To find $V_a$ in this circuit, let’s analyze the circuit step-by-step. This circuit contains two voltage sources (8V and 12V) and multiple resistors connected in a combination of series and parallel arrangements. Here’s the approach:

Step 1: Identify the Loops

1. Loop 1 (Top loop): Contains the 8V source, 125Ω resistor, and 250Ω resistor.
2. Loop 2 (Bottom loop): Contains the 12V source, 500Ω resistor, and $V_a$ resistor (500Ω).
3. Shared Branch: The 125Ω and 250Ω resistors connect the two loops together.

Step 2: Set Up Kirchhoff’s Voltage Law (KVL) Equations

Using Kirchhoff's Voltage Law (KVL), we can set up equations for each loop:

1. For Loop 1:

   • Apply KVL in the clockwise direction, starting from the 8V source: $8V - I_1 \times 125\Omega - I_2 \times 250\Omega = 0$

2. For Loop 2:

   • Apply KVL in the clockwise direction, starting from the 12V source: $12V - I_2 \times 250\Omega - I_3 \times 500\Omega = 0$

Step 3: Set Up Node Equations

Let $I_1$ be the current in the top loop through the 125Ω resistor, $I_2$ be the current through the 250Ω resistor (common branch), and $I_3$ be the current in the bottom loop through the 500Ω resistor.

At the node between the 125Ω and 250Ω resistors: $I_1 = I_2 + I_3$

Step 4: Solve for $V_a$

Since $V_a$ is the voltage drop across the 500Ω resistor in Loop 2, we can express it as: $V_a = I_3 \times 500\Omega$

Step 5: Substitute and Solve the Equations

Now, you can substitute the values from the equations above and solve for $I_1$, $I_2$, and $I_3$, and ultimately calculate $V_a$.

Would you like me to go through the calculation step-by-step for each current, or do you have any specific questions on any part of this analysis?

Additional Questions for Further Understanding:

1. How would the circuit analysis change if the resistors were arranged differently?
2. What effect would increasing the resistance values have on $V_a$?
3. Can we solve for $V_a$ using Thevenin’s theorem?
4. How would we find the total current supplied by each voltage source?
5. What if we removed the 12V source; how would $V_a$ be affected?

Tip: Breaking down a circuit into smaller loops and applying Kirchhoff's laws can simplify complex problems, especially with multiple sources.