Nikiya Anton Bettey 2021: 09-14-2021-0316

Tuesday, September 14, 2021

09-14-2021-0316 - quadrupole or quadrapole

A quadrupole or quadrapole is one of a sequence of configurations of things like electric charge or current, or gravitational mass that can exist in ideal form, but it is usually just part of a multipole expansion of a more complex structure reflecting various orders of complexity.

Mathematical definition[edit]

The quadrupole moment tensor Q is a rank-two tensor—3×3 matrix. There are several definitions, but it is normally stated in the traceless form (i.e. $Q_{xx}+Q_{yy}+Q_{zz}=0$ ). The quadrupole moment tensor has thus 9 components, but because of transposition symmetry and zero-trace property, in this form only 5 of these are independent.

For a discrete system of point charges or masses in the case of a gravitational quadrupole, each with charge $q_{l}$ , or mass $m_{l}$ , and position ${\vec {r_{l}}}=\left(r_{xl},r_{yl},r_{zl}\right)$ relative to the coordinate system origin, the components of the Q matrix are defined by:

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

The indices $i,j$ run over the Cartesian coordinates $x,y,z$ and $\delta _{ij}$ is the Kronecker delta. This means that $x,y,z$ must be equal, up to sign, to distances from the point to $n$ mutually perpendicular hyperplanes for the Kronecker delta to equal 1.

In the non-traceless form, the quadrupole moment is sometimes stated as:

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

with this form seeing some usage in the literature regarding the fast multipole method. Conversion between these two forms can be easily achieved using a detracing operator.^[1]

For a continuous system with charge density, or mass density, $\rho (x,y,z)$ , the components of Q are defined by integral over the Cartesian space r:^[2]

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 multipole moment, if a lower-order moment, monopole or dipole in this case, is non-zero, then the value of the quadrupole moment depends on the choice of the coordinate origin. For example, a dipole of two opposite-sign, same-strength point charges, which has no monopole moment, can have a nonzero quadrupole moment if the origin is shifted away from the center of the configuration exactly between the two charges; or the quadrupole moment can be reduced to zero with the origin at the center. In contrast, if the monopole and dipole moments vanish, but the quadrupole moment does not, e.g. four same-strength charges, arranged in a square, with alternating signs, then the quadrupole moment is coordinate independent.

If each charge is the source of a " $1/r$ potential" field, like the electric or gravitational field, the contribution to the field's potentialfrom the quadrupole moment is:

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 system of charges and R̂ is the unit vector in the direction of R. Here, $k$ is a constant that depends on the type of field, and the units being used. The factors ${\hat {R}}_{i},{\hat {R}}_{j}$ are components of the unit vector from the point of interest to the location of the quadrupole moment.

Electric quadrupole [edit]

Contour plot of the equipotential surfaces of an electric quadrupole field

The simplest example of an electric quadrupole consists of alternating positive and negative charges, arranged on the corners of a square. The monopole moment (just the total charge) of this arrangement is zero. Similarly, the dipole moment is zero, regardless of the coordinate origin that has been chosen. But the quadrupole moment of the arrangement in the diagram cannot be reduced to zero, regardless of where we place the coordinate origin. The electric potential of an electric charge quadrupole is given by^[3]

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 $\epsilon _{0}$ is the electric permittivity, and $Q_{ij}$ follows the definition above.

https://en.wikipedia.org/wiki/Quadrupole

Nikiya Anton Bettey 2021

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Tuesday, September 14, 2021

09-14-2021-0316 - quadrupole or quadrapole

Mathematical definition[edit]

Electric quadrupole [edit]

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