Two identical small dust particles, each of mass 3.0 10⁻⁶ kg , are separated by a certain distance in free space. They are given identical charges q such that the electrostatic repulsion between them is exactly 10¹² times their gravitational attraction. What is the approximate value of the charge q on each particle? (Given: G = 6.67 10⁻¹¹ N m ^2 kg ⁻² , 1 4 ₀ = 9 10^9 N m ^2 C ⁻² )

Step-by-step solution

The electrostatic force of repulsion between the two particles is given by Coulomb's law: F_e = 1 4 ₀ q^2 r^2 The gravitational force of attraction between them is given by Newton's law of gravitation: F_g = G m^2 r^2 Given that F_e = 10¹² F_g , we have: 1 4 ₀ q^2 r^2 = 10¹² G m^2 r^2 Substituting the given values: 9 10^9 q^2 = 10¹² 6.67 10⁻¹¹ (3.0 10⁻⁶)^2 9 10^9 q^2 = 10¹² 6.67 10⁻¹¹ 9.0 10⁻¹² 9 10^9 q^2 = 6.67 9 10⁻¹¹ q^2 = 6.67 9 10⁻¹¹ 9 10^9 q^2 = 6.67 10⁻²⁰ Taking the square root on both sides: q = 6.67 10⁻¹⁰