Determine the potential difference V_a - V_b between the points a and b for the circuit labeled (a) in the given figure.

Step-by-step solution

In the circuit labeled (a), let the potential at the bottom wire be V_b = 0 . The 12 V battery fixes the potential at the node directly above it to 12 V . Let the potential at the node between the 12 V battery and node a be x . Applying the principle of charge conservation at node x yields 2(x - 12) + 4(x - 0) + 2(x - V_a) = 0 , which simplifies to 8x - 24 - 2V_a = 0 or 4x - V_a = 12 . Applying charge conservation at node a gives 2(V_a - x) + 4(V_a - 0) = 0 , resulting in 6V_a - 2x = 0 , or x = 3V_a . Substituting