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