https://file.aiquickdraw.com/mathbot/file/1725074908682cub0ghv9.jpg

Find the following:

a. The equivalent Resistor

b. The current source Is

c. The voltage across the 3-ohm resistor

d. Verify KCL.

E. Verify KVL on the left side loop.

Let's work through each part of the problem systematically.

### Given Circuit:
- Voltage source \( V_s = 100 \, \text{V} \)
- Resistors: \( R_1 = 5 \, \Omega \), \( R_2 = 10 \, \Omega \), \( R_3 = 7 \, \Omega \), \( R_4 = 3 \, \Omega \)

### Step 1: Find the Equivalent Resistance (\( R_{eq} \))

1. **Combine \( R_3 \) and \( R_4 \) (which are in series):**

\[
R_{34} = R_3 + R_4 = 7 \, \Omega + 3 \, \Omega = 10 \, \Omega
\]

2. **Combine \( R_{34} \) with \( R_2 \) (which are in parallel):**

\[
\frac{1}{R_{234}} = \frac{1}{R_2} + \frac{1}{R_{34}} = \frac{1}{10 \, \Omega} + \frac{1}{10 \, \Omega} = \frac{1}{5 \, \Omega}
\]

Thus:

\[
R_{234} = 5 \, \Omega
\]

3. **Combine \( R_{234} \) with \( R_1 \) (which are in series):**

\[
R_{eq} = R_1 + R_{234} = 5 \, \Omega + 5 \, \Omega = 10 \, \Omega
\]

**Equivalent Resistance \( R_{eq} = 10 \, \Omega \)**

### Step 2: Find the Source Current \( I_s \)

Using Ohm’s law:

\[
I_s = \frac{V_s}{R_{eq}} = \frac{100 \, \text{V}}{10 \, \Omega} = 10 \, \text{A}
\]

**Source Current \( I_s = 10 \, \text{A} \)**

### Step 3: Find the Voltage across the \( 3 \, \Omega \) Resistor

- The current flowing through the \( 3 \, \Omega \) resistor is the same as the current flowing through the parallel combination of \( R_2 \) and \( R_{34} \).

\[
V_{34} = I_{234} \times R_4
\]

Since \( R_2 \) and \( R_{34} \) are in parallel and have equal resistance:

\[
I_{234} = \frac{I_s}{2} = \frac{10 \, \text{A}}{2} = 5 \, \text{A}
\]

Therefore, the voltage across the \( 3 \, \Omega \) resistor is:

\[
V_4 = I_{234} \times R_4 = 5 \, \text{A} \times 3 \, \Omega = 15 \, \text{V}
\]

**Voltage across \( 3 \, \Omega \) resistor \( V_4 = 15 \, \text{V} \)**

### Step 4: Verify Kirchhoff's Current Law (KCL)

KCL states that the sum of currents entering a junction must equal the sum of currents leaving the junction.

- At the junction between \( R_1 \), \( R_2 \), and \( R_{34} \):
  - Current entering: \( I_s = 10 \, \text{A} \)
  - Currents leaving:
    - Through \( R_2 \): \( 5 \, \text{A} \)
    - Through \( R_{34} \): \( 5 \, \text{A} \)
  
\[
I_s = I_{R_2} + I_{R_{34}} = 5 \, \text{A} + 5 \, \text{A} = 10 \, \text{A}
\]

KCL is verified.

### Step 5: Verify Kirchhoff's Voltage Law (KVL) on the Left Loop

KVL states that the sum of the voltage drops in a closed loop must equal the sum of the voltage rises.

- Consider the left loop including the source \( V_s \), \( R_1 \), and \( R_2 \):

\[
V_s = I_s \times R_1 + V_{R_2} = 10 \, \text{A} \times 5 \, \Omega + 50 \, \text{V} = 50 \, \text{V} + 50 \, \text{V} = 100 \, \text{V}
\]

KVL is verified.

---

### Summary of Answers:
1. **Equivalent Resistance**: \( R_{eq} = 10 \, \Omega \)
2. **Source Current**: \( I_s = 10 \, \text{A} \)
3. **Voltage across 3-ohm resistor**: \( V_{4} = 15 \, \text{V} \)
4. **KCL**: Verified
5. **KVL on Left Loop**: Verified

Would you like more details on any of these calculations, or have any questions? 

Here are five related questions for further exploration:

1. What happens to the equivalent resistance if the \( 7 \, \Omega \) resistor is removed?
2. How would the current distribution change if the \( 10 \, \Omega \) resistor was replaced with a \( 20 \, \Omega \) resistor?
3. What is the power dissipated by the \( 3 \, \Omega \) resistor?
4. How does the circuit behavior change if the voltage source is doubled?
5. What would be the impact on \( I_s \) if all resistances were halved?

**Tip**: Always double-check your current and voltage calculations when applying KVL and KCL to ensure accuracy.

Explore how to solve complex electrical circuit problems involving voltage sources, resistors, and the application of Kirchhoff's Laws (KCL and KVL). This detailed guide covers finding equivalent resistance, determining current sources, calculating voltage drops across resistors, and verifying Kirchhoff's Laws. Suitable for college-level students and electrical engineering enthusiasts.

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