A charge particle `q_(1)`, is at position (2, -1, 3). The electrostatic force on another charged particle `q_(2)` at (0, 0, 0) is :

Options

A. `(q_1q_2)/(56piepsilon_0)(2hati-hatj+3hatk)`
B. `(q_1q_2)/(56sqrt(14)epsilon_0)(2hati-hatj-3hatk)`
C. `(q_1q_2)/(56piepsilon_0)(hati-hatj-3hatk)`
D. `(q_1q_2)/(56sqrt(14)piepsilon_0)(-2hati+hatj-3hatk)`

Answer

To find the electrostatic force on a charged particle \( q_2 \) 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: Determine the Position Vectors
The position of \( q_1 \) is given as \( \vec{r_1} = (2, -1, 3) \) and the position of \( q_2 \) is \( \vec{r_2} = (0, 0, 0) \).

### Step 2: Calculate the Displacement Vector
The displacement vector \( \vec{r} \) from \( q_1 \) to \( q_2 \) is calculated as:
\[ \vec{r} = \vec{r_2} - \vec{r_1} = (0 - 2) \hat{i} + (0 - (-1)) \hat{j} + (0 - 3) \hat{k} \]
\[ \vec{r} = -2 \hat{i} + 1 \hat{j} - 3 \hat{k} \]

### Step 3: Calculate the Magnitude of the Displacement Vector
The magnitude \( r \) of the displacement vector \( \vec{r} \) is given by:
\[ r = |\vec{r}| = \sqrt{(-2)^2 + (1)^2 + (-3)^2} = \sqrt{4 + 1 + 9} = \sqrt{14} \]

### Step 4: Determine the Unit Vector
The unit vector \( \hat{r} \) in the direction of \( \vec{r} \) is:
\[ \hat{r} = \frac{\vec{r}}{|\vec{r}|} = \frac{-2 \hat{i} + 1 \hat{j} - 3 \hat{k}}{\sqrt{14}} \]

### Step 5: Apply Coulomb's Law
According to Coulomb's law, the electrostatic force \( \vec{F} \) between two point charges is given by:
\[ \vec{F} = k \frac{q_1 q_2}{r^2} \hat{r} \]
where \( k = \frac{1}{4 \pi \epsilon_0} \).

### Step 6: Substitute Values
Substituting the values:
\[ \vec{F} = k \frac{q_1 q_2}{(\sqrt{14})^2} \hat{r} = k \frac{q_1 q_2}{14} \hat{r} \]
Substituting \( \hat{r} \):
\[ \vec{F} = k \frac{q_1 q_2}{14} \left(\frac{-2 \hat{i} + 1 \hat{j} - 3 \hat{k}}{\sqrt{14}}\right) \]
\[ \vec{F} = \frac{k q_1 q_2}{14 \sqrt{14}} (-2 \hat{i} + 1 \hat{j} - 3 \hat{k}) \]

### Step 7: Final Expression
Thus, the electrostatic force on \( q_2 \) due to \( q_1 \) is:
\[ \vec{F} = \frac{k q_1 q_2}{14 \sqrt{14}} (-2 \hat{i} + 1 \hat{j} - 3 \hat{k}) \]