https://file.aiquickdraw.com/mathbot/file/1730058695330zv7qbn0b.jpg

To solve for the mesh current \( I_1 \) in this circuit, let's apply mesh analysis using Kirchhoff's Voltage Law (KVL) in each loop.

### Given Data
- Voltage source, \( V = 20 \, \text{V} \)
- Current source in the second loop, \( I_2 = 2 \, \text{A} \)
- Resistor values: \( R_1 = 6 \, \Omega \), \( R_2 = 1 \, \Omega \), and an additional resistor of \( 3 \, \Omega \) shared between both meshes.

### Analysis
1. **Define Mesh Currents**: Let the mesh currents be \( I_1 \) for the left loop and \( I_2 = 2 \, \text{A} \) for the right loop, as given by the current source.

2. **Apply KVL to the Left Loop (Loop 1)**:
   - The voltage drop across \( R_1 \) due to \( I_1 \) is \( R_1 \cdot I_1 = 6 I_1 \).
   - The shared resistor (3 Ω) has a voltage drop based on the difference in current, \( (I_1 - I_2) \cdot 3 \).
   
   Applying KVL around the left loop:
   \[
   20 - 6I_1 - 3(I_1 - I_2) = 0
   \]

3. **Substitute \( I_2 = 2 \, \text{A} \)**:
   \[
   20 - 6I_1 - 3(I_1 - 2) = 0
   \]

4. **Simplify the Equation**:
   Expand and simplify:
   \[
   20 - 6I_1 - 3I_1 + 6 = 0
   \]
   Combine terms:
   \[
   26 - 9I_1 = 0
   \]
   \[
   9I_1 = 26
   \]
   \[
   I_1 = \frac{26}{9} \approx 2.89 \, \text{A}
   \]

### Final Answer
The mesh current \( I_1 \) is approximately:
\[
I_1 \approx 2.89 \, \text{A}
\]

Would you like a detailed explanation of each step or have any questions?

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Here are five related questions for further exploration:
1. How would the solution change if \( R_1 \) was different?
2. What if there were additional resistors in the circuit—how would that affect \( I_1 \)?
3. Can we solve this circuit using node analysis instead of mesh analysis?
4. How does the presence of the current source affect the mesh current calculation?
5. What are the implications of changing the voltage source to a higher or lower value?

**Tip:** Mesh analysis is useful in planar circuits, where it can simplify solving for currents by focusing on loops rather than individual components.

For the circuit below, determine the mesh current I1 if R1 = 6 Ω and R2 = 1 Ω.

Learn how to calculate the mesh current I1 in an electrical circuit using mesh analysis and Kirchhoff's Voltage Law (KVL). This problem involves a voltage source of 20V, a 2A current source, and resistor values of R1 = 6 Ω and R2 = 1 Ω. Suitable for college-level students studying circuit analysis.

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