír Šumichrast
Keyword(s):  
Electromagnetic Field ◽  
Shock Waves ◽  
Vector Potential ◽  
Point Charge ◽  
Solvable Problems

Abstract Scalar and vector potential as well as the electromagnetic field of a moving point charge is a nice example how the application of symbolic functions (distributions) in electromagnetics makes it easier to obtain and interpret solutions of otherwise hardly solvable problems.

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