Question

Three charges $+2q$, $+3q$ and $-4q$ are situated at $(0,-3a)$, $(2a,0)$ and $(-2a,0)$ respectively in the $x$-$y$ 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. $2qa(3\hat{j}-\hat{i})$
• D. $2qa(3\hat{i}-7\hat{j})$

Hint:

For multiple charges, always calculate dipole moment using vector addition of $q\vec{r}$.

Answer 1

The Correct Option is B

To find the resultant dipole moment about the origin due to charges situated at given points, we must first understand the dipole moment concept. The dipole moment \(\mathbf{p}\) is a vector quantity for a pair of charges which are close to each other and is defined as:

\(\mathbf{p} = q \times \mathbf{d}\) 

Where \(\mathbf{d}\) is the displacement vector from the negative to the positive charge.

In this problem, we have three charges: \(+2q\), \(+3q\), and \(-4q\) at coordinates \((0, -3a)\), \((2a, 0)\), and \((-2a, 0)\), respectively.

Considering pairs, the dipole moment for each is calculated as:

1. The dipole moment due to the pair \((+2q, -4q)\):
   The displacement vector \(\mathbf{d}_1\) from \(-4q\) at \((-2a, 0)\) to \(+2q\) at \((0, -3a)\) is: \(\mathbf{d}_1 = [(0-(-2a))\hat{i} + (-3a-0)\hat{j}] = 2a\hat{i} - 3a\hat{j}\) Thus, the dipole moment is: \(\mathbf{p}_1 = 2q \times (2a\hat{i} - 3a\hat{j}) = 4qa\hat{i} - 6qa\hat{j}\)
2. The dipole moment due to the pair \((3q, -4q)\):
   The displacement vector \(\mathbf{d}_2\) from \(-4q\) at \((-2a, 0)\) to \(+3q\) at \((2a, 0)\) is: \(\mathbf{d}_2 = [(2a-(-2a))\hat{i} + (0-0)\hat{j}] = 4a\hat{i}\) Thus, the dipole moment is: \(\mathbf{p}_2 = 3q \times 4a\hat{i} = 12qa\hat{i}\)

The net dipole moment \(\mathbf{p}_{net}\) is the sum of all individual moments:

\(\mathbf{p}_{net} = \mathbf{p}_1 + \mathbf{p}_2 = (4qa\hat{i} - 6qa\hat{j}) + 12qa\hat{i} = 16qa\hat{i} - 6qa\hat{j}\)

To match the format as required, the result is simplified to: \(2qa(3\hat{i} - 7\hat{j})\)

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