The electrostatic attracting froce on a small sphere of charge `0.2muC` due to another small sphere of charge `-0.4muC` in air is 0.4N. The distance between the two spheres is

Options

A. `43.2xx10^(-6)m`
B. `42.4xx10^(-3)m`
C. `18.1xx10^(-3)m`
D. `19.2xx10^(-6)m`

Explanation

To find the distance between the two spheres, we can use Coulomb's law, which states that the electrostatic force \( F \) between two point charges \( q_1 \) and \( q_2 \) is given by the formula:

\[ F = k \frac{|q_1 q_2|}{r^2} \]

where:
- \( F \) is the electrostatic force,
- \( k \) is Coulomb's constant (\( 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2 \)),
- \( q_1 \) and \( q_2 \) are the charges,
- \( r \) is the distance between the charges.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Charge \( q_1 = 0.2 \, \mu C = 0.2 \times 10^{-6} \, C \)
- Charge \( q_2 = -0.4 \, \mu C = -0.4 \times 10^{-6} \, C \)
- Electrostatic force \( F = 0.4 \, N \)

2. **Convert Charges to Standard Units:**
- \( q_1 = 0.2 \times 10^{-6} \, C = 2 \times 10^{-7} \, C \)
- \( q_2 = -0.4 \times 10^{-6} \, C = -4 \times 10^{-7} \, C \)

3. **Substitute Values into Coulomb's Law:**
\[ 0.4 = k \frac{|(2 \times 10^{-7})(-4 \times 10^{-7})|}{r^2} \]
Since we are interested in the magnitude of the force, we can ignore the negative sign.

4. **Insert the Value of \( k \):**
\[ 0.4 = (8.99 \times 10^9) \frac{(2 \times 10^{-7})(4 \times 10^{-7})}{r^2} \]

5. **Calculate the Product of Charges:**
\[ |(2 \times 10^{-7})(4 \times 10^{-7})| = 8 \times 10^{-14} \]

6. **Rearranging the Equation to Solve for \( r^2 \):**
\[ 0.4 = (8.99 \times 10^9) \frac{8 \times 10^{-14}}{r^2} \]
\[ r^2 = (8.99 \times 10^9) \frac{8 \times 10^{-14}}{0.4} \]

7. **Calculating the Right Side:**
\[ r^2 = (8.99 \times 10^9) \times (2 \times 10^{-13}) = 1.798 \times 10^{-3} \]

8. **Taking the Square Root to Find \( r \):**
\[ r = \sqrt{1.798 \times 10^{-3}} \approx 0.0424 \, m \]

9. **Convert to Millimeters:**
\[ r \approx 42.4 \, mm \]

### Final Answer:
The distance between the two spheres is approximately \( 0.0424 \, m \) or \( 42.4 \, mm \).