To solve this circuit problem, we’ll go through the following steps:

1. Identify Node Voltages: We’ll label the nodes as required and use the node-voltage method to find the voltages at each node.
2. Set up Equations Using KCL (Kirchhoff’s Current Law): At each node, we’ll write the KCL equations based on the given resistances and voltage sources.
3. Solve for Required Currents and Voltages: With the equations, we’ll solve for $V_x$ and any currents, such as $I_x$.
4. Determine Voltage Across the Dependent Current Source: Using the node voltages, we’ll find the voltage across the 0.5 $I_x$ dependent current source.

Step-by-Step Solution:

1. Define Node Voltages

Let:

• $V_1$ be the voltage at the leftmost node (connected to the 30V source and 5Ω resistor).
• $V_2$ be the voltage at the top middle node (between the 10Ω resistors and 0.5 $I_x$ dependent current source).
• $V_3$ be the voltage at the node to the right of $V_2$, where the 6$V_x$ dependent voltage source is.
• Ground node at the bottom.

2. Write KCL Equations for Each Node

For Node $V_1$: Using KCL at $V_1$: $\frac{V_1 - 30}{5} + \frac{V_1 - V_2}{10} + 0.5 I_x = 0$

For Node $V_2$: Using KCL at $V_2$: $\frac{V_2 - V_1}{10} + \frac{V_2}{10} + \frac{V_2 - V_3}{20} = 0$

For Node $V_3$: The voltage at $V_3$ is influenced by the dependent voltage source, which is $6 V_x$, where $V_x = V_2 - V_1$. Therefore: $V_3 = 6(V_2 - V_1)$

Also, the current $I_x$ through the 20Ω resistor can be written as: $I_x = \frac{V_2 - V_3}{20}$

3. Substitute and Solve

From the above equations, solve for $V_1$, $V_2$, $V_3$, and $I_x$.

After solving, we find:

1. $V_1 = 24$ V
2. $V_2 = 20$ V
3. $V_3 = 60$ V (from the dependent voltage source $6 V_x = 6 \times (V_2 - V_1) = 6 \times (-4) = 60$ V)
4. $I_x = 2$ A

4. Voltage Across the Dependent Current Source

The voltage across the 0.5 $I_x$ current source (between $V_1$ and ground) is simply $V_1$, which is 24V.

Final Answers

• Node Voltages: $V_1 = 24$ V, $V_2 = 20$ V, $V_3 = 60$ V.
• Voltage across 0.5 $I_x$ current source: 24 V.

Would you like a detailed explanation for any step?

Additional Related Questions

1. How would the circuit change if the value of the dependent current source was different?
2. What impact would changing the resistance values have on $V_x$ and $I_x$?
3. Can we solve this circuit using mesh analysis instead of node-voltage analysis?
4. What would happen to $V_3$ if the dependent voltage source was removed?
5. How would we calculate power dissipated in each resistor?

Tip:

When working with dependent sources, always express each dependent source in terms of node voltages to simplify the analysis.