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Maxwell’s Equations and Electromagnetic Waves
Published in Martin J. N. Sibley, Introduction to Electromagnetism, 2021
Gauss’ law states that the electrostatic flux flowing through a closed surface is equal to the enclosed charge. We encountered this when we considered electrostatic flux flow in Chapter 2. We can write φ=qor,D×area=qand so,∮surfD⋅ds=q
Time-Varying Fields
Published in Ahmad Shahid Khan, Saurabh Kumar Mukerji, Electromagnetic Fields, 2020
Ahmad Shahid Khan, Saurabh Kumar Mukerji
Gauss’ law, which states that the electric flux through any closed surface is equal to the total charge enclosed, is the first of Maxwell’s equations. It holds for a closed surface of any shape and size, and for any charge distribution inside that surface, moving or stationary. It is more general than Coulomb’s law, which only holds for charges whose accelerations are small or zero. It can be derived from the first two of Maxwell’s equations for static conditions, i.e. when all charges are at rest or moving with uniform velocity. The first two of Maxwell's equations, Equations 6.19 and 15.7, allow estimation of E due to any static charge distribution. Once the field is known the force that this charge distribution exerts on any charge can be estimated. If a charge distribution has a high degree of symmetry, then Gauss’ law, Equation 6.21, the first Maxwell’s equation, alone can be used to determine the magnitude of E. The direction of E may be deduced from the symmetry of situation.
Electrostatic and Thermodynamic Potentials of Electrons in Materials
Published in Juan Bisquert, The Physics of Solar Energy Conversion, 2020
In principle, we neglect the extension of the surface and take the charge sheets as infinitely large. Since d can be a fraction of a nanometer, this is a good approximation provided that we do not move the test charge a macroscopic distance away from the surface. Adopting this restriction, the electric field E is normal to the surface as shown in Figure 2.9a. Gauss law provides the connection of the electric field to the amount of surface charge. We consider a surface that encloses a total quantity of charge Q in the positive sheet. The area of the test surface that cuts the lines of the electric field is A. Therefore, E=σqε
Finite Maxwell field and electric displacement Hamiltonians derived from a current dependent Lagrangian
Published in Molecular Physics, 2018
A uniform electric field cannot be specified in terms of a charge density using Gauss law. Neither is it accounted for by Ewald summation in the standard form (‘tin foil’ boundary conditions). The uniform electric field under Ewald periodic boundary conditions will have to be treated as an additional dynamical degree of freedom E with an equation of motion derived from an extended Lagrangian. This requires defining a velocity for E. The alternative suggested by electrodynamics [28] is to use the time integral A of E as the basic dynamical variable. The sign is for reasons of consistency with the vector potential A(r) of electrodynamics [17,28]. A of Equation (7) is, however, by definition uniform. Hence, which eliminates magnetic induction B. Substituted in Faraday’s law this would imply consistent with Equation (7) with c set to unity. In this quasistatic nonmagnetic limit, the electrodynamical Lagrangian is simplified to jt is the total uniform current density consisting of an external (controlled) current j0 and the internal current j of Equation (4) As we will argue below, rather than corresponding to a flow of explicit external charge, j0 must be regarded as a displacement current.