To find the electrostatic force on a charged particle \( q_2 \) located at the origin (0, 0, 0) due to another charged particle \( q_1 \) located at (2, -1, 3), we can follow these steps:
### Step 1: Identify the positions of the charges
- The position of charge \( q_1 \) is given as \( \mathbf{r_1} = (2, -1, 3) \).
- The position of charge \( q_2 \) is given as \( \mathbf{r_2} = (0, 0, 0) \).
### Step 2: Calculate the displacement vector \( \mathbf{r} \) from \( q_1 \) to \( q_2 \)
The displacement vector \( \mathbf{r} \) is calculated as:
\[
\mathbf{r} = \mathbf{r_2} - \mathbf{r_1} = (0, 0, 0) - (2, -1, 3) = (-2, 1, -3)
\]
### Step 3: Calculate the magnitude of the displacement vector \( r \)
The magnitude \( r \) is given by:
\[
r = \sqrt{(-2)^2 + (1)^2 + (-3)^2} = \sqrt{4 + 1 + 9} = \sqrt{14}
\]
### Step 4: Calculate the unit vector \( \hat{r} \)
The unit vector \( \hat{r} \) in the direction of \( \mathbf{r} \) is:
\[
\hat{r} = \frac{\mathbf{r}}{r} = \frac{(-2, 1, -3)}{\sqrt{14}} = \left(-\frac{2}{\sqrt{14}}, \frac{1}{\sqrt{14}}, -\frac{3}{\sqrt{14}}\right)
\]
### Step 5: Use Coulomb's Law to find the force \( \mathbf{F} \)
Coulomb's Law states that the electrostatic force \( \mathbf{F} \) between two point charges is given by:
\[
\mathbf{F} = k \frac{q_1 q_2}{r^2} \hat{r}
\]
where \( k = \frac{1}{4 \pi \epsilon_0} \).
Substituting the values:
\[
\mathbf{F} = k \frac{q_1 q_2}{(\sqrt{14})^2} \hat{r} = k \frac{q_1 q_2}{14} \hat{r}
\]
### Step 6: Substitute \( \hat{r} \) into the force equation
\[
\mathbf{F} = k \frac{q_1 q_2}{14} \left(-\frac{2}{\sqrt{14}}, \frac{1}{\sqrt{14}}, -\frac{3}{\sqrt{14}}\right)
\]
This can be simplified to:
\[
\mathbf{F} = \frac{k q_1 q_2}{14 \sqrt{14}} (-2, 1, -3)
\]
### Final Result
Thus, the electrostatic force on charge \( q_2 \) due to charge \( q_1 \) is:
\[
\mathbf{F} = \frac{k q_1 q_2}{14 \sqrt{14}} (-2, 1, -3)
\]
