A variable inductor can be placed in series with a lightbulb to act as a dimmer. (a) What inductance would reduce the current through a \(100-\) W lightbulb to \(75 \%\) of its maximum value? Assume a \(120-\mathrm{V}\) rms, \(60-\mathrm{Hz}\) source. (b) Could a variable resistor be used in place of the variable inductor to reduce the current? Why is the inductor a much better choice for a dimmer?

We are given the power rating of the lightbulb (100 W) and the RMS voltage (120 V). Let's use the formula for power in an AC circuit to find the maximum current. Power (P) = V_rms * I_max Rearrange the formula for I_max: I_max = Power / V_rms Plug in the given values: I_max = 100 W / 120 V = 0.8333 A

Now, we need to find the current when it's reduced to 75% of its maximum value. I_reduced = 0.75 * I_max = 0.75 * 0.8333 A = 0.625 A

We will now calculate the impedance needed to achieve the reduced current in the circuit. Using Ohm's law: V = I * Z Where V is the RMS voltage, I is the current, and Z is the impedance. Rearrange the formula for Z: Z = V / I Plug in the known values for the reduced current: Z = 120 V / 0.625 A = 192 Ω

We know the impedance required, and we are assuming that it's purely inductive (ignoring any resistance in the coil), so we will calculate the required inductance for the circuit at a frequency of 60 Hz. The formula for the inductive reactance (X_L) is: X_L = 2 * pi * f * L Where f is the frequency, and L is the inductance. To find the inductance, we will rearrange the formula: L = X_L / (2 * pi * f) Since we found that the impedance (Z) required is 192 Ω and we assume that it's purely inductive, X_L = Z: L = 192 Ω / (2 * pi * 60 Hz) = 0.5304 H (a) The inductance needed to reduce the current to 75% of its maximum value is 0.5304 H.

A variable resistor could also be used to reduce the current in the circuit. However, the inductor is a more efficient choice than a resistor as a dimmer, mainly for two reasons: 1. A resistor dissipates the energy in the form of heat as it reduces the current flow. This could make the dimmer quite hot and waste energy. 2. An inductor, on the other hand, stores energy in its magnetic field when it restricts the flow of current. It releases that energy back into the circuit when the current is allowed to increase. This makes the inductor-based dimmer more energy-efficient and safer than a resistor-based dimmer. In conclusion, while a variable resistor could potentially be used as a dimmer, a variable inductor is a better choice due to its energy efficiency and safety.