In mathematical physics, a multipole expansion is a series expansion of the effect produced by localized source terms in a given partial differential equation, most commonly Poisson's equation (for electrostatics and gravity), in spherical coordinates or cylindrical coordinates. Typically, the expansion is in terms of spherical harmonics or related angular functions multiplied by an appropriate radial dependence. In order for the expansion to be convergent and useful, one relies on the property that the higher-order terms in the expansion decay increasingly quickly far away from the sources. In this case, the leading terms in a multipole expansion are generally the most significant, and the low-order behavior of the system at large distances can be approximated by the first few terms of the expansion, which are usually much easier to compute than the general solution. An expert is someone widely recognized as a reliable source of knowledge, technique, or skill whose judgment is accorded authority and status by the public or their peers. ... Mathematical physics is the scientific discipline concerned with the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories1. ... As the degree of the taylor series rises, it approaches the correct function. ... In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ... In mathematics, Poissons equation is a partial differential equation with broad utility in electrostatics, mechanical engineering and theoretical physics. ... This article describes some of the common coordinate systems that appear in elementary mathematics. ... This article describes some of the common coordinate systems that appear in elementary mathematics. ... In mathematics, the spherical harmonics are an orthogonal set of solutions to Laplaces equation represented in a system of spherical coordinates. ... For a discussion of convergence and convergent series, see limit (mathematics). ...

Multipole expansions are most widely used in problems involving gravitational fields of systems of masses, electric and magnetic fields of charge and current distributions, and the propagation of electromagnetic waves, and are the basis of the fast multipole method for numerical simulation. The gravitational field is a field that causes bodies with mass to attract each other. ... Mass is a property of a physical object that quantifies the amount of matter it contains. ... It has been suggested that Magnetic field density be merged into this article or section. ... Electromagnetic radiation is a propagating wave in space with electric and magnetic components. ...

Mathematically, multipole expansions are related to the underlying rotational symmetry of the physical laws and their associated differential equations. Even though the source terms (such as the masses, charges, or currents) may not be symmetrical, one can expand them in terms of irreducible representations of the rotational symmetry group, which leads to spherical harmonics and related sets of orthogonal functions. One then uses the technique of separation of variables to extract the corresponding solutions for the radial dependencies. Group representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ... The symmetry group of an object (e. ... In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ... In mathematics, separation of variables is any of several methods of solving ordinary and partial differential equations. ...

Multipole expansion for electric potentials

The (scalar) potential at the point x for an arbitrary charge distribution ρ(x) is given by

This can be expanded in (negative) powers of , obtaining (after some work) the multipole expansion

where this integral, like the previous one, is over all of space, P_n is the degree-n Legendre polynomial, and θ is the angle between the vectors x and y. Note: The term Legendre polynomials is sometimes used (wrongly) to indicate the associated Legendre polynomials. ...

The first couple of terms in the expansion are familiar:

where is the unit vector parallel to x. The first term here is the field of a point charge equal to the total charge, located at the origin. The second is the field of an electric dipole; the integral is the dipole moment of the configuration of charges. The Earths magnetic field, which is approximately a dipole. ... This article is about the electromagnetic phenomenon. ...

Higher terms in the expansion include higher powers of 1/|x|, and therefore become less and less important at large distances. Hence the multipole expansion is a practical tool for the approximation of fields; far away from a given configuration of charges, the first few terms are typically dominant.

We may term the charge a "monopole moment"; it is a scalar. The dipole moment is a vector. In general, the order-n term in the sum is 1/|x|ⁿ⁺¹ times the contraction of a certain nth-rank tensor with n copies of ; the tensor is the 2ⁿ-pole moment of the configuration of charges.

The gravitational field is formally identical to the electrical field, so there is also a multipole expansion for gravitational potentials.

Multipole expansion for electric fields

We may take gradients of the expansion above to yield an expansion of the electric or gravitational field.

Multipole expansion for magnetic vector potentials

Suppose we have a current loop with a current I in it. Then the vector potential of the induced magnetic field is

and as before we can expand in negative powers of , obtaining another multipole expansion:

The n = 0 term is always zero, since it equals the integral of a constant function around a closed loop. (This term, if present, would describe magnetic monopoles; if those existed, there would be no such thing as a magnetic field's vector potential.)

The n = 1 term is the dipole term; applying Stokes' theorem we recover its usual form in terms of the area of the loop. The Earths magnetic field, which is approximately a dipole. ... The Stokes theorem in differential geometry is a statement about the integration of differential forms which generalizes several theorems from vector calculus. ...

Multipole expansion for magnetic fields

We may take curls of the expansion above to yield an expansion of the magnetic field. This article is about the cURL command line tool. ...

Multipole expansions in electrodynamics

Although the above multipole expansions apply to electrostatic potentials and related quantities, there are also multipole expansions in the study of electromagnetic waves radiated from localized sources (such as antennas). In this case, expands the radiated field from a localized time-harmonic current source (i.e. with time-dependence exp( − iωt)) in terms of vector spherical harmonics. A complete description can be found in, e.g., Jackson's Classical Electrodynamics.

Applications of the multipole expansion

The fast multipole method [1] of Greengard and Rokhlin is a general technique for accelerating computer simulations of particle dynamics and electrostatics. The idea is to decompose the force on a particle, or the potential at a given point, into two terms: one comes from nearby particles and can be computed quickly because there aren't too many of them, and the other comes from distant particles and can be computed quickly (with known bounds on the error) by aggregating many distant particles and using only the first few terms in a multipole expansion.