Show how you would connect three resistors, each of resistance \(6 \Omega\), so that the combination has a resistance of (i) \(9 \Omega\), (ii) \(4 \Omega\).

When resistors are connected in series, their resistances simply add up. This means that if you have multiple resistors in a single path or line, you can find the total resistance by summing their individual resistances. For example, if you have three resistors each with a resistance of \(6 \, \text{Ω}\), then the total resistance is: \ R_{\text{total}} = R_1 + R_2 + R_3 = 6 \, \text{Ω} + 6 \, \text{Ω} + 6 \, \text{Ω} = 18 \, \text{Ω}\. This concept is fundamental in understanding how electric circuits divide voltage across different components. Series connections are common in applications where you need a higher total resistance or want to control the current uniformly through all parts.

When resistors are connected in parallel, the total resistance decreases. This is because parallel connections create multiple paths for the current to travel. The formula to calculate the total resistance \(R_{\text{total}}\) for resistors in parallel is: \[ \frac{1}{R_{\text{total}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \]. For example, if you have three resistors of \(6 \, \text{Ω}\) each in parallel, the total resistance is: \[ \frac{1}{R_{\text{total}}} = \frac{1}{6} + \frac{1}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2} \] \ R_{\text{total}} = 2 \, \text{Ω}\. Parallel connections are useful in circuits needing to reduce total resistance and to ensure that components share the same voltage.

Ohm's Law is a fundamental principle in electrical engineering that relates the current flowing through a conductor to the voltage and the resistance. It is represented as \[ V = IR \], where \(V\) is the voltage (in volts), \(I\) is the current (in amperes), and \(R\) is the resistance (in ohms). This simple yet powerful equation helps determine one of the three variables if the other two are known. For example, if you have a resistor of \(6 \, \text{Ω}\) and a voltage of \(12 \, \text{V}\) across it, you can find the current using \[ I = \frac{V}{R} = \frac{12 \, \text{V}}{6 \, \text{Ω}} = 2 \, \text{A} \]. Understanding Ohm's Law is crucial for analyzing and designing electronic circuits, as it allows for predicting how changes in voltage or resistance affect the current.