What is multipole expansion in electrodynamics?
A multipole expansion is a mathematical series representing a function that depends on angles—usually the two angles on a sphere. Multipole expansions are very frequently used in the study of electromagnetic and gravitational fields, where the fields at distant points are given in terms of sources in a small region.
What is the use of multipole expansion?
Multipole expansions are 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.
What is multipole expansion of charge distribution?
In physics, the electric potential Φ, caused by—and outside—a non-central-symmetric charge distribution ρ(r), can be expanded in a series, the multipole expansion of Φ. This expansion is in terms of powers of 1/R, where R is the distance of a field point R to a point inside ρ.
Which polynomial series is multipole expansion?
Legendre polynomials
The natural basis functions for the multipole expansion are the Legendre polynomials, or spherical harmonics which have correspondence relation with the Legendre polynomials.
What does multi pole mean?
: having or involving more than one pole (see pole entry 3) : multipolar sense 1 multipole particles multipole interaction a multipole magnet/connector.
What is Octopole moment?
An extreme generalization (“point octopole”) would be: Eight alternating point charges at the eight corners of a parallelepiped, e.g. of a cube with edge length a. The “octopole moment” of this arrangement would correspond, in the “octopole limit” to a nonzero diagonal tensor of order three.
What is linear quadrupole?
A linear electric quadrupole can be created by superimposing two electric dipoles of opposite orientation so that their positive charges overlap. This case can be treated analytically and gives some insights into the nature of quadrupole fields.
What is dipole and quadrupole?
3.8: Quadrupole Moment While a single charge is a scalar quantity, and a dipole moment is a vector quantity, the quadrupole moment is a second order symmetric tensor. It has no net charge and no net dipole moment. Unlike a dipole, it will experience neither a net force nor a net torque in any uniform field.
Is co2 a quadrupole?
The symmetric molecule carbon dioxide has a permanent electrical quadrupole moment that can be described as two electrical dipoles sitting back-to-back and pointing in opposite directions. In contrast, nitrogen, hydrogen, and methane have much smaller quadrupole moments.
What is the dipole moment of CHCl3?
Therefore, the dipole moment of three C-Cl bonds on one side, gives a resultant moment that is equal and opposite to the dipole moment of the single C-Cl bond on the opposite side. So, μ=0. This explains CHCl3>CCl4.
How are multipole expansions used to study electromagnetic fields?
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.
What are the disadvantages of multipole expansion?
Multipole expansion of a potential outside an electrostatic charge distribution. Its disadvantage is that the derivations are fairly cumbersome (in fact a large part of it is the implicit rederivation of the Legendre expansion of 1/|r − R|, which was done once and for all by Legendre in the 1780s).
How are energies and forces calculated in multipole expansion?
The basic idea is to decompose the particles into groups; particles within a group interact normally (i.e., by the full potential), whereas the energies and forces between groups of particles are calculated from their multipole moments.
Why are multipole expansions useful for Taylor series?
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 .
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