Derivation of dipole-dipole interaction energy

[schattenjaeger]

if you have 2 point dipoles p1 and p2 located at r1 and r2 respectively(all four are vectors)

what's the energy of p2 in the field of p1?

I know the general expression for the energy of a dipole in an external field Eo is U=-p * Eo but from there I get confused

I can express the electric field of the dipole in terms of its radial and angular components, but I'm not sure how to express p2 in those same terms.

FYI the answer is Udd=1/(4pi*epsilon*R^3)*[p1*p2 - 3(p1*R)(p2*R)]

both those bold Rs are unit vectors, * is dot product if it's between two vectors

 

[siddharth]

Can you find the electric field of a dipole in a coordinate free form?

Use that and U = - \vec{p}.\vec{E}

 

[Meir Achuz]

Substitute E_0 for a dipole in vector notation:
E_0=3(p.r)r/r^5-p/r^3.