The Electric Field in Maxwell's Equations

1978 ◽  
Vol 15 (2) ◽  
pp. 169-171 ◽  
Author(s):  
Z. L. Budrikis
Keyword(s):  
Electric Field ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Coulomb Force ◽  
Maxwell’S Equation ◽  
Ohm’S Law ◽  
Ohm's Law ◽  
Maxwell's Equation

The field E in Maxwell's equation curl E = – δB/δ t is limited to induction and Coulomb force. It does not extend to all phenomena that are included in E of Ohm's law, J = σE. Maxwell's equation would need another term to account for additional vorticity of the E in Ohm's law.

On the Relativistic Invariance of Maxwell's Equation

1999 ◽  
Vol 54 (10-11) ◽  
pp. 637-644 ◽  
Author(s):  
Oleg D. Jefimenko
Keyword(s):  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Common Knowledge ◽  
Physical Significance ◽  
Relativistic Invariance ◽  
Maxwell’S Equation ◽  
Maxwell's Equation

It is common knowledge that Maxwell's electromagnetic equations are invariant under relativistic transformations. However the relativistic invariance of Maxwell's equations has certain heretofore over-looked peculiarities. These peculiarities point out to the need of reexamining the physical significance of some basic electromagnetic formulas and equations.

scholarly journals Gauge independent quantum gravitational corrections to Maxwell’s equation

2021 ◽  
Vol 2021 (10) ◽  
Author(s):  
S. Katuwal ◽  
R. P. Woodard
Keyword(s):  
Field Equation ◽  
Maxwell Equation ◽  
Maxwell’S Equations ◽  
Vacuum Polarization ◽  
Maxwell's Equations ◽  
Flat Space ◽  
Effective Field ◽  
Maxwell’S Equation ◽  
Gauge Dependence ◽  
Maxwell's Equation

Abstract We consider quantum gravitational corrections to Maxwell’s equations on flat space background. Although the vacuum polarization is highly gauge dependent, we explicitly show that this gauge dependence is canceled by contributions from the source which disturbs the effective field and the observer who measures it. Our final result is a gauge independent, real and causal effective field equation that can be used in the same way as the classical Maxwell equation.

scholarly journals Time evolution of electric fields and currents and the generalized Ohm's law

2005 ◽  
Vol 23 (4) ◽  
pp. 1347-1354 ◽  
Author(s):  
V. M. Vasyliūnas
Keyword(s):  
Electric Field ◽  
Electric Current ◽  
Time Evolution ◽  
Time Scales ◽  
Current Density ◽  
Time Derivative ◽  
Electron Plasma ◽  
Displacement Current ◽  
Ohm’S Law ◽  
Ohm's Law

Abstract. Fundamentally, the time derivative of the electric field is given by the displacement-current term in Maxwell's generalization of Ampère's law, and the time derivative of the electric current density is given by the generalized Ohm's law. The latter is derived by summing the accelerations of all the plasma particles and can be written exactly, with no approximations, in a (relatively simple) primitive form containing no other time derivatives. When one is dealing with time scales long compared to the inverse of the electron plasma frequency and spatial scales large compared to the electron inertial length, however, the time derivative of the current density becomes negligible in comparison to the other terms in the generalized Ohm's law, which then becomes the equation that determines the electric field itself. Thus, on all scales larger than those of electron plasma oscillations, neither the time evolution of J nor that of E can be calculated directly. Instead, J is determined by B through Ampère's law and E by plasma dynamics through the generalized Ohm's law. The displacement current may still be non-negligible if the Alfvén speed is comparable to or larger than the speed of light, but it no longer determines the time evolution of E, acting instead to modify J. For theories of substorms, this implies that, on time scales appropriate to substorm expansion, there is no equation from which the time evolution of the current could be calculated, independently of ∇xB. Statements about change (disruption, diversion, wedge formation, etc.) of the electric current are merely descriptions of change in the magnetic field and are not explanations.

scholarly journals OPTIMAL FOCUSING FOR MONOCHROMATIC SCALAR AND ELECTROMAGNETIC WAVES

2011 ◽  
Vol 23 (08) ◽  
pp. 839-863
Author(s):  
JEFFREY RAUCH
Keyword(s):  
Wave Equation ◽  
Electric Field ◽  
Electromagnetic Waves ◽  
Field Energy ◽  
Scalar Case ◽  
Far Field ◽  
Maxwell’S Equation ◽  
Maximum Field ◽  
Optimal Focusing ◽  
Maxwell's Equation

For monochromatic solutions of D'Alembert's wave equation and Maxwell's equations, we obtain sharp bounds on the sup norm as a function of the far field energy. The extremizer in the scalar case is radial. In the case of Maxwell's equation, the electric field maximizing the value at the origin follows longitude lines on the sphere at infinity. In dimension d = 3, the highest electric field for Maxwell's equation is smaller by a factor 2/3 than the highest corresponding scalar waves. The highest electric field densities on the balls BR(0) occur as R → 0. The density dips to half max at R approximately equal to one third the wavelength. For these small R, the extremizing fields are identical to those that attain the maximum field intensity at the origin.

A Decomposition of the Electromagnetic Field — Application to the Darwin Model

1997 ◽  
Vol 07 (08) ◽  
pp. 1085-1120 ◽  
Author(s):  
P. Ciarlet ◽  
E. Sonnendrücker
Keyword(s):  
Electromagnetic Field ◽  
Electric Field ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Computational Cost ◽  
Field Application ◽  
Displacement Current ◽  
Computational Domain ◽  
Numerical Resolution ◽  
Simply Connected

In many cases, the numerical resolution of Maxwell's equations is very expensive in terms of computational cost. The Darwin model, an approximation of Maxwell's equations obtained by neglecting the divergence free part of the displacement current, can be used to compute the solution more economically. However, this model requires the electric field to be decomposed into two parts for which no straightforward boundary conditions can be derived. In this paper, we consider the case of a computational domain which is not simply connected. With the help of a functional framework, a decomposition of the fields is derived. It is then used to characterize mathematically the solutions of the Darwin model on such a domain.

scholarly journals Comparative Analysis of the Various Generalized Ohm's Law Terms in Magnetosheath Turbulence as Observed by Magnetospheric Multiscale

2021 ◽  
Author(s):  
Julia Stawarz ◽  
Lorenzo Matteini ◽  
Tulasi Parashar ◽  
Luca Franci ◽  
Jonathan Eastwood ◽  
...  
Keyword(s):  
Electric Field ◽  
Electric Fields ◽  
Electron Mass ◽  
Turbulent Fluctuation ◽  
Electron Pressure ◽  
Ohm’S Law ◽  
Magnetospheric Multiscale ◽  
Ohm's Law ◽  
Turbulent Plasmas ◽  
The Impact

<p><span>Electric fields (<strong>E</strong>) play a fundamental role in facilitating the exchange of energy between the electromagnetic fields and the changed particles within a plasma. </span>Decomposing <strong>E</strong> into the contributions from the different terms in generalized Ohm's law, therefore, provides key insight into both the nonlinear and dissipative dynamics across the full range of scales within a plasma. Using the unique, high‐resolution, multi‐spacecraft measurements of three intervals in Earth's magnetosheath from the Magnetospheric Multiscale mission, the influence of the magnetohydrodynamic, Hall, electron pressure, and electron inertia terms from Ohm's law, as well as the impact of a finite electron mass, on the turbulent electric field<strong> </strong>spectrum are examined observationally for the first time. The magnetohydrodynamic, Hall, and electron pressure terms are the dominant contributions to <strong>E</strong> over the accessible length scales, which extend to scales smaller than the electron gyroradius at the greatest extent, with the Hall and electron pressure terms dominating at sub‐ion scales. The strength of the non‐ideal electron pressure contribution is stronger than expected from linear kinetic Alfvén waves and a partial anti‐alignment with the Hall electric field is present, linked to the relative importance of electron diamagnetic currents within the turbulence. The relative contributions of linear and nonlinear electric fields scale with the turbulent fluctuation amplitude, with nonlinear contributions playing the dominant role in shaping <strong>E</strong> for the intervals examined in this study. Overall, the sum of the Ohm's law terms and measured <strong>E</strong> agree to within ∼ 20% across the observable scales. The results both confirm a number of general expectations about the behavior of <strong>E</strong> within turbulent plasmas, as well as highlight additional features that may help to disentangle the complex dynamics of turbulent plasmas and should be explored further theoretically.</p>

Solving Maxwell's Equation in Meta-Materials by a CG-DG Method

2016 ◽  
Vol 19 (5) ◽  
pp. 1242-1264 ◽  
Author(s):  
Ziqing Xie ◽  
Jiangxing Wang ◽  
Bo Wang ◽  
Chuanmiao Chen
Keyword(s):  
Error Estimate ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Numerical Results ◽  
Time Dependent ◽  
Spatial Discretization ◽  
Maxwell’S Equation ◽  
Temporal Discretization ◽  
Dg Method ◽  
Theoretical Results

AbstractIn this paper, an approach combining the DG method in space with CG method in time (CG-DG method) is developed to solve time-dependent Maxwell's equations when meta-materials are involved. Both the unconditional L2-stability and error estimate of order are obtained when polynomials of degree at most r is used for the temporal discretization and at most k for the spatial discretization. Numerical results in 3D are given to validate the theoretical results.

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