Question

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

• A. $\sqrt{2}qd$
• B. $\sqrt{3}qd$
• C. $\sqrt{5}qd$
• D. $\sqrt{6}qd$

Answer 1

The correct option is B $\sqrt{3}qd$

Here, assuming two charges of same magnitude $+q$ are placed at the vertex $\text{B}$ of the triangle.


From formula, dipole moment $\overrightarrow{p}$ is
$\overrightarrow{p}=q\left(2\overrightarrow{l}\right)$

Now from figure,

$p_1=p_2=p=qd$

Angle between $\overrightarrow{p_1}$ and $\overrightarrow{p_2}$,

$\theta ={60}^{\circ }$

So, net electric dipole of the pair of dipoles,

$p_{\text{net}}=\sqrt{p_1^2+p_2^2+2p_1{p}_2\cos \theta }$

$\Rightarrow p_{\text{net}}=\sqrt{(qd{)}^2+(qd{)}^2+2(qd{)}^2\times \cos {60}^{\circ }}$

$\Rightarrow p_{\text{net}}=\sqrt{3}qd$

Hence, option $\left(b\right)$ is correct.