Three charges +2q, +3q and −4q are situated at (0, −3a), (2a, 0) and (−2a, 0) respectively in the xy plane. The resultant dipole moment about origin is
Q. Three charges +2q, +3q and −4q are situated at (0, −3a), (2a, 0) and (−2a, 0) respectively in the xy plane. The resultant dipole moment about origin is ____ .

(A) \(2qa(7\hat{i} - 3\hat{j})\)

(B) \(2qa(3\hat{j} - 7\hat{i})\)

(C) \(2qa(3\hat{i} - 7\hat{j})\)

(D) \(2qa(3\hat{j} - \hat{i})\)

Correct Answer: \(2qa(7\hat{i} - 3\hat{j})\)

Explanation (Complete Vector Calculation)

The electric dipole moment of a system of point charges about origin is given by:

\[ \vec{p} = \sum q_i \vec{r}_i \]

Position vectors of charges are:

\[ \vec{r}_1 = 0\hat{i} - 3a\hat{j}, \quad \vec{r}_2 = 2a\hat{i}, \quad \vec{r}_3 = -2a\hat{i} \]

Dipole moment due to charge \(+2q\):

\[ \vec{p}_1 = 2q(0\hat{i} - 3a\hat{j}) = -6qa\hat{j} \]

Dipole moment due to charge \(+3q\):

\[ \vec{p}_2 = 3q(2a\hat{i}) = 6qa\hat{i} \]

Dipole moment due to charge \(−4q\):

\[ \vec{p}_3 = -4q(-2a\hat{i}) = 8qa\hat{i} \]

Resultant dipole moment:

\[ \vec{p} = \vec{p}_1 + \vec{p}_2 + \vec{p}_3 \]
\[ \vec{p} = (6qa + 8qa)\hat{i} - 6qa\hat{j} \]
\[ \vec{p} = 14qa\hat{i} - 6qa\hat{j} \]

Taking common factor \(2qa\):

\[ \vec{p} = 2qa(7\hat{i} - 3\hat{j}) \]

Hence, the resultant dipole moment is \(2qa(7\hat{i} - 3\hat{j})\).

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