Electrostatics 4 Question 2
2. Determine the electric dipole moment of the system of three charges, placed on the vertices of an equilateral triangle as shown in the figure.
(Main 2019, 12 Jan I)
(a) $\sqrt{3} q l \frac{\hat{\mathbf{j}}-\hat{\mathbf{i}}}{\sqrt{2}}$
(b) $2 q l \hat{\mathbf{j}}$
(c) $-\sqrt{3} q l \hat{\mathbf{j}}$
(d) $(q l) \frac{\hat{\mathbf{i}}+\hat{\mathbf{j}}}{\sqrt{2}}$
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Solution:
- Given system is equivalent to two dipoles inclined at $60^{\circ}$ to each other as shown in the figure below,
Now, magnitude of resultant of these dipole moments is
$$ p _{\text {net }}=\sqrt{p^{2}+p^{2}+2 p \cdot p \cos 60^{\circ}}=\sqrt{3} p=\sqrt{3} q l $$
As, resultant is directed along negative $y$-direction
$$ p _{\text {net }}=-\sqrt{3} p \hat{\mathbf{j}}=-\sqrt{3} q l \hat{\mathbf{j}} $$
