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Ewald–Oseen Extinction Theorem
In optics, the Ewald–Oseen extinction theorem, sometimes referred to as just the extinction theorem, is a theorem that underlies the common understanding of scattering (as well as refraction, reflection, and diffraction). It is named after Paul Peter Ewald and Carl Wilhelm Oseen, who proved the theorem in crystalline and isotropic media, respectively, in 1916 and 1915. Originally, the theorem applied to scattering by an isotropic dielectric objects in free space. The scope of the theorem was greatly extended to encompass a wide variety of bianisotropic media. Overview An important part of optical physics theory is starting with microscopic physics—the behavior of atoms and electrons—and using it to ''derive'' the familiar, macroscopic, laws of optics. In particular, there is a derivation of how the refractive index works and where it comes from, starting from microscopic physics. The Ewald–Oseen extinction theorem is one part of that derivation (as is the Lorentz–Lorenz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Optics
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviolet, and infrared light. Because light is an electromagnetic wave, other forms of electromagnetic radiation such as X-rays, microwaves, and radio waves exhibit similar properties. Most optical phenomena can be accounted for by using the classical electromagnetic description of light. Complete electromagnetic descriptions of light are, however, often difficult to apply in practice. Practical optics is usually done using simplified models. The most common of these, geometric optics, treats light as a collection of rays that travel in straight lines and bend when they pass through or reflect from surfaces. Physical optics is a more comprehensive model of light, which includes wave effects such as diffraction and interference that cannot be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electric Susceptibility
In electricity (electromagnetism), the electric susceptibility (\chi_; Latin: ''susceptibilis'' "receptive") is a dimensionless proportionality constant that indicates the degree of polarization of a dielectric material in response to an applied electric field. The greater the electric susceptibility, the greater the ability of a material to polarize in response to the field, and thereby reduce the total electric field inside the material(and store energy). It is in this way that the electric susceptibility influences the electric permittivity of the material and thus influences many other phenomena in that medium, from the capacitance of capacitors to the speed of light. Definition for linear dielectrics If a dielectric material is a linear dielectric, then electric susceptibility is defined as the constant of proportionality (which may be a matrix) relating an electric field E to the induced dielectric polarization density P such that \mathbf P =\varepsilon_0 \chi_, wher ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
De Sitter Double Star Experiment
The de Sitter effect was described by Willem de Sitter in 1913 (as well as by Daniel Frost Comstock in 1910) and used to support the special theory of relativity against a competing 1908 emission theory by Walther Ritz that postulated a variable speed of light dependent on the velocity of the emitting object. De Sitter showed what Ritz's theory have predicted that the orbits of binary stars would appear more eccentric than consistent with experiment and with the laws of mechanics, however, the experimental result was negative. This was confirmed by Brecher in 1977 by observing the x-rays spectrum. For other experiments related to special relativity, see tests of special relativity. The effect According to simple emission theory, light thrown off by an object should move at a speed of c with respect to the emitting object. If there are no complicating dragging effects, the light would then be expected to move at this same speed until it eventually reached an observer. For an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Special Relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws of physics are invariant (that is, identical) in all inertial frames of reference (that is, frames of reference with no acceleration). # The speed of light in vacuum is the same for all observers, regardless of the motion of the light source or the observer. Origins and significance Special relativity was originally proposed by Albert Einstein in a paper published on 26 September 1905 titled "On the Electrodynamics of Moving Bodies".Albert Einstein (1905)''Zur Elektrodynamik bewegter Körper'', ''Annalen der Physik'' 17: 891; English translatioOn the Electrodynamics of Moving Bodiesby George Barker Jeffery and Wilfrid Perrett (1923); Another English translation On the Electrodynamics of Moving Bodies by Megh Nad Saha (1920). The incompa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Calculus Identities
The following are important identities involving derivatives and integrals in vector calculus. Operator notation Gradient For a function f(x, y, z) in three-dimensional Cartesian coordinate variables, the gradient is the vector field: \operatorname(f) = \nabla f = \begin \frac,\ \frac,\ \frac \end f = \frac \mathbf + \frac \mathbf + \frac \mathbf where i, j, k are the standard unit vectors for the ''x'', ''y'', ''z''-axes. More generally, for a function of ''n'' variables \psi(x_1, \ldots, x_n), also called a scalar field, the gradient is the vector field: \nabla\psi = \begin\frac, \ldots,\ \frac \end\psi = \frac \mathbf_1 + \dots + \frac\mathbf_n . where \mathbf_ are orthogonal unit vectors in arbitrary directions. As the name implies, the gradient is proportional to and points in the direction of the function's most rapid (positive) change. For a vector field \mathbf = \left(A_1, \ldots, A_n\right) written as a 1 × ''n'' row vector, also called a tenso ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cylindrical Coordinate System
A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis ''(axis L in the image opposite)'', the direction from the axis relative to a chosen reference direction ''(axis A)'', and the distance from a chosen reference plane perpendicular to the axis ''(plane containing the purple section)''. The latter distance is given as a positive or negative number depending on which side of the reference plane faces the point. The ''origin'' of the system is the point where all three coordinates can be given as zero. This is the intersection between the reference plane and the axis. The axis is variously called the ''cylindrical'' or ''longitudinal'' axis, to differentiate it from the ''polar axis'', which is the ray that lies in the reference plane, starting at the origin and pointing in the reference direction. Other directions perpendicular to the longitudinal axis are called ''radial lines''. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 (all ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Green's Function
In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if \operatorname is the linear differential operator, then * the Green's function G is the solution of the equation \operatorname G = \delta, where \delta is Dirac's delta function; * the solution of the initial-value problem \operatorname y = f is the convolution (G \ast f). Through the superposition principle, given a linear ordinary differential equation (ODE), \operatorname y = f, one can first solve \operatorname G = \delta_s, for each , and realizing that, since the source is a sum of delta functions, the solution is a sum of Green's functions as well, by linearity of . Green's functions are named after the British mathematician George Green, who first developed the concept in the 1820s. In the modern study of linear partial differential equations, Green's functions are s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homogeneous Differential Equation
A differential equation can be homogeneous in either of two respects. A first order differential equation is said to be homogeneous if it may be written :f(x,y) \, dy = g(x,y) \, dx, where and are homogeneous functions of the same degree of and . In this case, the change of variable leads to an equation of the form :\frac = h(u) \, du, which is easy to solve by integration of the two members. Otherwise, a differential equation is homogeneous if it is a homogeneous function of the unknown function and its derivatives. In the case of linear differential equations, this means that there are no constant terms. The solutions of any linear ordinary differential equation of any order may be deduced by integration from the solution of the homogeneous equation obtained by removing the constant term. History The term ''homogeneous'' was first applied to differential equations by Johann Bernoulli in section 9 of his 1726 article ''De integraionibus aequationum differentialium'' (On t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hertz Vector
Hertz vectors, or the Hertz vector potentials, are an alternative formulation of the electromagnetic potentials. They are most often introduced in electromagnetic theory textbooks as practice problems for students to solve.E.A. Essex"Hertz vector potentials of electromagnetic theory" ''American Journal of Physics'' 45, 1099 (1977); doi: 10.1119/1.10955 There are multiple cases where they have a practical use, including antennas and waveguides.H. R. L. Lamont, Wave Guides, (Metheun, London, 1963). Though they are sometimes used in such practice problems, they are still rarely mentioned in most electromagnetic theory courses, and when they are they are often not practiced in a manner that demonstrates when they may be useful or provide a simpler method to solving a problem than more commonly practiced methods. Overview Hertz vectors can be advantageous when solving for the electric and magnetic fields in certain scenarios, as they provide an alternative way to define the scalar po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inhomogeneous Electromagnetic Wave Equation
In electromagnetism and applications, an inhomogeneous electromagnetic wave equation, or nonhomogeneous electromagnetic wave equation, is one of a set of wave equations describing the propagation of electromagnetic waves generated by nonzero source charges and currents. The source terms in the wave equations make the partial differential equations ''inhomogeneous'', if the source terms are zero the equations reduce to the homogeneous electromagnetic wave equations. The equations follow from Maxwell's equations. Maxwell's equations For reference, Maxwell's equations are summarized below in SI units and Gaussian units. They govern the electric field E and magnetic field B due to a source charge density ''ρ'' and current density J: : where ''ε''0 is the vacuum permittivity and ''μ''0 is the vacuum permeability. Throughout, the relation :\varepsilon_0\mu_0 = \dfrac is also used. SI units E and B fields Maxwell's equations can directly give inhomogeneous wave equati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Einstein Notation
In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of indexed terms in a formula, thus achieving brevity. As part of mathematics it is a notational subset of Ricci calculus; however, it is often used in physics applications that do not distinguish between tangent and cotangent spaces. It was introduced to physics by Albert Einstein in 1916. Introduction Statement of convention According to this convention, when an index variable appears twice in a single term and is not otherwise defined (see Free and bound variables), it implies summation of that term over all the values of the index. So where the indices can range over the set , : y = \sum_^3 c_i x^i = c_1 x^1 + c_2 x^2 + c_3 x^3 is simplified by the convention to: : y = c_i x^i The upper indices are not exponents but are indices ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
