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: Apply Kirchhoff's Current Law (KCL) at Node V_1. Assume currents leaving the node are positive. The current through the 2 resistor to ground is (V_1 - 0)/(2). The current through the 4 resistor to V_2 is (V_1 - V_2)/(4). The 5A current source is directed from V_1 to V_2, meaning 5A is leaving V_1. So, KCL at V_1: (V_1)/(2) + (V_1 - V_2)/(4) + 5 = 0 Multiply the entire equation by 4 to eliminate denominators: 2V_1 + (V_1 - V_2) + 20 = 0 3V_1 - V_2 + 20 = 0 Equation (1): 3V_1 - V_2 = -20 Step 2: Apply Kirchhoff's Current Law (KCL) at Node V_2. Assume currents leaving the node are positive. The current through the 4 resistor to V_1 is (V_2 - V_1)/(4). The current through the 6 resistor to ground is (V_2 - 0)/(6). The 5A current source is directed from V_1 to V_2, meaning 5A is entering V_2. So, the current leaving V_2 due to this source is -5A. The 10A current source is directed upwards into V_2, meaning 10A is entering V_2. So, the current leaving V_2 due to this source is -10A. So, KCL at V_2: (V_2 - V_1)/(4) + (V_2)/(6) - 5 - 10 = 0 (V_2 - V_1)/(4) + (V_2)/(6) - 15 = 0 Multiply the entire equation by 12 (the least common multiple of 4 and 6) to eliminate denominators: 3(V_2 - V_1) + 2V_2 - 180 = 0 3V_2 - 3V_1 + 2V_2 - 180 = 0 -3V_1 + 5V_2 - 180 = 0 Equation (2): -3V_1 + 5V_2 = 180 Step 3: Solve the system of linear equations. We have two equations: 1) 3V_1 - V_2 = -20 2) -3V_1 + 5V_2 = 180 Add Equation (1) and Equation (2): (3V_1 - V_2) + (-3V_1 + 5V_2) = -20 + 180 4V_2 = 160 V_2 = (160)/(4) V_2 = 40 V Substitute the value of V_2 into Equation (1): 3V_1 - 40 = -20 3V_1 = -20 + 40 3V_1 = 20 V_1 = (20)/(3) V The node voltages are: V_1 = (20)/(3) V V_2 = 40 V