Question

Two point charges \(q_1 = 3\) $\mu$C and \(q_2 = -4\) $\mu$C are placed at points \((2\hat{i} + 3\hat{j} + 3\hat{k})\) and \((\hat{i} + \hat{j} + \hat{k})\) respectively. Force on charge \(q_2\) is \underline{\hspace{2cm}} N. (Take \(\frac{1}{4\pi\epsilon_0} = 9 \times 10^9\) SI Units)

• A. \((12\hat{i} + 24\hat{j} + 24\hat{k}) \times 10^{-3}\)
• B. \((4\hat{i} + 8\hat{j} + 8\hat{k}) \times 10^{-3}\)
• C. \((3\hat{i} + 6\hat{j} + 6\hat{k}) \times 10^{-3}\)
• D. \((-4\hat{i} - 8\hat{j} - 8\hat{k}) \times 10^{-3}\)

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
We need to calculate the electrostatic force exerted on one point charge by another using Coulomb's Law in vector form. The force vector will point along the line joining the two charges.
Step 2: Key Formula or Approach:
Coulomb's Law in vector form is given by:
$\vec{F}_{12} = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{|\vec{r}_{12}|^3} \vec{r}_{12}$
where $\vec{r}_{12} = \vec{r}_2 - \vec{r}_1$ is the position vector from charge 1 to charge 2.
Step 3: Detailed Explanation:
Given the position vectors:
$\vec{r}_1 = 2\hat{i} + 3\hat{j} + 3\hat{k}$
$\vec{r}_2 = \hat{i} + \hat{j} + \hat{k}$
The displacement vector from $q_1$ to $q_2$ is:
$\vec{r}_{12} = \vec{r}_2 - \vec{r}_1 = (\hat{i} + \hat{j} + \hat{k}) - (2\hat{i} + 3\hat{j} + 3\hat{k}) = -\hat{i} - 2\hat{j} - 2\hat{k}$.
The magnitude of this distance vector is:
$|\vec{r}_{12}| = \sqrt{(-1)^2 + (-2)^2 + (-2)^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3\text{ m}$.
Substitute the values into the Coulomb's Law formula:
$\vec{F}_{12} = (9 \times 10^9) \frac{(3 \times 10^{-6})(-4 \times 10^{-6})}{3^3} (-\hat{i} - 2\hat{j} - 2\hat{k})$.
Calculate the scalar part:
$\text{Scalar} = \frac{9 \times 10^9 \times (-12 \times 10^{-12})}{27} = \frac{-108 \times 10^{-3}}{27} = -4 \times 10^{-3}\text{ N}$.
Multiply the scalar part by the vector:
$\vec{F}_{12} = -4 \times 10^{-3} (-\hat{i} - 2\hat{j} - 2\hat{k}) = (4\hat{i} + 8\hat{j} + 8\hat{k}) \times 10^{-3}\text{ N}$.
Step 4: Final Answer:
The force on charge $q_2$ is $(4\hat{i} + 8\hat{j} + 8\hat{k}) \times 10^{-3}$ N.