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Question

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

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Solution

## 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, ${P}_{\mathrm{net}}=\sqrt{{P}^{2}+{P}^{2}+2P\mathrm{cos}60°}\phantom{\rule{0ex}{0ex}}=\sqrt{\left(qd{\right)}^{2}+\left(qd{\right)}^{2}+2{\left(qd\right)}^{2}\mathrm{cos}60°}\phantom{\rule{0ex}{0ex}}=\left[{d}^{2}{q}^{2}+{d}^{2}{q}^{2}+2{q}^{2}{d}^{2}×\frac{1}{2}{\right]}^{1/2}\phantom{\rule{0ex}{0ex}}=\left[3{q}^{2}{d}^{2}{\right]}^{1/2}=\sqrt{3}qd$ The net dipole moment is along the bisector of the angle at +q, away from the triangle.

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