Show with the help of diagrams, how you would connet three resistors each of resistance `6 Omega`, so that the combination has resistance of (i) `9Omega` (ii) `4Omega`.

Answer

To solve the problem of connecting three resistors of 6 ohms each to achieve a total resistance of (i) 9 ohms and (ii) 4 ohms, we will follow these steps:

### Part (i): Achieving a Total Resistance of 9 Ohms

1. **Connect Two Resistors in Parallel**:
- Take two 6-ohm resistors and connect them in parallel.
- The formula for equivalent resistance \( R_{parallel} \) for two resistors in parallel is given by:
\[ \frac{1}{R_{parallel}} = \frac{1}{R_1} + \frac{1}{R_2} \]
- Substituting the values:
\[ \frac{1}{R_{parallel}} = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3} \]
- Therefore, \( R_{parallel} = 3 \, \Omega \).

2. **Connect the Third Resistor in Series**:
- Now, connect the equivalent resistance of 3 ohms (from the parallel connection) in series with the third 6-ohm resistor.
- The total resistance \( R_{total} \) in series is given by:
\[ R_{total} = R_{parallel} + R_3 = 3 + 6 = 9 \, \Omega \]

3. **Diagram**:
- The diagram for this connection would look like this:
```
+---[6Ω]---+
| |
| |
+---[6Ω]---+
| |
+---[6Ω]---+
```

### Part (ii): Achieving a Total Resistance of 4 Ohms

1. **Connect Two Resistors in Series**:
- Take two 6-ohm resistors and connect them in series.
- The total resistance \( R_{series} \) for two resistors in series is given by:
\[ R_{series} = R_1 + R_2 = 6 + 6 = 12 \, \Omega \]

2. **Connect the Third Resistor in Parallel**:
- Now, connect the equivalent resistance of 12 ohms (from the series connection) in parallel with the third 6-ohm resistor.
- The formula for equivalent resistance \( R_{net} \) for one resistor in parallel with another is:
\[ \frac{1}{R_{net}} = \frac{1}{R_{series}} + \frac{1}{R_3} \]
- Substituting the values:
\[ \frac{1}{R_{net}} = \frac{1}{12} + \frac{1}{6} = \frac{1}{12} + \frac{2}{12} = \frac{3}{12} = \frac{1}{4} \]
- Therefore, \( R_{net} = 4 \, \Omega \).

3. **Diagram**:
- The diagram for this connection would look like this:
```
+---[6Ω]---[6Ω]---+
| |
+------------------+
```

### Summary of Diagrams:
- For **9 ohms**: Two resistors in parallel (3 ohms) in series with one resistor (6 ohms).
- For **4 ohms**: Two resistors in series (12 ohms) in parallel with one resistor (6 ohms).