The equivalent resistance of network of three `4Omega ` resistors can not be

Options

A. `12 Omega `
B. `1.33 Omega `
C. `6 Omega `
D. `4 Omega `

Answer

To determine the equivalent resistance of a network of three 4-ohm resistors, we will analyze different configurations: series, parallel, and a combination of both.

### Step-by-Step Solution:

1. **Understanding Series Connection:**
- In a series connection, the total or equivalent resistance (R_eq) is the sum of the individual resistances.
- Formula:
\[ R_{\text{eq}} = R_1 + R_2 + R_3 \]
- For three 4-ohm resistors:
\[ R_{\text{eq}} = 4 \, \Omega + 4 \, \Omega + 4 \, \Omega = 12 \, \Omega \]
- **Conclusion:** The equivalent resistance can be 12 ohms.

2. **Understanding Parallel Connection:**
- In a parallel connection, the total or equivalent resistance is given by:
- Formula:
\[ \frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \]
- For three 4-ohm resistors:
\[ \frac{1}{R_{\text{eq}}} = \frac{1}{4 \, \Omega} + \frac{1}{4 \, \Omega} + \frac{1}{4 \, \Omega} = \frac{3}{4 \, \Omega} \]
- Therefore,
\[ R_{\text{eq}} = \frac{4}{3} \, \Omega \approx 1.33 \, \Omega \]
- **Conclusion:** The equivalent resistance can be approximately 1.33 ohms.

3. **Combination of Series and Parallel:**
- We can also have combinations of series and parallel connections. For example, if two resistors are in parallel and one is in series:
- First, calculate the equivalent resistance of the two resistors in parallel:
\[ R_{\text{parallel}} = \frac{1}{\left(\frac{1}{4} + \frac{1}{4}\right)} = \frac{4}{2} = 2 \, \Omega \]
- Then, add the series resistor:
\[ R_{\text{eq}} = R_{\text{parallel}} + R = 2 \, \Omega + 4 \, \Omega = 6 \, \Omega \]
- **Conclusion:** The equivalent resistance can also be 6 ohms.

4. **Final Consideration:**
- We have found the possible equivalent resistances:
- 12 ohms (series)
- 1.33 ohms (parallel)
- 6 ohms (two in parallel and one in series)
- The only value that cannot be achieved with any configuration of three 4-ohm resistors is **4 ohms**.

### Final Answer:
The equivalent resistance of a network of three 4-ohm resistors cannot be **4 ohms**.