Question

A charge \(Q\) is divided into two parts and placed at a given distance so that the force between them is maximum. The value of each charge is:

• A. \(\frac{Q}{2},\frac{Q}{2}\)
• B. \(\frac{3Q}{4},\frac{Q}{4}\)
• C. \(\frac{Q}{3},\frac{2Q}{3}\)
• D. \(\frac{2Q}{3},\frac{Q}{3}\)

Hint:

The force is proportional to the product of the two charges, which is largest when they are equal.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
Let the parts be \(q\) and \(Q-q\). At fixed distance \(r\), \(F=\dfrac{1}{4\pi\varepsilon_{0}}\dfrac{q(Q-q)}{r^{2}}\).

Step 2: Maximise
\(F\) is largest when \(q(Q-q)\) is largest. Differentiating, \(Q-2q=0\), so \(q=\dfrac{Q}{2}\).

Step 3: Check
The second derivative is \(-2\), negative, so it is a maximum. Each part is \(\dfrac{Q}{2}\).

Final Answer:
Each part should be Q by 2. \[ \boxed{\text{(A) }\dfrac{Q}{2},\dfrac{Q}{2}} \]