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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PROBLEM 1 Step 1: Understand the relationship between forces. According to Newton's Third Law, when two objects exert forces on each other, these forces are equal in magnitude and opposite in direction. This applies to electrostatic forces as well. The force exerted by object A on object B (F_AB) is equal in magnitude and opposite in direction to the force exerted by object B on object A (F_BA). Step 2: Apply Newton's Third Law. The mathematical representation of this relationship is: F_AB = -F_BA The magnitudes of the charges (+2 mC and +6 mC) determine the magnitude of the force, but they do not change this fundamental vector relationship. The correct statement is F_AB = -F_BA. PROBLEM 2 Step 1: State Coulomb's Law for the initial force. The initial force F_1 between two charges q_1 and q_2 separated by a distance d is given by Coulomb's Law: F_1 = k (|q_1 q_2|)/(d^2) where k is Coulomb's constant. Step 2: Determine the new force when q_1 is tripled. If q_1 is tripled, the new charge is q_1' = 3q_1. The new force F_2 will be: F_2 = k (|(3q_1) q_2|)/(d^2) Step 3: Compare the new force to the initial force. We can rewrite F_2 by factoring out the constant 3: F_2 = 3 · k (|q_1 q_2|)/(d^2) Since F_1 = k (|q_1 q_2|)/(d^2), we can substitute F_1 into the equation for F_2: F_2 = 3F_1 The new force will be three times the original force. The new force F_2 will be 3F_1. PROBLEM 3 Given values: Distance r = 5.3 × 10^-11 m Elementary charge e = 1.602 × 10^-19 C Mass of electron m_e = 9.109 × 10^-31 kg Mass of proton m_p = 1.672 × 10^-27 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 a) Find the magnitudes of the electric force and the gravitational force, and their ratio. Step 1: Calculate the electric force (F_e). The electric force between the electron (q_e = -e) and proton (q_p = +e) is given by Coulomb's Law: F_e = k (|q_e q_p|)/(r^2) = k (e^2)/(r^2) F_e = (8.987 × 10^9 N · m^2/C^2) (1.602 × 10^-19 C)^2(5.3 × 10^-11 m)^2 F_e = (8.987 × 10^9) 2.566404 × 10^-382.809 × 10^-21 N F_e = 8.24 × 10^-8 N Step 2: Calculate the gravitational force (F_g). The gravitational force between the electron and proton is given by Newton's Law of Universal Gravitation: F_g = G (m_e m_p)/(r^2) F_g = (6.674 × 10^-11 N · m^2/kg^2) (9.109 × 10^-31 kg)(1.672 × 10^-27 kg)(5.3 × 10^-11 m)^2 F_g = (6.674 × 10^-11) 1.523 × 10^-572.809 × 10^-21 N F_g = 3.63 × 10^-47 N Step 3: Calculate the ratio of the electric force to the gravitational force (F_e / F_g). (F_e)/(F_g) = 8.24 × 10^-8 N3.63 × 10^-47 N (F_e)/(F_g) = 2.27 × 10^39 The electric force is 8.24 × 10^-8 N, the gravitational force is 3.63 × 10^-47 N, and their ratio is 2.27 × 10^39. b) Compute the acceleration caused by the electric force of the proton on the electron. Step 4: Calculate the acceleration of the electron (a_e). Using Newton's Second Law, F_e = m_e a_e, the acceleration of the electron due to the electric force is: a_e = (F_e)/(m_e) a_e = 8.24 × 10^-8 N9.109 × 10^-31 kg a_e = 9.05 × 10^22 m/s^2 The acceleration caused by the electric force on the electron is 9.05 × 10^22 m/s^2. PROBLEM 4 Given values: Charge q_1 = 4.3 µC = 4.3 × 10^-6 C Charge q_2 = 4.3 µC = 4.3 × 10^-6 C Distance r = 0.25 m Coulomb's constant k = 8.987 × 10^9 N · m^2/C^2 Step 1: Determine the type of force. Since both charges are positive (+4.3 µC), they will repel each other. Step 2: Calculate the magnitude of the force using Coulomb's Law. F = k (|q_1 q_2|)/(r^2) F = (8.987 × 10^9 N · m^2/C^2) (4.3 × 10^-6 C)(4.3 × 10^-6 C)(0.25 m)^2 F = (8.987 × 10^9) 1.849 × 10^-110.0625 N F = (8.987 × 10^9) (2.9584 × 10^-10) N F = 2.66 N The size of the force is 2.66 N and the type of force is repulsive. Drop the next question.