Two equal point charges each of `5uC` are separated by a certain distance in metres.. If they are located at (i + j + k) and (2i + 3j +3k), then the electrostatic force between them is

Options

A. `19xx10^3N`
B. `25xx10^-3N`
C. `10^-3N`
D. `9xx10^-2N`

Answer

To find the electrostatic force between two equal point charges of \(5 \mu C\) located at the points \((1, 1, 1)\) and \((2, 3, 3)\), we will follow these steps:

### Step 1: Determine the positions of the charges
The positions of the charges are given as:
- Charge 1: \( \vec{r_1} = (1, 1, 1) \)
- Charge 2: \( \vec{r_2} = (2, 3, 3) \)

### Step 2: Calculate the distance between the charges
The distance \(R\) between the two charges can be calculated using the distance formula in three-dimensional space:
\[ R = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]
Substituting the coordinates:
\[ R = \sqrt{(2 - 1)^2 + (3 - 1)^2 + (3 - 1)^2} \]
Calculating each term:
\[ R = \sqrt{(1)^2 + (2)^2 + (2)^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3 \text{ meters} \]

### Step 3: Use Coulomb's Law to calculate the electrostatic force
Coulomb's Law states that the force \(F\) between two point charges is given by:
\[ F = k \frac{q_1 q_2}{R^2} \]
Where:
- \(k\) is Coulomb's constant, \(k = 9 \times 10^9 \, \text{N m}^2/\text{C}^2\)
- \(q_1 = q_2 = 5 \mu C = 5 \times 10^{-6} \, C\)
- \(R = 3 \, m\)

Substituting the values into the formula:
\[ F = 9 \times 10^9 \frac{(5 \times 10^{-6})(5 \times 10^{-6})}{(3)^2} \]
Calculating \(R^2\):
\[ R^2 = 3^2 = 9 \]
Now substituting back:
\[ F = 9 \times 10^9 \frac{25 \times 10^{-12}}{9} \]
Simplifying:
\[ F = 9 \times 10^9 \times \frac{25}{9} \times 10^{-12} \]
\[ F = 25 \times 10^{-3} \, N \]

### Final Answer
The electrostatic force between the two charges is:
\[ F = 0.025 \, N \text{ or } 25 \, mN \]