Question

Find dipole moment of a system consisting of charge \(q_1 = 3\,\mu C\) and \(q_2 = -9\,\mu C\) with position coordinates \(\vec r_1 = 2\hat i + 3\hat j + 3\hat k\) and \(\vec r_2 = \hat i + \hat j + \hat k\) respectively.

• A. \(-3\hat i\ \mu C\text{-m}\)
• B. \(-9\hat i\ \mu C\text{-m}\)
• C. \(-6\hat i\ \mu C\text{-m}\)
• D. \(-5\hat i\ \mu C\text{-m}\)

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The electric dipole moment of a system of point charges is given by the vector sum of the product of each charge and its position vector relative to an origin.
Step 2: Key Formula or Approach:
The formula for the dipole moment of a system of charges is $\vec{P} = \sum q_i \vec{r}_i$.
For two charges, $\vec{P} = q_1 \vec{r}_1 + q_2 \vec{r}_2$.
Step 3: Detailed Explanation:
Given values:
$q_1 = 3\ \mu\text{C}$, $\vec{r}_1 = 2\hat{i} + 3\hat{j} + 3\hat{k}$
$q_2 = -9\ \mu\text{C}$, $\vec{r}_2 = \hat{i} + \hat{j} + \hat{k}$
Calculating the dipole moment vector $\vec{P}$:
\[ \vec{P} = 3(2\hat{i} + 3\hat{j} + 3\hat{k}) + (-9)(\hat{i} + \hat{j} + \hat{k}) \]
Expanding the terms:
\[ \vec{P} = (6\hat{i} + 9\hat{j} + 9\hat{k}) - (9\hat{i} + 9\hat{j} + 9\hat{k}) \]
Grouping the vector components:
\[ \vec{P} = (6-9)\hat{i} + (9-9)\hat{j} + (9-9)\hat{k} \]
\[ \vec{P} = -3\hat{i}\ \mu\text{C-m} \]
Step 4: Final Answer:
The dipole moment is $-3\hat{i}\ \mu\text{C-m}$.