Three charges are arranged on the vertices of an equilateral triangle. Find the dipole moment of the combination.
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( \sqrt{3}\,qd \)
The dipole moment of a system of charges is \( \vec{p} = \sum q_i\vec{r}_i \). The charge \( +2q \) can be split into two charges \( +q \), one paired with each \( -q \). The system is then the vector sum of two dipoles, each of magnitude \( qd \) (charge \( q \) times the side \( d \)), both pointing towards the vertex of \( +2q \).
The two dipole moments make an angle of \( 60^\circ \) with each other (the angle of the equilateral triangle at that vertex).
\( p = \sqrt{p_1^{2} + p_2^{2} + 2p_1p_2\cos60^\circ} = \sqrt{(qd)^{2} + (qd)^{2} + (qd)^{2}} \)
\( p = 2qd\cos30^\circ = \sqrt{3}\,qd \)