Charge Division for Maximum Repulsion – Detailed Solution

Charge Division for Maximum Repulsion

Question: A certain charge \(Q\) is divided into two parts \(q\) and \(Q-q\). How should the charges be divided so that, when placed a fixed distance apart, they experience maximum electrostatic repulsion?

Possible Answers:

• (A) \( Q = \frac{q}{2} \)
• (B) \( Q = 2q \)
• (C) \( Q = 4q \)
• (D) \( Q = 3q \)

Select the Correct Option

(A) \( Q = \frac{q}{2} \)

(B) \( Q = 2q \)

(C) \( Q = 4q \)

(D) \( Q = 3q \)

Step‑by‑Step Explanation

1. Formulate the Expression for Electrostatic Force:
   According to Coulomb's law, the force between two charges is given by: \[ F = k \frac{q(Q-q)}{r^2}, \] where \(k\) and \(r\) are constants for this problem. Thus, for maximum force, we maximize the product: \[ f(q) = q(Q-q). \]
2. Maximize the Function \(f(q)\):
   Write: \[ f(q) = Qq - q^2. \] Differentiate \(f(q)\) with respect to \(q\): \[ \frac{df}{dq} = Q - 2q. \] Setting this derivative equal to zero for a maximum: \[ Q - 2q = 0 \quad \Longrightarrow \quad q = \frac{Q}{2}. \]
3. Conclude the Charge Division:
   If \(q = \frac{Q}{2}\), then the other part is: \[ Q - q = Q - \frac{Q}{2} = \frac{Q}{2}. \] So the charges are equal. This condition can also be written as: \[ Q = 2q. \]
4. Final Answer: The maximum electrostatic repulsion is achieved when the charge is divided equally, i.e., when \(Q = 2q\), which corresponds to option (B).