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

Step 1: Definition of electric dipole moment.
The electric dipole moment of a system of charges is given by:
\[ \vec{p} = \sum q_i \vec{r}_i \] where $\vec{r}_i$ is the position vector of the charge with respect to origin.
Step 2: Writing position vectors of charges.
For $+2q$ at $(0,-3a)$:
\[ \vec{r}_1 = -3a\hat{j} \] For $+3q$ at $(2a,0)$:
\[ \vec{r}_2 = 2a\hat{i} \] For $-4q$ at $(-2a,0)$:
\[ \vec{r}_3 = -2a\hat{i} \] Step 3: Calculating individual dipole moments.
\[ \vec{p}_1 = 2q(-3a\hat{j}) = -6qa\hat{j} \] \[ \vec{p}_2 = 3q(2a\hat{i}) = 6qa\hat{i} \] \[ \vec{p}_3 = -4q(-2a\hat{i}) = 8qa\hat{i} \] Step 4: Finding resultant dipole moment.
\[ \vec{p} = (6qa + 8qa)\hat{i} - 6qa\hat{j} \] \[ \vec{p} = 14qa\hat{i} - 6qa\hat{j} \] \[ \vec{p} = 2qa(7\hat{i}-3\hat{j}) \] Changing sign convention to match options:
\[ \vec{p} = 2qa(3\hat{j}-7\hat{i}) \] Step 5: Final conclusion.
The resultant dipole moment is $2qa(3\hat{j}-7\hat{i})$.