Question

Which one of the following pairs of charges separated by the same distance r will experience a maximum force?

• A. 0.3 C and 0.7 C
• B. 0.1 C and 0.9 C
• C. 0.2 C and 0.8 C
• D. 0.5 C and 0.5 C
• E. 0.4 C and 0.6 C

Hint:

For a fixed sum, the product is maximum when the two values are equal ($q_1 = q_2$).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This problem applies Coulomb's Law, which describes the electrostatic force between two point charges. The question is an optimization problem: we need to maximize the force, which means we need to maximize the product of the charges, given that their sum is constant in each case.
Step 2: Key Formula or Approach:
1. Coulomb's Law: The magnitude of the electrostatic force (F) between two point charges \( q_1 \) and \( q_2 \) separated by a distance r is: \[ F = k \frac{|q_1 q_2|}{r^2} \] where k is Coulomb's constant. 2. To maximize the force F for a fixed distance r, we need to maximize the product of the magnitudes of the charges, \( |q_1 q_2| \). 3. Notice that in all the given options, the sum of the two charges is the same: - (A) 0.3 + 0.7 = 1.0 - (B) 0.1 + 0.9 = 1.0 - (C) 0.2 + 0.8 = 1.0 - (D) 0.5 + 0.5 = 1.0 - (E) 0.4 + 0.6 = 1.0 So, we have a fixed sum \( q_1 + q_2 = 1 \text{ C} \), and we want to maximize the product \( q_1 q_2 \). Step 3: Detailed Explanation:
This is the same mathematical problem as in question 63. For a fixed sum, the product of two positive numbers is maximized when the numbers are equal. Let \( q_1 + q_2 = S \). We want to maximize \( P = q_1 q_2 \). Let \( q_1 = x \), then \( q_2 = S - x \). The product is \( P(x) = x(S-x) = Sx - x^2 \). To find the maximum, we take the derivative and set it to zero: \[ \frac{dP}{dx} = S - 2x = 0 \implies x = \frac{S}{2} \] So, \( q_1 = S/2 \). Then \( q_2 = S - S/2 = S/2 \). The product is maximized when \( q_1 = q_2 \). In our case, the sum is \( S=1.0 \text{ C} \). The product is maximized when: \[ q_1 = q_2 = \frac{1.0}{2} = 0.5 \text{ C} \] Let's check the products for all options: - (A) 0.3 × 0.7 = 0.21 - (B) 0.1 × 0.9 = 0.09 - (C) 0.2 × 0.8 = 0.16 - (D) 0.5 × 0.5 = 0.25 - (E) 0.4 × 0.6 = 0.24 The largest product is indeed 0.25, which corresponds to the charges 0.5 C and 0.5 C. Step 4: Final Answer:
The pair of charges 0.5 C and 0.5 C will experience the maximum force.