Question

Given that \(q_1 + q_2 = q\). For what ratio \(q_1/q_2\) will the force between \(q_1\) and \(q_2\) be maximum ?

• A. 0.25
• B. 0.5
• C. 1
• D. 2

Hint:

Express ratio as a variable and optimize the function carefully.

Answer 1

The Correct Option is B

Concept: \[ F = k \frac{q_1 q_2}{r^2} \]

Step 1: Assume ratio.
Let: \[ \frac{q_1}{q_2} = x \Rightarrow q_1 = x q_2 \]

Step 2: Use sum condition.
\[ x q_2 + q_2 = q \Rightarrow q_2 = \frac{q}{x+1} \] \[ q_1 = \frac{xq}{x+1} \]

Step 3: Force expression.
\[ F \propto q_1 q_2 = \frac{xq}{x+1} \cdot \frac{q}{x+1} = \frac{xq^2}{(x+1)^2} \]

Step 4: Maximize.
Maximize: \[ \frac{x}{(x+1)^2} \] By inspection (or differentiation), maximum occurs at: \[ x = \frac{1}{2} \] \[ \Rightarrow \frac{q_1}{q_2} = 0.5 \]