Quadrupole

A Quadrupole or quadrilateral is one of a sequence of configurations of things such as electric charge or current, or gravitational mass that may exist ideally, but it is usually multi– spanning of a more complex structure reflecting different orders of complexity. is part of.

mathematical definition

The tensor quadrupole moment Q is a rank-two tensor -3×3 matrix. There are many definitions, but it is commonly called in traceless form (i.e. ) the quadratic moment tensor thus has 9 components, but due to the transposition symmetry and zero-trace property, only 5 of these are independent in this form.

{\displaystyle Q_{xx}+Q_{yy}+Q_{zz}=0}

For a discrete system of point charges or masses in the case of a gravitational quadrilateral , each with a charge , or mass , and position relative to the origin of the coordinate system, the components of the Q matrix are defined as:

q_{l}m_{l}{\displaystyle {\vec {r_{l}}}=\left(r_{xl},r_{yl},r_{zl}\right)}

{\displaystyle Q_{ij}=\sum _{l}q_{l}\left(3r_{il}r_{jl}-\left\|{\vec {r_{l}}}\right\|^{2}\delta _{ij}\right)}.

index run on cartesian coordinatesi,j x,y,zAnd there’s the Kronecker Delta . This means that there must be equal, in order to sign up, the distance from the point to the Kronecker delta equal to 1 for a mutually perpendicular hyperplane .

\delta _{ij}x,y,zn

In the non-traceless form, the quadruple moment is sometimes referred to as:

{\displaystyle Q_{ij}=\sum _{l}q_{l}r_{il}r_{jl}}

eeing some use in the literature regarding the fast multipole method with this form. The conversion between these two forms can easily be achieved using a detracing operator. [1]

For a continuous system with charge density, or mass density, , the components of Q are defined by the integral over the Cartesian space R:

{\displaystyle Q_{ij}=\int \,\rho (\mathbf {r} )\left(3r_{i}r_{j}-\left\|{\vec {r}}\right\|^{2}\delta _{ij}\right)\,d^{3}\mathbf {r} }

As with any polypole moment, if the lower-order moment, in this case the monopole or dipole , is non-zero , the value of the quadrupole moment depends on the choice of the coordinate origin . For example, a dipole of two opposite-sign, equal-strength point charges, which has no monopolar moment, can have a non-zero quadrilateral moment if the origin is shifted away from the center of configuration between the two charges; Or the quadrupole moment can be reduced to zero with the origin point at the centre. Conversely, if the monopole and dipole moments disappear but the quadrupole moment does not, for example four equal-power charges, arranged in a square, with alternating signs, then the quadrupole moment is coordinate independent.

If each charge is the source of a “potential” field, like an electric or gravitational field , the contribution to the field potential by the quadrupole moment is: 1/r

{\displaystyle V_{\text{q}}(\mathbf {R} )={\frac {k}{|\mathbf {R} |^{3}}}\sum _{i,j}{\frac {1}{2}}Q_{ij}\,{\hat {R}}_{i}{\hat {R}}_{j}\ ,}

where r is a vector with origin in the arrangement of charges and r is the unit vector in the direction of r . Here, there is a constant that depends on the type of field and the units used. The factors are the components of the unit vector from the point of interest to the locus of the quadrupole moment.

k{\displaystyle {\hat {R}}_{i},{\hat {R}}_{j}}

electric quadruple

The simplest example of an electric quadrilateral consists of positive and negative charges arranged at the corners of a square. The monopole moment (only the total charge) of this arrangement is zero. Similarly, the dipole moment is zero even if the coordinate origin is chosen. But the quadrupole moment of the arrangement in the diagram cannot be reduced to zero, even if we are at the coordinate origin. The electric potential of an electric charge quadrupole is given [3]

{\displaystyle V_{\text{q}}(\mathbf {R} )={\frac {1}{4\pi \epsilon _{0}}}{\frac {1}{|\mathbf {R} |^{3}}}\sum _{i,j}{\frac {1}{2}}Q_{ij}\,{\hat {R}}_{i}{\hat {R}}_{j}\ ,}

Where is the electric permittivity , and follows the definition above.

\epsilon _{0}Q_{ij}

Generalization: high polyhedron

An extreme generalization (“point octopole “) would be: eight alternating point charges at the eight corners of a parallelogram , with edge lengths similar to that of a cube . The “octopole moment” of this arrangement would

correspond to the “octopole limit” for a non-zero diagonal tensor of order three. Still higher multiples, for example 2 ^l , dipoles (quadrangular, octopolar, …) would be obtained by arrangement of point dipoles (quadrangle, octopole, …), not lower order point monopoles, e.g. 2 ^{l− 1} 

{\displaystyle \lim _{a\to 0}{a^{3}\cdot Q}\to {\text{const. }}}

magnetic quadrilateral

All known magnetic sources give a dipole field. However, it is possible to create a magnetic quadruple by placing four identical bar magnets perpendicular to each other such that the north pole of one is next to the south of the other. Such a configuration cancels out the dipole moment and gives a quadruple moment, and its field will decrease faster over a larger distance than that of the dipole.

An example of a magnetic quadruple, consisting of permanent magnets, is shown on the right. Electromagnets of similar conceptual design (called quadruple magnets ) are commonly used in particle accelerators and beam transport lines to focus beams of charged particles , a method known as strong focusing . There are four steel pole tips, two anti-magnetic north poles and two anti-magnetic south poles. a large electrocution of steelMagnetized by current that flows in coils of tubing wrapped around the poles. Furthermore, the quaternary–dipole intersection can be found by multiplying the spin of the unpaired nucleon by that of its parent atom.

A changing magnetic quadrupole moment generates electromagnetic radiation .

gravitational quadrilateral

Mass quadrupling corresponds to electric charge quadrupling, where charge density is simply replaced by mass density and a negative sign is added because the masses are always positive and the force is attractive. The gravitational potential is then expressed as:

{\displaystyle V_{\text{q}}(\mathbf {R} )=-{\frac {G}{2|\mathbf {R} |^{3}}}\sum _{i,j}Q_{ij}\,{\hat {R}}_{i}{\hat {R}}_{j}\ .}

For example, because the Earth is rotating, it is flattened (flattened at the poles). This gives it a non-zero quadratic moment. While the contribution to Earth’s gravitational field from this quadrant is extremely important for artificial satellites close to Earth, it is less important for the Moon because the period drops off quickly.

\frac{1}{|\mathbf{R}|^3}

The mass quadrupling moment is also important in general relativity because, if it changes in time, it can generate gravitational radiation similar to electromagnetic radiation produced by oscillating electric or magnetic dipoles and high multiples . However, only quadrupole and higher moments can radiate gravity. The mass monopole represents the total mass-energy in a system, which is conserved – thus giving off no radiation. Similarly, the mass dipole corresponds to the center of mass of a system and its first derivative represents the momentum which is also a conserved quantity so the mass dipole also does not emit any radiation. Quadruple the mass, however, can change over time, and is the lowest order contribution to gravitational radiation. [4]

The simplest and most important example of a radiation system is a pair of points of equal mass that orbit each other in a circular orbit, for example the special case of a binary black hole . Since the dipole moment is constant, we can place the coordinate origin exactly between the two points for convenience. Then the dipole moment will be zero, and if we also measure the coordinates so that the points are at a unit distance from the center, in the opposite direction, the quadrupole moment of the system will then simply be

{\displaystyle Q_{ij}=M\left(3x_{i}x_{j}-|\mathbf {x} |^{2}\delta _{ij}\right)}

where M is the mass of each point, and are the components of the (unit) position vector of a single point. This x -vector will rotate as they orbit , which means it will have a non-zero first, and a second time derivative (this is of course true regardless of the choice of coordinate system). Therefore the system will radiate gravitational waves. The energy lost in this way was first estimated in the changing period of the Hulse–Taylor binary , a pulsar in orbit with another neutron star of similar mass.x_{i}

Just as electric charge and current polypoles contribute to the electromagnetic field, mass and mass-current polypoles contribute to the gravitational field in general relativity, leading to the so-called gravitational – magnetic effects. Gravitational radiation can also be released by changing the mass-current multipole. However, the contribution from the current multipole will typically be much smaller than from a quadrupole of the mass.