# Potential energy of quadrupole moments

hi

i am looking for an equation that gives me the potential energy of 2 quadrupole moments and their mutual distance is assumed to be large.

and they have no total charge or dipole moments.

does such an equation exist?

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Bill_K
Are you sure you want to see this? It's rather complicated. Lets write down the dipole-dipole interaction first. An electric dipole moment is described by a vector p. The electrostatic potential energy between two dipoles p and p' a distance r apart is

W = (p.p' -3 (n.p)(n.p'))/r3

where n is the unit vector pointing from one dipole to the other.

Ok similarly, the quadrupole moment of an object is described by a symmetric traceless tensor Q. The electrostatic potential energy between two quadrupoles Q and Q' is given by

W = (Q:Q' -10 n.Q.Q'.n + 35/3 (n.Q.n)(n.Q'.n))/r5

Maybe you'd rather see this written out explicitly with all the indices.

Φ = (QijQ'ij - 10 niQijQ'jknk + 35/3 niQijnj nkQ'klnl)/r5

Are you sure you want to see this? It's rather complicated. Lets write down the dipole-dipole interaction first. An electric dipole moment is described by a vector p. The electrostatic potential energy between two dipoles p and p' a distance r apart is

W = (p.p' -3 (n.p)(n.p'))/r3

where n is the unit vector pointing from one dipole to the other.

Ok similarly, the quadrupole moment of an object is described by a symmetric traceless tensor Q. The electrostatic potential energy between two quadrupoles Q and Q' is given by

W = (Q:Q' -10 n.Q.Q'.n + 35/3 (n.Q.n)(n.Q'.n))/r5

Maybe you'd rather see this written out explicitly with all the indices.

Φ = (QijQ'ij - 10 niQijQ'jknk + 35/3 niQijnj nkQ'klnl)/r5
thank you this exactly what i needed!!!

oh,

could you explain to me, why there are 4 indices? i,j,k and l

i thought that this tensor would only use 3 cartesian coordinates?

then why are there 4 different indices?