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 given the charges and the electrostatic force, we can use Coulomb's Law, which states:

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

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

### Step 1: Identify the given values
- Charge \( q_1 = 0.2 \, \mu C = 0.2 \times 10^{-6} \, C = 2 \times 10^{-7} \, C \)
- Charge \( q_2 = -0.4 \, \mu C = -0.4 \times 10^{-6} \, C = -4 \times 10^{-7} \, C \)
- Force \( F = 0.4 \, N \)

### Step 2: Substitute the values into Coulomb's Law
Using the absolute values of the charges in Coulomb's Law:

\[ 0.4 = 8.99 \times 10^9 \frac{(2 \times 10^{-7})(4 \times 10^{-7})}{r^2} \]

### Step 3: Simplify the equation
Calculate the product of the charges:

\[ (2 \times 10^{-7})(4 \times 10^{-7}) = 8 \times 10^{-14} \]

Now substitute this back into the equation:

\[ 0.4 = 8.99 \times 10^9 \frac{8 \times 10^{-14}}{r^2} \]

### Step 4: Rearrange the equation to solve for \( r^2 \)
Multiply both sides by \( r^2 \) and divide by \( 0.4 \):

\[ r^2 = 8.99 \times 10^9 \frac{8 \times 10^{-14}}{0.4} \]

### Step 5: Calculate the right-hand side
First, calculate \( \frac{8 \times 10^{-14}}{0.4} \):

\[ \frac{8 \times 10^{-14}}{0.4} = 2 \times 10^{-13} \]

Now substitute this back into the equation:

\[ r^2 = 8.99 \times 10^9 \times 2 \times 10^{-13} \]

### Step 6: Final calculation
\[ r^2 = 17.98 \times 10^{-4} = 1.798 \times 10^{-3} \]

Now take the square root to find \( r \):

\[ r = \sqrt{1.798 \times 10^{-3}} \approx 0.0424 \, m \]

### Step 7: Convert to centimeters
To convert meters to centimeters, multiply by 100:

\[ r \approx 4.24 \, cm \]

### Final Answer
The distance between the two spheres is approximately \( 4.24 \, cm \).

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