Quadrupole

The quadrupole , in electrostatic, is a charge distribution, having for characteristic that the barycentres of the respectively positive and negative loads are confused.

Quadrupole analyzes

That is to say a distribution (D) of loads $q\_i$ at the points $P\_i$. This distribution (D) with compact support creates at a long distance from the loads (for R >> has, with has length characteristic of the distribution) a potential V1 (R).

One defines:

• $q\; =\; \backslash \; Sigma\; q\_i$ the sum of the loads
• $\backslash \; vec\; \{p\}\; (O)$ = $\backslash \; Sigma\; q\_i\; \backslash \; vec\; \{OP\_i\}$, independent of O if q=0, no one if O is selected barycentre of the loads
• $J\_O\; =\; \backslash \; Sigma\; q\_i\; \{OP\_i\}\; ^2$, moment of inertia compared to O
• $\backslash \; hat\; \{J\}\; (\backslash \; vec\; \{X\})\; =\; \backslash \; Sigma\; q\_i\; \backslash \; vec\; \{OP\_i\}\; \backslash \; times\; (\backslash \; vec\; \{X\}\; \backslash \; times\; \backslash \; vec\; \{OP\_i\})$, the linear operator of inertia compared to O
• $\backslash hat\; \{Q\}\; =\; 2\; J\_o\; X\; -3\; \backslash \; hat\; \{J\}\; X$, the quadrupolar linear operator out of O

One can check that trace ($\backslash \; hat\; \{Q\}$) = 0.

Quadrupolar development

Theorem:

$V1\; (M)\; =\; \backslash \; frac\; \{Q\}\; \{R\}\; +\; \backslash \; frac\; \{\backslash \; vec\; \{p\}.\; \backslash \; vec\; \{U\}\}\; \{r^2\}\; +\; \backslash \; frac\; \{\backslash \; vec\; \{U\}.\; (\backslash \; hat\; \{Q\}\; \backslash \; vec\; \{U\})\}\{2\; \backslash \; times\; r^3\}\; +\; O\; (\backslash \; frac\; \{1\}\; \{r^3\})$, with $\backslash \; vec\; \{U\}\; =\; \backslash \; vec\; \{R\}\; /r$

Particular case: axis of symmetry

(D) has the symmetry of revolution around an axis, say OZ.

Then the matrix of $\backslash \; hat\; \{Q\}$ is diagonal, with $Q\_\; \{X,\; X\}\; =\; Q\_\; \{there,\; there\}\; =\; -\; Q\_o/2$ and $Q\_\; \{Z,\; Z\}\; =\; Q\_o$ which is called quadrupolar moment out of O of the distribution. If Q is not null, one chooses O in G, and then:

$V1\; (M)\; =\; \backslash \; frac\; \{Q\}\; \{R\}\; +\; \backslash \; frac\; \{Q\_o\}\; \{2\; \backslash \; times\; r^3\}\; \backslash \; times\; \backslash \; theta)\; +\; O\; (\backslash \; frac\; \{1\}\; \{r^3\})$, with $P\_2\; (X)\; =\; 1/2.\; (3x^2-1)$ (2nd polynomial of Legendre).

This theorem is worth in gravimetry for the supposed Earth of revolution. In this case, $Q\_o\; =\; 2\; (AC)$ < 0; the use is to pose $J\_2\; =\; \backslash \; frac\; \{CA\}\; \{Ma^2\}\; =\; 1.08263\; \backslash \; times\; 10^\; \{-\; 3\}$.

The terrestrial potential is thus $V\; (M)\; =\; -\; \backslash \; frac\; \{GM\}\; \{R\}\; +\; \backslash \; frac\; \{GMa\; J\_2\; P\_2\; (cos\; \backslash \; theta)\}\{r^3\}$.

This development can be further thorough (development in spherical harmonics; terms in J4 (octupolaire), J6, etc).

See too

• Electrostatic
• electrostatic Dipole
• Gravimetry

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