Question

The sum of two point positive charges separated by a distance of 1.5 m in air is \( 25 \, \mu\text{C} \). If the electrostatic force between the two charges is 0.6 N, then the difference between the two charges is

• A. \( 5 \, \mu\text{C} \)
• B. \( 8 \, \mu\text{C} \)
• C. \( 3 \, \mu\text{C} \)
• D. \( 6 \, \mu\text{C} \)
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Hint:

Solve for charge differences using \( (q_1 - q_2)^2 = (q_1 + q_2)^2 - 4 q_1 q_2 \) with Coulomb’s law.

Answer 1

The Correct Option is A

Let the charges be \( q_1 \) and \( q_2 \), with: \[ q_1 + q_2 = 25 \times 10^{-6} \, \text{C} = 25 \, \mu\text{C} \] The electrostatic force is given by Coulombs law: \[ F = \frac{k q_1 q_2}{r^2} \] where \( k = 9 \times 10^9 \, \text{N·m}^2/\text{C}^2 \), \( F = 0.6 \, \text{N} \), and \( r = 1.5 \, \text{m} \). Compute: \[ q_1 q_2 = \frac{F r^2}{k} = \frac{0.6 \cdot (1.5)^2}{9 \times 10^9} = \frac{0.6 \cdot 2.25}{9 \times 10^9} = \frac{1.35}{9 \times 10^9} = 1.5 \times 10^{-10} \, \text{C}^2 \] Use the identity for the difference of charges: \[ (q_1 - q_2)^2 = (q_1 + q_2)^2 - 4 q_1 q_2 \] Calculate: \[ (q_1 + q_2)^2 = (25 \times 10^{-6})^2 = 625 \times 10^{-12} \, \text{C}^2 \] \[ 4 q_1 q_2 = 4 \cdot 1.5 \times 10^{-10} = 6 \times 10^{-10} = 600 \times 10^{-12} \, \text{C}^2 \] \[ (q_1 - q_2)^2 = 625 \times 10^{-12} - 600 \times 10^{-12} = 25 \times 10^{-12} \, \text{C}^2 \] \[ |q_1 - q_2| = \sqrt{25 \times 10^{-12}} = 5 \times 10^{-6} \, \text{C} = 5 \, \mu\text{C} \] Solve for \( q_1, q_2 \): \[ q_1 + q_2 = 25, \quad q_1 q_2 = \frac{1.5 \times 10^{-10}}{10^{-12}} = 150 \] Quadratic equation: \( t^2 - (q_1 + q_2)t + q_1 q_2 = 0 \): \[ t^2 - 25t + 150 = 0 \] Discriminant: \[ \Delta = 25^2 - 4 \cdot 150 = 625 - 600 = 25 \] Roots: \[ t = \frac{25 \pm \sqrt{25}}{2} = \frac{25 \pm 5}{2} \] \[ t_1 = \frac{30}{2} = 15, \quad t_2 = \frac{20}{2} = 10 \] Thus, \( q_1 = 15 \, \mu\text{C} \), \( q_2 = 10 \, \mu\text{C} \), and \( |q_1 - q_2| = 15 - 10 = 5 \, \mu\text{C} \). Check options: - (1) 5 \( \mu\text{C} \): Correct. - (2) 8 \( \mu\text{C} \): Incorrect. - (3) 3 \( \mu\text{C} \): Incorrect. - (4) 6 \( \mu\text{C} \): Incorrect. Option (1) is correct.