A typical light dimmer used to dim the stage lights in a...

A typical light dimmer used to dim the stage lights in a theater consists of a variable inductor $L$ (whose inductance is adjustable between zero and $L_{\max }$ connected in series with a lightbulb $\mathrm{B},$ as shown in Fig. $31-34$ . The electrical supply is 120 $\mathrm{V}(\mathrm{rms})$ at 60.0 $\mathrm{Hz}$ , the lightbulb is rated at $120 \mathrm{V}, 1000 \mathrm{W}$ . (a) What $L_{\mathrm{max}}$ is required if the rate of energy dissipation in the lightbulb is to be varied by a factor of 5 from its upper limit of 1000 $\mathrm{w} ?$ Assume that the resistance of the lightbulb is independent of its temperature. (b) Could one use a variable resistor (adjustable between zero and $R_{\text { max }}$ instead of an inductor? (c) If so, what $R_{\text { max }}$ is required? (d) Why isn't this done?

Key Concepts

Power Dissipation in Resistive Loads

Power dissipation in resistive elements occurs when electrical energy is converted into heat. It is governed by formulas such as P = V²/R and is important in scenarios where the control of energy output, like dimming lights, is achieved by varying circuit resistance.

Efficiency of Reactive versus Resistive Control

This concept delineates the differences between using reactive components, like inductors, and resistive components, such as variable resistors, in controlling circuit current. Reactive elements adjust the current flow without significant energy loss, while resistive elements reduce current by wasting energy as heat, making the former more efficient for applications like light dimming.

AC Circuit Analysis

This concept focuses on the study of circuits powered by alternating current, where voltages and currents vary sinusoidally. It involves the use of phasors and impedance, which is a combination of resistance and reactance, to analyze how circuit elements respond to AC signals.

Inductive Reactance

Inductive reactance represents the opposition that an inductor offers to the flow of AC, calculated as 2?fL, where f is the frequency and L is the inductance. This concept is essential for understanding how a variable inductor modulates the current in a circuit without dissipating energy as heat.

Transcript

00:01 31 .56 we're considering one possible way of doing light dimming.

00:05 This is evidently used in theaters for dimming stage lights.

00:13 The more typical light dimmer circuits for things in your house are more complicated in some ways than this, although they are cheaper to produce it uses a solid state component that we you know don't need to worry about right now because that's not what this problem is anyway so our voltage supply here is 120 volts rms at a frequency of 60 hertz and our light bulb is rated for 120 volts so that's good we aren't going to make it explode and its power rating is one kilowatt so first we want to find what the maximum inductance of this inductor needs to to be in order for the minimum power in this light bulb to be 200 watts, one -fifth of its maximum value.

01:09 We want to know if you could use a resistor instead of an inductor, and if so, what is its maximum value? and then we want to discuss why you wouldn't actually use a resistor.

01:31 So the power in the bulb b is its current squared times its resistance, which we're told doesn't change with temperature.

01:43 So we don't need to worry about it.

01:44 That that would make the problem mode not that much more complicated but it would be enough to be obnoxious and obscure the point of what we're trying to do here this is the rms power so this is the rms current this is the rms power if you wanted to use the the peak for one of them or both of them that you get a factor of the square root of two but that's going to cancel out because we want pmax over pmin, which is going to be i over imin squared to be equal to five.

02:45 And so the, you know, whatever factors we have to account for whether you're using the maximum or the rms voltage ends up, they end up canceling out, it doesn't matter.

03:12 So in each case, the current is going to be the emf divided by the impedance.

03:19 So you'll have the maximum power with the minimum impedance and the minimum power with the maximum impedance.

03:37 And then these cancel out.

03:42 So we get z -max over zmin squared.

04:04 So just rewriting this in this new page so that we don't lose track of what we're doing.

04:10 Z -max over a z -min quantity squared.

04:19 So the light bulb has a resistance of r.

04:25 So the maximum impedance is going to be r squared plus omega l max, which is what we want to find.

04:38 Squared.

04:38 You take a square root of that because impedances add in this way, or their magnitudes do at least.

04:49 And then the minimum impedance is just the resistance of the light bulb...

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