Let's solve the problem step by step! --- ## **Step 1: Understanding the Circuit** - \( I_1, I_2, I_3 \) are currents entering or leaving the main node. - \( i_1, i_2 \) are currents in the branches inside the loop. **Given:** - \( I_1 = -2 \) A - \( I_2 = 4 \) A - \( I_3 = -3 \) A --- ## **Step 2: Apply Kirchhoff's Current Law (KCL)** KCL states: **The sum of currents entering a node is equal to the sum of currents leaving the node.** At the top node: \[ I_1 + I_2 + I_3 = 0 \] Let's confirm with the values: \[ -2 + 4 + (-3) = -1 \] There might be a sign confusion or the direction of current as shown in the diagram. Let's make sure we use the correct signs when setting up equations for \(i_1\) and \(i_2\). --- ## **Step 3: Relate \(I_1, I_2, I_3\) to \(i_1, i_2\)** By looking at the diagram: - \( I_1 \) splits into \( i_1 \) and the rest. - \( I_2 \) splits into \( i_2 \) and the rest. - \( I_3 \) comes out of the bottom node. Let's write KCL at the top node: \[ I_1 \text{ (in)} + I_2 \text{ (in)} + I_3 \text{ (in)} = 0 \] But let's focus on the branches containing \(i_1\) and \(i_2\): - The **left branch** has current \(i_1\) - The **right branch** has current \(i_2\) At the left junction (top): \[ I_1 = i_1 + i_2 \] At the bottom junction: \[ i_1 + i_2 = I_3 \] But notice, due to the directions, we need to check the signs. Assume all currents entering the node are positive, leaving are negative. --- ## **Step 4: Write Equations** Let's use KCL at both nodes: ### **At the Top Node:** \[ I_1 + I_2 = i_1 \] - \(I_1\) and \(I_2\) both enter the node, and \(i_1\) leaves. ### **At the Right Node:** \[ i_2 = I_2 \] - \(i_2\) leaves, \(I_2\) enters. ### **At the Bottom Node:** \[ i_1 + i_2 = -I_3 \] - Both \(i_1\) and \(i_2\) enter the bottom node, and \(I_3\) leaves. --- ## **Step 5: Plug in Values and Solve** We have: - \(I_1 = -2\) A - \(I_2 = 4\) A - \(I_3 = -3\) A ### **Equation 1:** \[ i_1 + i_2 = -I_3 \] \[ i_1 + i_2 = -(-3) = 3 \] ### **Equation 2:** \[ I_1 + I_2 = i_1 \] \[ -2 + 4 = i_1 \] \[ i_1 = 2 \] Now substitute \(i_1\) into Equation 1: \[ 2 + i_2 = 3 \] \[ i_2 = 1 \] --- ## **Step 6: Final Answers** \[ \boxed{i_1 = 2\, \text{A}} \] \[ \boxed{i_2 = 1\, \text{A}} \] --- ### **Summary Table** | Variable | Value | |----------|-------| | \(i_1\) | 2 A | | \(i_2\) | 1 A | --- ### **Explanation** - We applied Kirchhoff's Current Law at two nodes. - Substituted given values. - Solved the two equations to find the unknown branch currents. Let me know if you need further clarification!