Step 1: Understanding the Concept:
Electrostatic force between two charges is given by Coulomb's law: \( F = k \frac{q_1 q_2}{r^2} \). For a fixed total charge and distance, the force is maximum when the product of the individual charges is maximum.
: Key Formula or Approach:
1. Total charge \( q = q_1 + q_2 \implies q_2 = q - q_1 \).
2. Force \( F \propto q_1 q_2 = q_1(q - q_1) \).
3. To find maximum force, use differentiation: \( \frac{dF}{dq_1} = 0 \).
Step 2: Detailed Explanation:
Let \( P = q_1 q_2 = q_1(q - q_1) = q q_1 - q_1^2 \).
Differentiating with respect to \( q_1 \):
\[ \frac{dP}{dq_1} = q - 2q_1 \]
Set the derivative to zero for maximum value:
\[ q - 2q_1 = 0 \implies q_1 = \frac{q}{2} \]
Substituting back to find \( q_2 \):
\[ q_2 = q - q_1 = q - \frac{q}{2} = \frac{q}{2} \]
The required ratio is:
\[ \frac{q_1}{q_2} = \frac{q/2}{q/2} = 1 \]
Note: While the provided PDF solution key marks (B) and calculates \( q_1/q = 0.5 \), the question specifically asks for the ratio of the two charges, which is 1.
Step 3: Final Answer:
The ratio \( q_1/q_2 \) for maximum force is 1.