https://file.aiquickdraw.com/mathbot/file/1725418040931nfnjmsgg.jpg

To solve for \( v_a \), \( v_b \), and \( v_c \) in this circuit, we'll need to analyze the voltage drops and apply Kirchhoff's Voltage Law (KVL) around the loops.

### Step 1: Label the Points
Let’s label the nodes where the components connect:

- Node at \( v_a \): Voltage across the component labeled \( v_a \).
- Node at \( v_b \): Voltage across the component labeled \( v_b \).
- Node at \( v_c \): Voltage across the component labeled \( v_c \).

### Step 2: Analyze the Voltages

**1. Loop Analysis (left loop including \( v_a \) and the 5V source):**

In the left loop, the 5V source is directly connected to \( v_a \). So:
\[ v_a = 5\text{ V} \]

**2. Loop Analysis (middle loop including \( v_a \), the 15V source, and the 10V source):**

Going clockwise:
\[ v_a + 15\text{ V} + v_b = 10\text{ V} \]

Since \( v_a = 5\text{ V} \):
\[ 5\text{ V} + 15\text{ V} + v_b = 10\text{ V} \]
\[ v_b = 10\text{ V} - 20\text{ V} \]
\[ v_b = -10\text{ V} \]

**3. Outer Loop Analysis (including \( v_c \)):**

Considering the loop from the 5V source, through \( v_c \), and back:
\[ v_a + v_c = 0 \]

Given \( v_a = 5\text{ V} \):
\[ 5\text{ V} + v_c = 0 \]
\[ v_c = -5\text{ V} \]

### Final Answers:

- \( v_a = 5 \text{ V} \)
- \( v_b = -10 \text{ V} \)
- \( v_c = -5 \text{ V} \)

Would you like any further details or explanations?

Here are some related questions you might find helpful:
1. What are the basic principles of Kirchhoff's Voltage Law (KVL)?
2. How do you apply KVL in more complex circuits?
3. What is the significance of negative voltages in a circuit analysis?
4. How can superposition theorem help in analyzing this circuit?
5. What are the differences between series and parallel circuits?

**Tip:** When analyzing circuits, always be mindful of the sign conventions (positive/negative) for voltage drops and sources. This is crucial for correctly applying KVL and obtaining the right answers.

This page provides a detailed solution for finding the voltages Va, Vc, and Vb in a given circuit using Kirchhoff's Voltage Law (KVL). The problem involves analyzing voltage drops and applying KVL to different loops in the circuit, suitable for undergraduate electrical engineering students.

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