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The given image is of a circuit (Figure 2.19) with resistors and sources, and the task is to analyze the circuit using the **mesh analysis** method. Specifically, the goal is to calculate the current through the 5Ω resistor.

Here's how to approach the problem:

1. **Identify the Mesh Currents:**
   - In part (b), two mesh currents \( I_1 \) and \( I_2 \) are defined, flowing clockwise in the respective loops.

2. **Write the Mesh Equations:**
   - For each mesh, we will use **Kirchhoff's Voltage Law (KVL)** to write equations that sum the voltage drops around the loops.

### Mesh Equations for the circuit in (b):

- **Mesh 1 (left loop):**
  \[
  16V - (10Ω \cdot I_1) - (6Ω \cdot (I_1 - I_2)) = 0
  \]
  Simplifying:
  \[
  16 = 16I_1 - 6I_2 \quad \text{(Equation 1)}
  \]

- **Mesh 2 (right loop):**
  \[
  - (6Ω \cdot (I_2 - I_1)) - (9Ω \cdot I_2) - (5Ω \cdot I_2) = 0
  \]
  Simplifying:
  \[
  -6I_1 + 20I_2 = 0 \quad \text{(Equation 2)}
  \]

### Solving the system of equations:

We now have a system of two linear equations:
1. \( 16 = 16I_1 - 6I_2 \)
2. \( -6I_1 + 20I_2 = 0 \)

Let me solve this system to find \( I_1 \) and \( I_2 \).The solution to the system of mesh equations is:

- \( I_1 = \frac{80}{71} \, A \approx 1.127 \, A \)
- \( I_2 = \frac{24}{71} \, A \approx 0.338 \, A \)

### Current through the 5Ω Resistor:
The current through the 5Ω resistor is the same as the mesh current \( I_2 \), since the resistor is in the second mesh.

Thus, the current through the 5Ω resistor is approximately **0.338 A**.

Would you like more details on any part of the solution?

Here are 5 related questions you can explore:
1. How is Kirchhoff's Voltage Law (KVL) applied in mesh analysis?
2. What are the advantages of using the mesh analysis method in circuit problems?
3. How do you solve systems of linear equations for more complex circuits?
4. Can mesh analysis be used for circuits with dependent sources?
5. How would the solution change if the resistor values were different?

**Tip:** When analyzing circuits, always check the polarity of voltage drops and the direction of current carefully to avoid sign errors.

Analyze the circuit in Fig. 2.19(a) by the mesh method. From the results, calculate the current in the 5Ω resistance.

This solution explains how to analyze a circuit using mesh analysis to calculate the current through a 5Ω resistor. The problem involves Kirchhoff's Voltage Law and solving systems of linear equations with resistors. Suitable for university-level electrical engineering students.

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