To solve this circuit using nodal analysis, we need to set up equations for each node based on Kirchhoff's Current Law (KCL), which states that the sum of currents entering a node equals the sum of currents leaving that node. We’ll denote the node voltages as $V_1$, $V_2$, and $V_3$ as specified.

Step-by-Step Solution

1. Define the Nodes and Currents:

   • $V_1$: Voltage at the left node.
   • $V_2$: Voltage at the middle node (with the 12V source).
   • $V_3$: Voltage at the right node.

2. Apply KCL at Each Node:

   • Node 1 ($V_1$): $\frac{V_1 - V_2}{4} = 2i$ where $i$ is the current through the $4 \Omega$ resistor on the right side.

   • Node 2 ($V_2$): The 12V source is connected between ground and $V_2$, so: $V_2 = 12 \text{ V}$

   • Node 3 ($V_3$): At node $V_3$, we write KCL: $\frac{V_3 - V_2}{2} + \frac{V_3}{4} = i$

3. Substitute Known Values and Solve: Substitute $V_2 = 12$ V into the equations for $V_1$ and $V_3$ and solve for $V_1$, $V_3$, and $i$.

Would you like a detailed solution with calculations for each step?

Related Questions

1. How would the results change if the 12V source was replaced by another value?
2. What would happen if the controlled source $2i$ were removed?
3. How can we verify the solution using mesh analysis?
4. What effect does changing the resistor values have on node voltages?
5. Can this circuit be analyzed using Thevenin’s theorem?

Tip

Always double-check each step in nodal analysis to ensure each term reflects the right direction and values for currents and voltages.