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Spherical Multipole Moments
Spherical multipole moments are the coefficients in a series expansion of a potential that varies inversely with the distance R to a source, ''i.e.'', as 1/''R''. 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 R \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Series Expansion
In mathematics, a series expansion is an expansion of a function into a series, or infinite sum. It is a method for calculating a function that cannot be expressed by just elementary operators (addition, subtraction, multiplication and division). The resulting so-called ''series'' often can be limited to a finite number of terms, thus yielding an approximation of the function. The fewer terms of the sequence are used, the simpler this approximation will be. Often, the resulting inaccuracy (i.e., the partial sum of the omitted terms) can be described by an equation involving Big O notation (see also asymptotic expansion). The series expansion on an open interval will also be an approximation for non-analytic functions. Types of series expansions There are several kinds of series expansions, listed below. A ''Taylor series'' is a power series based on a function's derivatives at a single point. More specifically, if a function f: U\to\mathbb is infinitely differentiable around ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical Multipole Angles
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the centre of the sphere, and is the sphere's radius. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians. The sphere is a fundamental object in many fields of mathematics. Spheres and nearly-spherical shapes also appear in nature and industry. Bubbles such as soap bubbles take a spherical shape in equilibrium. The Earth is often approximated as a sphere in geography, and the celestial sphere is an important concept in astronomy. Manufactured items including pressure vessels and most curved mirrors and lenses are based on spheres. Spheres roll smoothly in any direction, so most balls used in sports and toys are spherical, as are ball bearings. Basic terminology As mentioned earlier is the sphere's r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electromagnetism
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] |
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. Cylindr ... [...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. Similarly to Taylor series, multipole expansions are useful because 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 featur ... [...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 nondimensional ... [...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. Similarly to Taylor series, multipole expansions are useful because 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 features ... [...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 gravitational potential or 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. 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''′) and ''r''> is max(''r'', ''r''′). The function Y^m_\ell is a normalized ... [...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 nondimensional ... [...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. Since the spherical harmonics form a complete set of orthogonal functions and thus an orthonormal basis, each 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 central role in the group theoretic discussion of SO(3). Spherical harmonics originate ... [...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. Since the spherical harmonics form a complete set of orthogonal functions and thus an orthonormal basis, each 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 central role in the group theoretic discussion of SO(3). Spherical harmonics originate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
