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Retarded Potential
In electrodynamics, the retarded potentials are the electromagnetic potentials for the electromagnetic field generated by time-varying electric current or charge distributions in the past. The fields propagate at the speed of light ''c'', so the delay of the fields connecting cause and effect at earlier and later times is an important factor: the signal takes a finite time to propagate from a point in the charge or current distribution (the point of cause) to another point in space (where the effect is measured), see figure below. In the Lorenz gauge The starting point is Maxwell's equations in the potential formulation using the Lorenz gauge: : \Box \varphi = \dfrac \,,\quad \Box \mathbf = \mu_0\mathbf where φ(r, ''t'') is the electric potential and A(r, ''t'') is the magnetic vector potential, for an arbitrary source of charge density ρ(r, ''t'') and current density J(r, ''t''), and \Box is the D'Alembert operator. Solving these gives the retarded potentials below (al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electrodynamics
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of atoms and molecules. Electromagnetism can be thought of as a combination of electricity and magnetism, two distinct but closely intertwined phenomena. In essence, electric forces occur between any two charged particles, causing an attraction between particles with opposite charges and repulsion between particles with the same charge, while magnetism is an interaction that occurs exclusively between ''moving'' charged particles. These two effects combine to create electromagnetic fields in the vicinity of charge particles, which can exert influence on other particles via the Lorentz force. At high energy, the weak force and electromagnetic force are unified as a single electroweak force. The electromagnetic force is responsible for many o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Position Vector
In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents the position of a point ''P'' in space in relation to an arbitrary reference origin ''O''. Usually denoted x, r, or s, it corresponds to the straight line segment from ''O'' to ''P''. In other words, it is the displacement or translation that maps the origin to ''P'': :\mathbf=\overrightarrow The term "position vector" is used mostly in the fields of differential geometry, mechanics and occasionally vector calculus. Frequently this is used in two-dimensional or three-dimensional space, but can be easily generalized to Euclidean spaces and affine spaces of any dimension.Keller, F. J, Gettys, W. E. et al. (1993), p 28–29 Relative position The relative position of a point ''Q'' with respect to point ''P'' is the Euclidean vector resulting from the subtraction of the two absolute position vectors (each with respect to the origin): :\Delta \mathb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inversion In A Point
In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invariant under point reflection through its center, it is said to possess central symmetry or to be centrally symmetric. Point reflection can be classified as an affine transformation. Namely, it is an isometric involutive affine transformation, which has exactly one fixed point, which is the point of inversion. It is equivalent to a homothetic transformation with scale factor equal to −1. The point of inversion is also called homothetic center. Terminology The term ''reflection'' is loose, and considered by some an abuse of language, with ''inversion'' preferred; however, ''point reflection'' is widely used. Such maps are involutions, meaning that they have order 2 – they are their own inverse: applying them twice yields the ident ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wheeler–Feynman Absorber Theory
The Wheeler–Feynman absorber theory (also called the Wheeler–Feynman time-symmetric theory), named after its originators, the physicists Richard Feynman and John Archibald Wheeler, is an interpretation of electrodynamics derived from the assumption that the solutions of the electromagnetic field equations must be invariant under time-reversal transformation, as are the field equations themselves. Indeed, there is no apparent reason for the time-reversal symmetry breaking, which singles out a preferential time direction and thus makes a distinction between past and future. A time-reversal invariant theory is more logical and elegant. Another key principle, resulting from this interpretation and reminiscent of Mach's principle due to Hugo Tetrode, is that elementary particles are not self-interacting. This immediately removes the problem of self-energies. T-symmetry and causality The requirement of time-reversal symmetry, in general, is difficult to reconcile with the princip ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Liénard–Wiechert Potential
The Liénard–Wiechert potentials describe the classical electromagnetic effect of a moving electric point charge in terms of a vector potential and a scalar potential in the Lorenz gauge. Stemming directly from Maxwell's equations, these describe the complete, relativistically correct, time-varying electromagnetic field for a point charge in arbitrary motion, but are not corrected for quantum mechanical effects. Electromagnetic radiation in the form of waves can be obtained from these potentials. These expressions were developed in part by Alfred-Marie Liénard in 1898 and independently by Emil Wiechert in 1900. Equations Definition of Liénard–Wiechert potentials The retarded time is defined, in the context of distributions of charges and currents, as :t_r(\mathbf,\mathbf, t) = t - \frac, \mathbf - \mathbf_s, , where \mathbf is the observation point, and \mathbf_s is the observed point subject to the variations of source charges and currents. For a moving point ch ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic Coordinate Condition
The harmonic coordinate condition is one of several coordinate conditions in general relativity, which make it possible to solve the Einstein field equations. A coordinate system is said to satisfy the harmonic coordinate condition if each of the coordinate functions ''x''α (regarded as scalar fields) satisfies d'Alembert's equation. The parallel notion of a harmonic coordinate system in Riemannian geometry is a coordinate system whose coordinate functions satisfy Laplace's equation. Since d'Alembert's equation is the generalization of Laplace's equation to space-time, its solutions are also called "harmonic". Motivation The laws of physics can be expressed in a generally invariant form. In other words, the real world does not care about our coordinate systems. However, for us to be able to solve the equations, we must fix upon a particular coordinate system. A coordinate condition selects one (or a smaller set of) such coordinate system(s). The Cartesian coordinates used in spe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linearized Gravity
In the theory of general relativity, linearized gravity is the application of perturbation theory to the metric tensor that describes the geometry of spacetime. As a consequence, linearized gravity is an effective method for modeling the effects of gravity when the gravitational field is weak. The usage of linearized gravity is integral to the study of gravitational waves and weak-field gravitational lensing. Weak-field approximation The Einstein field equation (EFE) describing the geometry of spacetime is given as (using natural units) :R_ - \fracRg_ = 8\pi GT_ where R_ is the Ricci tensor, R is the Ricci scalar, T_ is the energy–momentum tensor, and g_ is the spacetime metric tensor that represent the solutions of the equation. Although succinct when written out using Einstein notation, hidden within the Ricci tensor and Ricci scalar are exceptionally nonlinear dependencies on the metric which render the prospect of finding exact solutions impractical in most systems. H ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coulomb Gauge
In the physics of gauge theories, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. By definition, a gauge theory represents each physically distinct configuration of the system as an equivalence class of detailed local field configurations. Any two detailed configurations in the same equivalence class are related by a gauge transformation, equivalent to a shear along unphysical axes in configuration space. Most of the quantitative physical predictions of a gauge theory can only be obtained under a coherent prescription for suppressing or ignoring these unphysical degrees of freedom. Although the unphysical axes in the space of detailed configurations are a fundamental property of the physical model, there is no special set of directions "perpendicular" to them. Hence there is an enormous amount of freedom involved in taking a "cross section" representing each physical configuration b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poisson's Equation
Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with the potential field known, one can then calculate electrostatic or gravitational (force) field. It is a generalization of Laplace's equation, which is also frequently seen in physics. The equation is named after French mathematician and physicist Siméon Denis Poisson. Statement of the equation Poisson's equation is \Delta\varphi = f where \Delta is the Laplace operator, and f and \varphi are real or complex-valued functions on a manifold. Usually, f is given and \varphi is sought. When the manifold is Euclidean space, the Laplace operator is often denoted as and so Poisson's equation is frequently written as \nabla^2 \varphi = f. In three-dimensional Cartesian coordinates, it takes the form \left( \frac + \frac + \frac \right)\varp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplacian
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the nabla operator), or \Delta. In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems, such as cylindrical and spherical coordinates, the Laplacian also has a useful form. Informally, the Laplacian of a function at a point measures by how much the average value of over small spheres or balls centered at deviates from . The Laplace operator is named after the French mathematician Pierre-Simon de Laplace (1749–1827), who first applied the operator to the study of celestial mechanics: the Laplacian of the gravitational potential due to a given mass density distribution is a constant multiple of that den ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magnetostatic
Magnetostatics is the study of magnetic fields in systems where the currents are steady (not changing with time). It is the magnetic analogue of electrostatics, where the charges are stationary. The magnetization need not be static; the equations of magnetostatics can be used to predict fast magnetic switching events that occur on time scales of nanoseconds or less. Magnetostatics is even a good approximation when the currents are not static – as long as the currents do not alternate rapidly. Magnetostatics is widely used in applications of micromagnetics such as models of magnetic storage devices as in computer memory. Applications Magnetostatics as a special case of Maxwell's equations Starting from Maxwell's equations and assuming that charges are either fixed or move as a steady current \mathbf, the equations separate into two equations for the electric field (see electrostatics) and two for the magnetic field. The fields are independent of time and each other. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electrostatic
Electrostatics is a branch of physics that studies electric charges at rest ( static electricity). Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for amber, (), was thus the source of the word 'electricity'. Electrostatic phenomena arise from the forces that electric charges exert on each other. Such forces are described by Coulomb's law. Even though electrostatically induced forces seem to be rather weak, some electrostatic forces are relatively large. The force between an electron and a proton, which together make up a hydrogen atom, is about 36 orders of magnitude stronger than the gravitational force acting between them. There are many examples of electrostatic phenomena, from those as simple as the attraction of plastic wrap to one's hand after it is removed from a package, to the apparently spontaneous explosion of grain silos, the damage of electronic components during manufa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
