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.

How is multipole expansion calculated?

Expansion in Cartesian coordinates

1. Let satisfy .
2. If v(r − R) satisfies the Laplace equation.
3. Consider now the following form of v(r − R):
4. In order to derive this multipole expansion, we write rXY = rY − rX, which is a vector pointing from X towards Y.

What is the point of multipole expansion?

A multipole expansion is a mathematical series representing a function that depends on angles—usually the two angles on a sphere. These series are useful because they can often be truncated, meaning that only the first few terms need to be retained for a good approximation to the original function.

Why is multipole expansion of vector potential important?

The importance of such an expansion is that, in general, it is not possible to compute exactly the electric field generated by a static electric charge distribution (or the magnetic field due to static electric currents).

How do you calculate 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. px=∑Qixipy=∑Qiyipz=∑Qizi.

What is the formula of electric dipole moment?

The formula for electric dipole moment for a pair of equal & opposite charges is p = qd, the magnitude of the charges multiplied by the distance between the two.

What is the unit of quadrupole moment?

The electric quadrupole moment has the units of Coulomb x meter2. It is sometimes tabulated in units of e x 10-24 cm2 where e is the electron charge.

What is quadrupole effect?

A quadrupolar nucleus is one that has a quantum spin number greater than ½. Such nuclei have a lower symmetry than spin-½ nuclei. The quadrupole moment that varies between nuclei is a measure of this asymmetry. Their energies split upon the application of a magnetic field into multiple levels (fig. 1).

What is monopole moment?

A multipole expansion is a mathematical series representing a function that depends on angles—usually the two angles on a sphere. For example, the initial term—called the zeroth, or monopole, moment—is a constant, independent of angle.

How can we calculate dipole moment?

A dipole moment is the product of the magnitude of the charge and the distance between the centers of the positive and negative charges. It is denoted by the Greek letter ‘µ’. It is measured in Debye units denoted by ‘D’. 1 D = 3.33564 × 10-30 C.m, where C is Coulomb and m denotes a meter.

How is the multipole expansion related to the charge distribution?

Multipole Expansion Consider a bounded charge distribution that lies inside the sphere . It follows that in the region . According to the previous three equations, the electrostatic potential in the region takes the form (339) where the (340) are known as the multipole momentsof the charge distribution . Here, the integral is over all space.

When does a multipole expansion converge under two conditions?

In principle, a multipole expansion provides an exact description of the potential and generally converges under two conditions: if the sources (e.g. charges) are localized close to the origin and the point at which the potential is observed is far from the origin; or

How are multipole expansions used in vector calculus?

A multipole expansion provides a set of parameters that characterize the potential due to a charge distribution of finite size at large distances from that distribution. The goal is to represent the potential by a series expansion of the form: 1/28/2010 Electric Multipole Expansion MultiPol-2

What is the dipole moment of an electric multipole?

Electric multipoles. ‰The dipole term: where is called the dipole moment. As usual, for surface, line and point charges, we have The simplest dipole has two point charges, separated by a displacement vector d that points from –q. to +q.

https://www.youtube.com/watch?v=QNGYKWVOV54