Charge `Q` is divided into two parts which are then kept some distance apart . The force between them will be maximum if the two parts are

Options

A. `(Q)/(2)` and `(Q)/(2)`
B. `(Q)/(2)` and `(3Q)/(4)`
C. `(Q)/(2)` and `(2Q)/(3)`
D. `e` and `(Q - e)`

Explanation

To solve the problem of maximizing the electrostatic force between two charges obtained by dividing a total charge \( Q \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have a total charge \( Q \) that is divided into two parts, \( q \) and \( Q - q \). We need to find the values of \( q \) and \( Q - q \) such that the electrostatic force \( F \) between them is maximized when they are separated by a distance \( r \).

2. **Electrostatic Force Formula**:
The electrostatic force \( F \) between two point charges is given by Coulomb's law:
\[ F = k \frac{q (Q - q)}{r^2} \]
where \( k \) is Coulomb's constant, \( q \) is one part of the charge, and \( Q - q \) is the other part.

3. **Rewriting the Force**:
We can express the force as:
\[ F = k \frac{q (Q - q)}{r^2} = \frac{k}{r^2} (qQ - q^2) \]
Here, \( \frac{k}{r^2} \) is a constant since \( r \) is fixed.

4. **Finding the Maximum Force**:
To find the maximum force, we need to differentiate \( F \) with respect to \( q \) and set the derivative equal to zero:
\[ \frac{dF}{dq} = \frac{k}{r^2} (Q - 2q) = 0 \]

5. **Solving the Derivative**:
Setting the derivative to zero gives:
\[ Q - 2q = 0 \]
Solving for \( q \) yields:
\[ 2q = Q \implies q = \frac{Q}{2} \]

6. **Finding the Other Charge**:
The other part of the charge is:
\[ Q - q = Q - \frac{Q}{2} = \frac{Q}{2} \]

7. **Conclusion**:
Thus, to maximize the force between the two charges, both charges should be equal:
\[ q = \frac{Q}{2} \quad \text{and} \quad Q - q = \frac{Q}{2} \]

### Final Answer:
The two parts should be \( \frac{Q}{2} \) and \( \frac{Q}{2} \) for the force between them to be maximum.

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