Let's revisit the calculation of \( I_3 \) to verify the correct value.

### Step-by-Step Analysis

Given:
- \( I_2 = 0.2 \, \text{A} \) (This is the current through \( R_2 \)).
- All resistances \( R_1 = R_2 = R_3 = R_4 = 10 \, \Omega \).
- \( E_1 = 1 \, \text{V} \), \( E_3 = 1 \, \text{V} \).

### Kirchhoff's Current Law (KCL):
At the junction where \( I_1 \), \( I_2 \), and \( I_3 \) meet:
\[
I_1 = I_2 + I_3
\]

### Kirchhoff's Voltage Law (KVL):

**Loop 1** (involving \( R_1 \), \( R_2 \), and \( E_1 \)):
\[
E_1 - I_1R_1 - I_2R_2 = 0
\]
Substituting the values:
\[
1V - 10\Omega \cdot I_1 - 10\Omega \cdot 0.2A = 0
\]
This simplifies to:
\[
I_1 = 0.1A - 0.2A = -0.1A
\]

**Loop 2** (involving \( R_2 \), \( R_3 \), and \( E_2 \)):
\[
-E_2 + I_2R_2 + I_3R_3 = 0
\]
Substituting the values:
\[
-E_2 + 10\Omega \cdot 0.2A + 10\Omega \cdot I_3 = 0
\]
Simplifying:
\[
E_2 = 2V + 10\Omega \cdot I_3
\]

Given \( I_1 = -0.1A \), and from KCL:
\[
-0.1A = 0.2A + I_3
\]
Solving for \( I_3 \):
\[
I_3 = -0.1A - 0.2A = -0.3A
\]

### Conclusion
You are correct: \( I_3 = -0.3A \). This negative sign indicates that the direction of \( I_3 \) is opposite to the direction assumed in the analysis.

Would you like to explore any further details? Here are some related questions:

1. How would the negative value of \( I_3 \) affect the calculation of power dissipation in \( R_3 \)?
2. If \( E_2 \) were given a specific value, how would that influence \( I_3 \)?
3. What physical interpretation can be made when currents have a negative sign?
4. How does the presence of multiple voltage sources influence the direction of current flow?
5. What would happen to \( I_3 \) if the resistance \( R_3 \) was doubled?

**Tip:** When dealing with complex circuits, always check your sign conventions to ensure consistency in your final answers.

Explore the solution to a circuit analysis problem involving Kirchhoff's Laws. Learn how to calculate I3 in a complex circuit with detailed step-by-step analysis. Suitable for college-level students and anyone interested in electrical circuit analysis.

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