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In theoretical classical electrodynamics we defined the quadrupol tensor of $n$ charges $q_k$ at positions (from origin or center of charge, see below) $\vec r_k$ like so: $$Q_{ij} = \sum_{k=1}^n q_k \left( 3 r_{ki} r_{kj} - r_k^2 \delta_{ij} \right)$$

I assumed that the $\vec r_k$ should be from the center of charge, so that the quadrupol tensor is translation invariant, which would seem logical to me. Out tutor said that we should just use the origin of our coordinate system. That would make the quadrupol tensor translation variant, which does not make sense to me.

Is $\vec r_k$ from the center of charge or from the coordinate origin?

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up vote 2 down vote accepted

Per wikipedia:

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.

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So there is not really a canonical quadrupole moment, but rather a quadrupole moment with respect to the origin? It does not make too much sense to me know, but that will come eventually. – Martin Ueding Nov 3 '12 at 9:09

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