Two insulated charged metallic spheres P and Q have their centres separated by a distance of 60 cm. The radii of P and Q are negligible compared to the distance of separation. The mutual force of electrostatic repulsion if the charge on each is `3.2xx10^(-7)C` is

Options

A. `5.2xx10^(-4)N`
B. `2.5xx10^(-3)N`
C. `1.5xx10^(-3)N`
D. `3.5xx10^(-4)N`

Explanation

To solve the problem of finding the mutual force of electrostatic repulsion between two charged metallic spheres P and Q, we can use Coulomb's Law. Here’s the step-by-step solution:

### Step 1: Understand the given data
- Distance between the centers of the spheres (r) = 60 cm = 0.6 m (conversion from cm to m)
- Charge on each sphere (q1 and q2) = \(3.2 \times 10^{-7} \, C\)

### Step 2: Write down Coulomb's Law
Coulomb's Law states that the electrostatic force (F) between two point charges is given by the formula:
\[ F = \frac{1}{4 \pi \epsilon_0} \cdot \frac{q_1 \cdot q_2}{r^2} \]
where:
- \(F\) = electrostatic force between the charges,
- \(q_1\) and \(q_2\) = magnitudes of the charges,
- \(r\) = distance between the centers of the charges,
- \(\epsilon_0\) = permittivity of free space = \(8.85 \times 10^{-12} \, C^2/(N \cdot m^2)\).

### Step 3: Substitute the values into the formula
Since both spheres have the same charge, we can set \(q_1 = q_2 = q = 3.2 \times 10^{-7} \, C\). Thus, we can rewrite the formula as:
\[ F = \frac{1}{4 \pi \epsilon_0} \cdot \frac{q^2}{r^2} \]

Substituting the known values:
\[ F = \frac{1}{4 \pi (8.85 \times 10^{-12})} \cdot \frac{(3.2 \times 10^{-7})^2}{(0.6)^2} \]

### Step 4: Calculate \(q^2\) and \(r^2\)
Calculate \(q^2\):
\[ q^2 = (3.2 \times 10^{-7})^2 = 1.024 \times 10^{-13} \, C^2 \]
Calculate \(r^2\):
\[ r^2 = (0.6)^2 = 0.36 \, m^2 \]

### Step 5: Substitute these values into the formula
Now substitute \(q^2\) and \(r^2\) into the force equation:
\[ F = \frac{1}{4 \pi (8.85 \times 10^{-12})} \cdot \frac{1.024 \times 10^{-13}}{0.36} \]

### Step 6: Calculate the force
First, calculate the denominator:
\[ 4 \pi (8.85 \times 10^{-12}) \approx 1.11265 \times 10^{-10} \]
Now calculate:
\[ F = \frac{1.024 \times 10^{-13}}{0.36} \approx 2.8444 \times 10^{-13} \]
Finally, calculate the force:
\[ F \approx \frac{2.8444 \times 10^{-13}}{1.11265 \times 10^{-10}} \approx 2.55 \times 10^{-3} \, N \]

### Final Answer
The mutual force of electrostatic repulsion between the two spheres is approximately \(2.55 \times 10^{-3} \, N\).