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The circuit presented involves resistors, a current source, and a voltage source. To find the current \( i \), we can use **mesh analysis**, particularly focusing on creating a **supermesh** around loops with the current source.

### Steps for solving:

1. **Identify the loops and meshes:**
   - There are three primary loops in the circuit:
     - **Loop 1:** The leftmost loop containing the 30 V voltage source, the \( i \) current, and the 2 Ω resistor.
     - **Loop 2:** The middle loop involving the 4 A current source, the 4 Ω and 6 Ω resistors.
     - **Loop 3:** The rightmost loop containing the 8 Ω, 6 Ω, 3 Ω, and \( R = 7 \) Ω resistor.

2. **Form a supermesh:**
   - The current source (4 A) is shared between Loop 2 and Loop 3. These loops form a **supermesh**, as the current source prevents writing a direct KVL (Kirchhoff’s Voltage Law) equation for these loops separately.

3. **Apply Kirchhoff's Voltage Law (KVL):**
   - For Loop 1 (without the current source):
     \[
     -30 + 2i_1 + 4(i_1 - i_2) = 0
     \]
     Simplifying:
     \[
     -30 + 6i_1 - 4i_2 = 0 \quad \text{(Equation 1)}
     \]
   - For the **supermesh** (Loop 2 and Loop 3 combined):
     \[
     6(i_2 - i_3) + 3i_3 + 7i_3 = 0
     \]
     Simplifying:
     \[
     6i_2 - 16i_3 = 0 \quad \text{(Equation 2)}
     \]
   - The current source constraint:
     \[
     i_2 - i_3 = 4 \quad \text{(Equation 3, from the 4 A current source)}
     \]

4. **Solve the system of equations:**
   Use the three equations derived to find the values of \( i_1 \), \( i_2 \), and \( i_3 \).

Let me compute these values.The solutions for the currents in the circuit are:

- \( i_1 = \frac{139}{15} \, \text{A} \approx 9.27 \, \text{A} \)
- \( i_2 = \frac{32}{5} \, \text{A} = 6.4 \, \text{A} \)
- \( i_3 = \frac{12}{5} \, \text{A} = 2.4 \, \text{A} \)

Thus, the current \( i \) in the circuit is \( i_1 \), which is approximately **9.27 A**.

Would you like a detailed explanation or have any further questions?

Here are some related questions you might find helpful:
1. How is a supermesh created in a circuit with a current source?
2. What are the advantages of using mesh analysis in circuit problems?
3. How can we apply Kirchhoff's Voltage Law (KVL) in a mesh with a current source?
4. What is the significance of the current source in determining loop currents?
5. How do resistor values affect the mesh current solutions in a circuit?

**Tip:** In mesh analysis, always look for current sources to form supermeshes to simplify solving loops.

In the circuit given below, R = 7 Ω. Find the current i.

Learn how to solve a circuit problem using mesh and supermesh analysis where R = 7 Ω. The problem involves calculating the current i using Kirchhoff's Voltage Law (KVL) and supermesh concepts. Suitable for students in Grades 10-12.

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