Question

Three charges are arranged on the vertices of an equilateral triangle, as shown in the figure. Find the dipole moment of the combination.

Answer 1

The system can be considered as a combination of two dipoles making an angle of 60^∘ with each other.

Length of each dipole = d
So, the dipole moment, P = q × d
So, the resultant dipole moment,
$$\begin{gathered} P_{\mathrm{net}}=\sqrt{P^2+P^2+2P\cos 60°} \\ =\sqrt{(qd{)}^2+(qd{)}^2+2{\left(qd\right)}^2\cos 60°} \\ =[d^2{q}^2+d^2{q}^2+2q^2{d}^2\times \frac{1}{2}{]}^{1/2} \\ =[3q^2{d}^2{]}^{1/2}=\sqrt{3}qd \end{gathered}$$
The net dipole moment is along the bisector of the angle at +q, away from the triangle.