This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.
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You're on a roll — To find the ratio of the Coulomb electric force (F_E) to the gravitational force (F_G) between two electrons in a vacuum, we will use Coulomb's Law and Newton's Law of Universal Gravitation. Given constants: • Charge of an electron, e = 1.602 × 10^-19 C • Mass of an electron, m_e = 9.109 × 10^-31 kg • Coulomb's constant, k = 8.987 × 10^9 N m^2/C^2 • Gravitational constant, G = 6.674 × 10^-11 N m^2/kg^2 Let r be the distance between the two electrons. Step 1: Calculate the Coulomb electric force (F_E) between two electrons. The formula for the electric force between two point charges q_1 and q_2 separated by a distance r is: F_E = (k q_1 q_2)/(r^2) For two electrons, q_1 = q_2 = e. F_E = (k e^2)/(r^2) Step 2: Calculate the gravitational force (F_G) between two electrons. The formula for the gravitational force between two masses m_1 and m_2 separated by a distance r is: F_G = (G m_1 m_2)/(r^2) For two electrons, m_1 = m_2 = m_e. F_G = (G m_e^2)/(r^2) Step 3: Find the ratio (F_E)/(F_G). (F_E)/(F_G) = (k e^2)/(r^2)(G m_e^2)/(r^2) The r^2 terms cancel out: (F_E)/(F_G) = (k e^2)/(G m_e^2) Step 4: Substitute the values and calculate the ratio. (F_E)/(F_G) = (8.987 × 10^9 N m^2/C^2) (1.602 × 10^-19 C)^2(6.674 × 10^-11 N m^2/kg^2) (9.109 × 10^-31 kg)^2 (F_E)/(F_G) = (8.987 × 10^9) (2.566404 × 10^-38)(6.674 × 10^-11) (82.973881 × 10^-62) (F_E)/(F_G) = 2.306 × 10^-285.538 × 10^-72 (F_E)/(F_G) ≈ 4.164 × 10^43 The ratio of the Coulomb electric force to the gravitational force between two electrons in a vacuum is approximately 4.164 × 10^43. 4.164 × 10^43 Got more? Send 'em!