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This problem involves applying **Kirchhoff's laws** to determine the current intensities \(i_1\), \(i_2\), and \(i_3\) in the given circuit.

### Kirchhoff's Laws:
1. **Kirchhoff's Current Law (KCL)**: The sum of currents entering a junction is equal to the sum of currents leaving the junction.
2. **Kirchhoff's Voltage Law (KVL)**: The sum of the potential differences (voltage drops and gains) around any closed loop is zero.

### Applying Ohm’s Law:
Ohm’s Law: \( U = IR \) relates the voltage \( U \), current \( I \), and resistance \( R \).

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### Circuit Analysis

The circuit has two voltage sources:
- **8V** on the left, and
- **9V** on the bottom.

It has three resistors:
- **2 ohms**,
- **3 ohms**,
- **4 ohms**.

### Step 1: Define Loop Currents
Let:
- \( i_1 \) be the current through the **3 ohm** resistor (left side),
- \( i_2 \) be the current through the **2 ohm** resistor (middle branch),
- \( i_3 \) be the current through the **4 ohm** resistor (right side).

### Step 2: Kirchhoff’s Voltage Law (KVL)
We apply KVL to each loop:

#### Loop 1 (Left Loop):
- Contains the **8V generator**, the **2 ohm** resistor, and the **3 ohm** resistor.
- The KVL equation for this loop is:
  \[
  8 - 2i_2 - 3i_1 = 0 \quad \text{(Equation 1)}
  \]

#### Loop 2 (Right Loop):
- Contains the **9V generator**, the **2 ohm** resistor, and the **4 ohm** resistor.
- The KVL equation for this loop is:
  \[
  -9 + 2i_2 + 4i_3 = 0 \quad \text{(Equation 2)}
  \]

#### Loop 3 (Total Loop Using KCL at the junction):
At the middle node, applying Kirchhoff's Current Law:
\[
i_1 = i_2 + i_3 \quad \text{(Equation 3)}
\]

### Step 3: Solve the Equations
Now we solve the system of equations:

1. \( 8 - 2i_2 - 3i_1 = 0 \)
2. \( -9 + 2i_2 + 4i_3 = 0 \)
3. \( i_1 = i_2 + i_3 \)

Let me calculate the values of \(i_1\), \(i_2\), and \(i_3\).The currents in the circuit are:

- \( i_1 = \frac{17}{7} \, \text{A} \approx 2.43 \, \text{A} \),
- \( i_2 = \frac{5}{14} \, \text{A} \approx 0.36 \, \text{A} \),
- \( i_3 = \frac{29}{14} \, \text{A} \approx 2.07 \, \text{A} \).

These are the intensities of the currents in the circuit as per Kirchhoff's laws and Ohm's law.

Do you have any questions or would you like further explanation?

Here are some related questions:
1. How do Kirchhoff's laws apply to complex circuits with multiple loops?
2. What are some common mistakes when using Kirchhoff's laws in circuit analysis?
3. How would the current values change if one of the resistances increased?
4. Can you explain how to identify junctions and loops in a circuit diagram?
5. How does the direction of current affect the sign in Kirchhoff’s Voltage Law?

**Tip**: Always check that the sum of voltage drops equals the voltage supplied when applying KVL, and ensure all current entering and leaving a node balance for KCL!

Using Kirchhoff's laws, determine the current intensities i1, i2, and i3 in the given circuit with two voltage sources (8V and 9V) and three resistors (2Ω, 3Ω, and 4Ω).

Learn how to apply Kirchhoff's Current and Voltage Laws to calculate the current intensities i1, i2, and i3 in a circuit with two voltage sources (8V and 9V) and three resistors (2Ω, 3Ω, 4Ω). Includes step-by-step solutions with Ohm's Law and KVL equations.

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