Question 14: Electrostats & Capacitance

Question

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 ______.

Accepted Answer

For a system of discrete point charges $q_i$ located at positions $\vec{r}_i$, the electric dipole moment $\vec{P}$ with respect to the origin is defined as the vector sum of the product of each charge and its position vector.$\$$\vec{P} = \sum_{i} q_i \vec{r}_i$$$$\Rightarrow \vec{P} = q_1\vec{r}_1 + q_2\vec{r}_2 + q_3\vec{r}_3 + \dots$$Step-by-Step Calculation:$\$For charge 1 ($q_1$):$q_1 = +2q$ at coordinates $(0, -3a)$Position vector: $\vec{r}_1 = 0\hat{i} - 3a\hat{j} = -3a\hat{j}$Individual dipole moment: $\vec{p}_1 = 2q(-3a\hat{j}) = -6aq\hat{j}$ $\$For charge 2 ($q_2$):$q_2 = +3q$ at coordinates $(2a, 0)$Position vector: $\vec{r}_2 = 2a\hat{i} + 0\hat{j} = 2a\hat{i}$Individual dipole moment: $\vec{p}_2 = 3q(2a\hat{i}) = 6aq\hat{i}$ $\$For charge 3 ($q_3$):$q_3 = -4q$ at coordinates $(-2a, 0)$Position vector: $\vec{r}_3 = -2a\hat{i} + 0\hat{j} = -2a\hat{i}$Individual dipole moment: $\vec{p}_3 = -4q(-2a\hat{i}) = 8aq\hat{i}$Net Electric Dipole Moment:The net moment is the sum of the individual moments:$$\vec{P} = \vec{p}_1 + \vec{p}_2 + \vec{p}_3$$ $$\Rightarrow \vec{P} = (-6qa\hat{j}) + (6qa\hat{i}) + (8qa\hat{i})$$ $$\Rightarrow \vec{P} = 14qa\hat{i} - 6qa\hat{j}$$ $$\Rightarrow \vec{P} = 2qa(7\hat{i} - 3\hat{j})$ $\$Hence, the correct option is (A).

Answer

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

Answer

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

Answer

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

Answer

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

Electrostats & Capacitance

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 ______.

Choose the correct answer: