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: Identify nodes and assign a reference node. We choose the bottom wire as the reference node, so its voltage is 0\,V. There is one unknown node voltage, which we will call V_X. This is the top node where resistors R_1, R_2, and R_3 connect. The voltage at the left end of R_1 is 28\,V (due to battery B_1). The voltage at the right end of R_3 is 7\,V (due to battery B_2). Step 2: Apply Kirchhoff's Current Law (KCL) at node V_X. According to KCL, the sum of currents entering a node must be equal to the sum of currents leaving the node. From the diagram, currents I_1 and I_3 are entering node V_X, and current I_2 is leaving node V_X. I_1 + I_3 = I_2 Now, express these currents using Ohm's Law (I = (V)/(R)): I_1 flows from 28\,V to V_X through R_1 = 4\,: I_1 = 28\,V - V_X4\, I_2 flows from V_X to 0\,V through R_2 = 2\,: I_2 = V_X - 0\,V2\, = (V_X)/(2\,) I_3 flows from 7\,V to V_X through R_3 = 1\,: I_3 = 7\,V - V_X1\, Substitute these expressions into the KCL equation: (28 - V_X)/(4) + (7 - V_X)/(1) = (V_X)/(2) Step 3: Solve the equation for V_X. To eliminate the denominators, multiply the entire equation by the least common multiple of 4, 1, and 2, which is 4: 4 ( (28 - V_X)/(4) ) + 4 ( (7 - V_X)/(1) ) = 4 ( (V_X)/(2) ) (28 - V_X) + 4(7 - V_X) = 2V_X Distribute the 4: 28 - V_X + 28 - 4V_X = 2V_X Combine like terms: (28 + 28) + (-V_X - 4V_X) = 2V_X 56 - 5V_X = 2V_X Add 5V_X to both sides: 56 = 2V_X + 5V_X 56 = 7V_X Divide by 7: V_X = (56)/(7) V_X = 8\,V The voltage at the top node is 8\,V. The final answer is 8 V.