Two identical small conducting spheres having unequal positive charges `q_1` and `q_2` are separated by a distance r. If they are now made to touch each other and then separated again to the same distance, the electrostatic force between them in this case will be

Options

A. less than before
B. same as before
C. more than before
D. zero

Explanation

To solve the problem, we will follow these steps:

### Step 1: Calculate the initial electrostatic force between the two spheres.
Given:
- Charges on the spheres: \( q_1 = 2 \, \mu C = 2 \times 10^{-6} \, C \) and \( q_2 = 4 \, \mu C = 4 \times 10^{-6} \, C \)
- Distance between the spheres: \( r \)
- Coulomb's constant: \( k = 9 \times 10^9 \, N \cdot m^2/C^2 \)

The electrostatic force \( F \) between the two spheres can be calculated using Coulomb's law:
\[ F = \frac{k \cdot |q_1 \cdot q_2|}{r^2} \]
Substituting the values:
\[ F = \frac{9 \times 10^9 \cdot (2 \times 10^{-6}) \cdot (4 \times 10^{-6})}{r^2} \]
\[ F = \frac{9 \times 10^9 \cdot 8 \times 10^{-12}}{r^2} \]
\[ F = \frac{72 \times 10^{-3}}{r^2} \, N \]

### Step 2: Determine the new charges after the spheres touch each other.
When two identical conducting spheres touch each other, they share their total charge equally. The total charge \( Q \) is:
\[ Q = q_1 + q_2 = 2 \times 10^{-6} + 4 \times 10^{-6} = 6 \times 10^{-6} \, C \]
After touching, each sphere will have:
\[ Q' = \frac{Q}{2} = \frac{6 \times 10^{-6}}{2} = 3 \times 10^{-6} \, C \]

### Step 3: Calculate the new electrostatic force after separation.
Now, both spheres have the same charge \( Q' = 3 \, \mu C = 3 \times 10^{-6} \, C \). The new electrostatic force \( F' \) can be calculated as:
\[ F' = \frac{k \cdot |Q' \cdot Q'|}{r^2} \]
Substituting the values:
\[ F' = \frac{9 \times 10^9 \cdot (3 \times 10^{-6}) \cdot (3 \times 10^{-6})}{r^2} \]
\[ F' = \frac{9 \times 10^9 \cdot 9 \times 10^{-12}}{r^2} \]
\[ F' = \frac{81 \times 10^{-3}}{r^2} \, N \]

### Step 4: Compare the initial and new forces.
We have:
- Initial force \( F = \frac{72 \times 10^{-3}}{r^2} \)
- New force \( F' = \frac{81 \times 10^{-3}}{r^2} \)

Since \( F' > F \), the electrostatic force between the spheres after they touch and are separated again is greater than the initial force.

### Final Answer:
The electrostatic force between the spheres after they touch and are separated again is greater than the initial force.

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