# Dipole Moment interaction

by raintrek
Tags: dipole, interaction, moment
 P: 76 1. The problem statement, all variables and given/known data Consider two dipoles with moments u1 and u2 arranged as in the following diagram. Each dipole is depicted as two charges of equal magnitude separated by a distance d. The centre-to-centre separation of the two dipoles is the distance r. The line joining the two dipole centres makes an angle theta with the lower dipole (ie. q1 and -q1). Derive an expression in terms of u1, u2, theta and r which describes the potential energy of interaction of these two dipoles which is valid when d<
 P: 76 OK, I've got a little bit further (this seems brutal!) I've been able to see that the separations of the charges mentioned above are: q1 & -q2: $$\sqrt{(d+rcos\theta)^{2}+(rsin\theta)^{2}}$$ -q1 & q2: $$\sqrt{(-d+rcos\theta)^{2}+(rsin\theta)^{2}}$$ Simplifying: q1 & -q2: $$\sqrt{d^{2}+r^{2}+2rdcos\theta}$$ -q1 & q2: $$\sqrt{d^{2}+r^{2}-2rdcos\theta}$$ However this quickly makes the dipole-dipole interaction energy horrible: $$U(r)=\frac{q_{1}q_{2}}{4\pi\epsilon_{0}}\left(\frac{2}{r}-\frac{1}{\sqrt{d^{2}+r^{2}+2rdcos\theta}}-\frac{1}{\sqrt{d^{2}+r^{2}-2rdcos\theta}}\right)$$ From that point I see no way to simplify the last two terms to get to a point where I can apply the Taylor expansion in the hint. I really am pulling my hair out over this now, if anyone can suggest anything I'd be ever grateful!
 P: 76 LOL, ok probably talking to myself here. Still playing around with this, taken it further, although I'm pretty sure my final answer here is wrong...

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