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

Options

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

Explanation

To determine the equivalent resistance of a network of three 4-ohm resistors and identify which resistance cannot be achieved, we can analyze different combinations of these resistors.

### Step-by-Step Solution:

1. **Connect All Resistors in Series:**
- When resistors are connected in series, the equivalent resistance (R_eq) is the sum of all resistances.
- Formula: \( R_{eq} = R_1 + R_2 + R_3 \)
- Calculation: \( R_{eq} = 4 \, \Omega + 4 \, \Omega + 4 \, \Omega = 12 \, \Omega \)

2. **Connect All Resistors in Parallel:**
- When resistors are connected in parallel, the equivalent resistance can be calculated using the formula:
- Formula: \( \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \)
- Calculation:
\[ \frac{1}{R_{eq}} = \frac{1}{4 \, \Omega} + \frac{1}{4 \, \Omega} + \frac{1}{4 \, \Omega} = \frac{3}{4} \Rightarrow R_{eq} = \frac{4}{3} \, \Omega \approx 1.33 \, \Omega \]

3. **Connect Two Resistors in Parallel and One in Series:**
- First, calculate the equivalent resistance of the two resistors in parallel:
- Calculation:
\[ \frac{1}{R_{parallel}} = \frac{1}{4 \, \Omega} + \frac{1}{4 \, \Omega} = \frac{2}{4} = \frac{1}{2} \Rightarrow R_{parallel} = 2 \, \Omega \]
- Now, add the third resistor in series:
- Calculation:
\[ R_{eq} = R_{parallel} + R_{3} = 2 \, \Omega + 4 \, \Omega = 6 \, \Omega \]

4. **Summary of Possible Equivalent Resistances:**
- From the calculations:
- Series: 12 Ω
- Parallel: 1.33 Ω
- Two in parallel and one in series: 6 Ω
- The possible equivalent resistances are 12 Ω, 1.33 Ω, and 6 Ω.

5. **Identify the Resistance that Cannot Be Achieved:**
- The possible equivalent resistances we can achieve are 1.33 Ω, 6 Ω, and 12 Ω.
- The resistance that cannot be achieved with these combinations is **4 Ω**.

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