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.
ScanSolve AI Answer
Step 1: The magnitude of the electrostatic force between an electron and a proton is F_e = (k q_e^2)/(r^2), and the gravitational force is F_g = (G m_e m_p)/(r^2). The ratio is (F_e)/(F_g) = (k q_e^2)/(G m_e m_p) where k = 9 × 10^9 \, N · m^2 / C^2, q_e = 1.6 × 10^-19 \, C, G = 6.7 × 10^-11 \, N · m^2 / kg^2, m_e = 9.1 × 10^-31 \, kg, m_p = 1.67 × 10^-27 \, kg. Step 2: Compute q_e^2: q_e^2 = (1.6 × 10^-19)^2 = 2.56 × 10^-38 \, C^2. Numerator: k q_e^2 = 9 × 10^9 × 2.56 × 10^-38 = 23.04 × 10^-29 = 2.304 × 10^-28 \, N · m^2. Step 3: Compute m_e m_p: m_e m_p = 9.1 × 10^-31 × 1.67 × 10^-27 = 15.197 × 10^-58 = 1.52 × 10^-57 \, kg^2. Denominator: G m_e m_p = 6.7 × 10^-11 × 1.52 × 10^-57 = 10.184 × 10^-68 = 1.02 × 10^-67 \, N · m^2. Step 4: Compute the ratio: (F_e)/(F_g) = 2.304 × 10^-281.02 × 10^-67 = (2.304)/(1.02) × 10^-28 - (-67) = 2.26 × 10^39 ≈ 2.3 × 10^39. (b) 2.3 × 10^39 (b)