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To solve this circuit problem using **Mesh Analysis**, we follow the following steps:

### Step 1: Define Mesh Currents
There are two meshes in this circuit:
- **Mesh 1 (Left loop)**: Includes the 12V voltage source, the 4Ω resistor, and the 6Ω resistor.
- **Mesh 2 (Right loop)**: Includes the 6Ω resistor, the 5Ω resistor, and the 3V voltage source.

Let's assume:
- **I₁** is the current in Mesh 1 (clockwise direction).
- **I₂** is the current in Mesh 2 (clockwise direction).

The current through the 6Ω resistor is the difference between these two mesh currents, i.e., \( I_{\text{6Ω}} = I₁ - I₂ \).

### Step 2: Apply Kirchhoff's Voltage Law (KVL)
We will now apply KVL to each loop.

#### For Mesh 1 (left loop):
Starting from the 12V voltage source:
\[
12V - 4I₁ - 6(I₁ - I₂) = 0
\]
This simplifies to:
\[
12 = 4I₁ + 6(I₁ - I₂)
\]
\[
12 = 4I₁ + 6I₁ - 6I₂
\]
\[
12 = 10I₁ - 6I₂ \quad \text{(Equation 1)}
\]

#### For Mesh 2 (right loop):
Starting from the 3V voltage source:
\[
-3V + 5I₂ + 6(I₂ - I₁) = 0
\]
This simplifies to:
\[
-3 = 5I₂ + 6(I₂ - I₁)
\]
\[
-3 = 5I₂ + 6I₂ - 6I₁
\]
\[
-3 = 11I₂ - 6I₁ \quad \text{(Equation 2)}
\]

### Step 3: Solve the System of Equations
We now have the system of equations:

1. \( 12 = 10I₁ - 6I₂ \)
2. \( -3 = -6I₁ + 11I₂ \)

Solving these equations will give us the values of \( I₁ \) and \( I₂ \).

Let me solve these equations now.The solutions for the mesh currents are:

- \( I₁ = \frac{57}{37} \, \text{A} \approx 1.54 \, \text{A} \)
- \( I₂ = \frac{21}{37} \, \text{A} \approx 0.57 \, \text{A} \)

### Step 4: Determine the Current through the 6Ω Resistor
The current through the 6Ω resistor is the difference between the mesh currents:
\[
I_{\text{6Ω}} = I₁ - I₂ = \frac{57}{37} - \frac{21}{37} = \frac{36}{37} \, \text{A} \approx 0.97 \, \text{A}
\]

Thus, the current flowing through the 6Ω resistor is approximately **0.97 A**.

Would you like a deeper explanation on any step? Let me know if you have more questions! Here are some related questions for you:

1. How does mesh analysis compare to nodal analysis?
2. What happens if the direction of current in the 6Ω resistor is reversed?
3. How can we solve this circuit using superposition theorem?
4. Can mesh analysis be applied to circuits with dependent sources?
5. What is the significance of negative values for mesh currents?

**Tip:** Always assume consistent current direction when solving with mesh analysis to avoid sign confusion later.

Determine the current flowing through 6Ω resistance in the circuit given below by applying Mesh analysis.

Learn how to determine the current flowing through a 6Ω resistor in a given electrical circuit using Mesh Analysis. This problem involves key concepts such as Kirchhoff's Voltage Law (KVL) and electrical circuit analysis. Suitable for college-level students.

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