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Spherical Multipole Moments
In physics, spherical multipole moments are the coefficients in a series expansion of a potential that varies inversely with the distance to a source, ''i.e.'', as Examples of such potentials are the electric potential, the magnetic potential and the gravitational potential. For clarity, we illustrate the expansion for a point charge, then generalize to an arbitrary charge density \rho(\mathbf r'). Through this article, the primed coordinates such as \mathbf r' refer to the position of charge(s), whereas the unprimed coordinates such as \mathbf refer to the point at which the potential is being observed. We also use spherical coordinates throughout, e.g., the vector \mathbf r' has coordinates ( r', \theta', \phi') where r' is the radius, \theta' is the colatitude and \phi' is the azimuthal angle. Spherical multipole moments of a point charge The electric potential due to a point charge located at \mathbf is given by \Phi(\mathbf) = \frac \frac = \frac \frac. where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physics
Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." It is one of the most fundamental scientific disciplines. "Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physics. (...) You will come to see physics as a towering achievement of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Law Of Cosines (spherical)
In spherical trigonometry, the law of cosines (also called the cosine rule for sides) is a theorem relating the sides and angles of spherical triangles, analogous to the ordinary law of cosines from plane trigonometry. Given a unit sphere, a "spherical triangle" on the surface of the sphere is defined by the great circles connecting three points , and on the sphere (shown at right). If the lengths of these three sides are (from to (from to ), and (from to ), and the angle of the corner opposite is , then the (first) spherical law of cosines states:Romuald Ireneus 'Scibor-MarchockiSpherical trigonometry ''Elementary-Geometry Trigonometry'' web page (1997).W. Gellert, S. Gottwald, M. Hellwich, H. Kästner, and H. Küstner, ''The VNR Concise Encyclopedia of Mathematics'', 2nd ed., ch. 12 (Van Nostrand Reinhold: New York, 1989). \cos c = \cos a \cos b + \sin a \sin b \cos C\, Since this is a unit sphere, the lengths , and are simply equal to the angles (in radians) subten ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interactions of atoms and molecules. Electromagnetism can be thought of as a combination of electrostatics and magnetism, which are distinct but closely intertwined phenomena. Electromagnetic forces occur between any two charged particles. Electric forces cause an attraction between particles with opposite charges and repulsion between particles with the same charge, while magnetism is an interaction that occurs between charged particles in relative motion. These two forces are described in terms of electromagnetic fields. Macroscopic charged objects are described in terms of Coulomb's law for electricity and Ampère's force law for magnetism; the Lorentz force describes microscopic charged particles. The electromagnetic force is responsible for ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cylindrical Multipole Moments
Cylindrical multipole moments are the coefficients in a series expansion of a potential that varies logarithmically with the distance to a source, i.e., as \ln \ R. Such potentials arise in the electric potential of long line charges, and the analogous sources for the magnetic potential and gravitational potential. For clarity, we illustrate the expansion for a single line charge, then generalize to an arbitrary distribution of line charges. Through this article, the primed coordinates such as (\rho^, \theta^) refer to the position of the line charge(s), whereas the unprimed coordinates such as (\rho, \theta) refer to the point at which the potential is being observed. We use cylindrical coordinates throughout, e.g., an arbitrary vector \mathbf has coordinates ( \rho, \theta, z) where \rho is the radius from the z axis, \theta is the azimuthal angle and z is the normal Cartesian coordinate. By assumption, the line charges are infinitely long and aligned with the z axis. Cylind ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axial Multipole Moments
Axial multipole moments are a series expansion of the electric potential of a charge distribution localized close to the origin along one Cartesian axis, denoted here as the ''z''-axis. However, the axial multipole expansion can also be applied to any potential or field that varies inversely with the distance to the source, i.e., as \frac. For clarity, we first illustrate the expansion for a single point charge, then generalize to an arbitrary charge density \lambda(z) localized to the ''z''-axis. Axial multipole moments of a point charge The electric potential of a point charge ''q'' located on the ''z''-axis at z=a (Fig. 1) equals \Phi(\mathbf) = \frac \frac = \frac \frac. If the radius ''r'' of the observation point is greater than ''a'', we may factor out \frac and expand the square root in powers of (a/r)<1 using |
Multipole Expansion
A multipole expansion is a mathematical series representing a function that depends on angles—usually the two angles used in the spherical coordinate system (the polar and azimuthal angles) for three-dimensional Euclidean space, \R^3. Multipole expansions are useful because, similar to Taylor series, oftentimes only the first few terms are needed to provide a good approximation of the original function. The function being expanded may be real- or complex-valued and is defined either on \R^3, or less often on \R^n for some other Multipole expansions are used frequently in the study of electromagnetic and gravitational fields, where the fields at distant points are given in terms of sources in a small region. The multipole expansion with angles is often combined with an expansion in radius. Such a combination gives an expansion describing a function throughout three-dimensional space. The multipole expansion is expressed as a sum of terms with progressively finer angular f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solid Harmonics
In physics and mathematics, the solid harmonics are solutions of the Laplace equation in spherical polar coordinates, assumed to be (smooth) functions \mathbb^3 \to \mathbb. There are two kinds: the ''regular solid harmonics'' R^m_\ell(\mathbf), which are well-defined at the origin and the ''irregular solid harmonics'' I^m_(\mathbf), which are singular at the origin. Both sets of functions play an important role in potential theory, and are obtained by rescaling spherical harmonics appropriately: R^m_(\mathbf) \equiv \sqrt\; r^\ell Y^m_(\theta,\varphi) I^m_(\mathbf) \equiv \sqrt \; \frac Derivation, relation to spherical harmonics Introducing , , and for the spherical polar coordinates of the 3-vector , and assuming that \Phi is a (smooth) function \mathbb^3 \to \mathbb, we can write the Laplace equation in the following form \nabla^2\Phi(\mathbf) = \left(\frac \fracr - \frac\right)\Phi(\mathbf) = 0 , \qquad \mathbf \ne \mathbf, where is the square of the angular moment ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multipole Moments
A multipole expansion is a mathematical series representing a function that depends on angles—usually the two angles used in the spherical coordinate system (the polar and azimuthal angles) for three-dimensional Euclidean space, \R^3. Multipole expansions are useful because, similar to Taylor series, oftentimes only the first few terms are needed to provide a good approximation of the original function. The function being expanded may be real- or complex-valued and is defined either on \R^3, or less often on \R^n for some other Multipole expansions are used frequently in the study of electromagnetic and gravitational fields, where the fields at distant points are given in terms of sources in a small region. The multipole expansion with angles is often combined with an expansion in radius. Such a combination gives an expansion describing a function throughout three-dimensional space. The multipole expansion is expressed as a sum of terms with progressively finer angular featu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplace Expansion (potential)
In physics, the Laplace expansion of potentials that are directly proportional to the inverse of the distance (1 / r ), such as Newton's law of universal gravitation#Gravity field, Newton's gravitational potential or Coulomb's law#Table of derived quantities, Coulomb's electrostatic potential, expresses them in terms of the spherical Legendre polynomials. In quantum mechanical calculations on atoms the expansion is used in the evaluation of integrals of the inter-electronic repulsion. Formulation The Laplace expansion is in fact the expansion of the inverse distance between two points. Let the points have position vectors \textbf and \textbf' , then the Laplace expansion is \frac = \sum_^\infty \frac \sum_^ (-1)^m \frac Y^_\ell(\theta, \varphi) Y^m_\ell(\theta', \varphi'). Here \textbf has the spherical polar coordinates (r, \theta, \varphi) and \textbf' has (r', \theta', \varphi') with homogeneous polynomials of degree \ell . Further ''r''< is min(''r'', ''r''′) an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solid Harmonics
In physics and mathematics, the solid harmonics are solutions of the Laplace equation in spherical polar coordinates, assumed to be (smooth) functions \mathbb^3 \to \mathbb. There are two kinds: the ''regular solid harmonics'' R^m_\ell(\mathbf), which are well-defined at the origin and the ''irregular solid harmonics'' I^m_(\mathbf), which are singular at the origin. Both sets of functions play an important role in potential theory, and are obtained by rescaling spherical harmonics appropriately: R^m_(\mathbf) \equiv \sqrt\; r^\ell Y^m_(\theta,\varphi) I^m_(\mathbf) \equiv \sqrt \; \frac Derivation, relation to spherical harmonics Introducing , , and for the spherical polar coordinates of the 3-vector , and assuming that \Phi is a (smooth) function \mathbb^3 \to \mathbb, we can write the Laplace equation in the following form \nabla^2\Phi(\mathbf) = \left(\frac \fracr - \frac\right)\Phi(\mathbf) = 0 , \qquad \mathbf \ne \mathbf, where is the square of the angular moment ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical Harmonics
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. The table of spherical harmonics contains a list of common spherical harmonics. Since the spherical harmonics form a complete set of orthogonal functions and thus an orthonormal basis, every function defined on the surface of a sphere can be written as a sum of these spherical harmonics. This is similar to periodic functions defined on a circle that can be expressed as a sum of circular functions (sines and cosines) via Fourier series. Like the sines and cosines in Fourier series, the spherical harmonics may be organized by (spatial) angular frequency, as seen in the rows of functions in the illustration on the right. Further, spherical harmonics are basis functions for irreducible representations of SO(3), the group of rotations in three dimensions, and thus play a cen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
