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Design a problem, complete with a solution, to help other students better understand Kirchhoff's Current Law. Design the problem by specifying values of ia, ib and ic, shown in picture, and asking them to solve for values of i1, i2, and i3. Be careful specify realistic currents.

### Problem Design: Understanding Kirchhoff's Current Law (KCL)

Given the circuit in the diagram, use **Kirchhoff's Current Law (KCL)** to solve for the unknown currents \( i_1 \), \( i_2 \), and \( i_3 \) in the circuit.

### Problem Setup:
The current values for the main branches are specified as follows:

- \( i_a = 12 \, \text{mA} \) (entering the top node)
- \( i_b = 8 \, \text{mA} \) (entering the left node)
- \( i_c = 9 \, \text{mA} \) (leaving the bottom node)

Using KCL, we know that the current entering and leaving any node must balance (the sum of currents into a node is zero). The relationships between the currents can be determined by analyzing the three different nodes in the circuit.

**Questions:**
1. Use KCL at the junction where \( i_a \), \( i_1 \), and \( i_b \) meet to set up an equation.
2. Apply KCL at the junction where \( i_1 \), \( i_2 \), and \( i_3 \) meet to set up a second equation.
3. Apply KCL at the junction where \( i_2 \), \( i_3 \), and \( i_c \) meet to set up a third equation.
4. Solve the system of equations to find \( i_1 \), \( i_2 \), and \( i_3 \).

### Solution:

**Step 1:** Apply Kirchhoff's Current Law at each junction.

At the first node (top junction where \( i_a \), \( i_b \), and \( i_1 \) meet):
\[
i_a + i_b = i_1
\]
Substitute the known values:
\[
12 \, \text{mA} + 8 \, \text{mA} = i_1
\]
Thus,
\[
i_1 = 20 \, \text{mA}
\]

**Step 2:** Apply Kirchhoff's Current Law at the second node (middle junction where \( i_1 \), \( i_2 \), and \( i_3 \) meet):
\[
i_1 = i_2 + i_3
\]
Substitute \( i_1 = 20 \, \text{mA} \):
\[
20 \, \text{mA} = i_2 + i_3 \quad \text{(Equation 1)}
\]

**Step 3:** Apply Kirchhoff's Current Law at the third node (bottom junction where \( i_2 \), \( i_3 \), and \( i_c \) meet):
\[
i_2 + i_c = i_3
\]
Substitute \( i_c = 9 \, \text{mA} \):
\[
i_2 + 9 \, \text{mA} = i_3 \quad \text{(Equation 2)}
\]

**Step 4:** Solve the system of equations.

From **Equation 2**, substitute \( i_3 = i_2 + 9 \, \text{mA} \) into **Equation 1**:
\[
20 \, \text{mA} = i_2 + (i_2 + 9 \, \text{mA})
\]
Simplify:
\[
20 \, \text{mA} = 2i_2 + 9 \, \text{mA}
\]
\[
20 \, \text{mA} - 9 \, \text{mA} = 2i_2
\]
\[
11 \, \text{mA} = 2i_2
\]
\[
i_2 = \frac{11}{2} \, \text{mA} = 5.5 \, \text{mA}
\]

Now substitute \( i_2 = 5.5 \, \text{mA} \) back into **Equation 2**:
\[
i_3 = 5.5 \, \text{mA} + 9 \, \text{mA} = 14.5 \, \text{mA}
\]

### Final Answers:
- \( i_1 = 20 \, \text{mA} \)
- \( i_2 = 5.5 \, \text{mA} \)
- \( i_3 = 14.5 \, \text{mA} \)

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### Do you have any questions, or would you like further details?

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Here are 5 related questions to practice:

1. What is Kirchhoff's Voltage Law (KVL), and how does it relate to KCL?
2. How would the current distribution change if \( i_a \) were reduced to \( 10 \, \text{mA} \)?
3. Can you derive a general formula for any arbitrary current entering and leaving the nodes using KCL?
4. How do resistors in the branches affect the current values in KCL problems?
5. How would the system of equations change if there were an additional current source in the middle branch?

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**Tip:** When applying Kirchhoff's Laws, always ensure the correct sign convention (current entering vs. current leaving) to avoid errors in your equations.

Design a problem, complete with a solution, to help other students better understand Kirchhoff's Current Law. Specify values of ia, ib, and ic shown in the picture, and ask students to solve for values of i1, i2, and i3.

Learn how to apply Kirchhoff's Current Law to solve for the unknown currents i1, i2, and i3 in a circuit with given values of ia, ib, and ic. This problem involves analyzing current flow at different nodes and setting up linear equations. Suitable for Grades 10-12.

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