Question

Three identical dipoles with charges q and -q and separation between the charges are placed at the corners of an equilateral triangle of side d as shown in Fig. 3.123. Find the interaction energy of the system (a<<d).

Answer 1

Electric field $E_1$ due to dipole at B is
$E_1=\frac{k2pcos{60}^{\circ }}{d^3}$ and $E_2=\frac{kpsin{60}^{\circ }}{d^3}$
Potential energy between A and B is
$-\overrightarrow{p}.\overrightarrow{E}=-\overrightarrow{p}.(\overrightarrow{E_1}+\overrightarrow{E_2})=-\overrightarrow{p}.\overrightarrow{E_1}(\therefore \overrightarrow{p}\perp \overrightarrow{E_2})$
or $-p{E}_{1}cos{180}^{\circ }=p{E}_1=\frac{k2p^{2}cos{60}^{\circ }}{d^3}=\frac{q^2{a}^3}{4\pi {\epsilon }_0{d}^3}$
Net energy of system is $3q^2{a}^2/4\pi {\epsilon }_0{d}^3$