Problem 66 - Physics for Scientists and Engineers with Modern Physics

Question

A three-way lightbulb can produce \(50 \mathrm{~W}, 100 \mathrm{~W},\) or \(150 \mathrm{~W}\) at \(120 \mathrm{~V}\). Such a bulb contains two filaments that can be connected to the \(120 \mathrm{~V}\) individually or in parallel. (a) Describe how the connections to the two filaments are made to give each of the three wattages. (b) What must be the resistance of each filament?

Step-by-Step Solution

Answer

Connect one filament for 50W, the other for 100W, and both parallel for 150W. Resistances are 288Ω and 144Ω.

Step 1: Understanding the Problem

This problem involves a three-way bulb with two filaments creating three different power outputs (50W, 100W, 150W) at a constant voltage of 120V. We need to determine how these filaments are connected to achieve each wattage and find the resistance of each filament.

Step 2: Determine Connections for Each Power Level

To achieve 50W, only one filament operates. At 100W, the other filament alone operates. For 150W, both are connected in parallel. In this way, turning on one filament gives 50W, the other gives 100W, and both together give 150W.

Step 3: Use Power Formula to Find Resistances

We use the power formula, \( P = \frac{V^2}{R} \). Solve for resistance, \( R = \frac{V^2}{P} \). For 50W filament, \( R_1 = \frac{120^2}{50} \). For 100W, \( R_2 = \frac{120^2}{100} \).

Step 4: Calculate Resistance for 50W Filament

Substitute into the formula, \( R_1 = \frac{120^2}{50} = \frac{14400}{50} = 288 \Omega \).

Step 5: Calculate Resistance for 100W Filament

Substitute into the formula, \( R_2 = \frac{120^2}{100} = \frac{14400}{100} = 144 \Omega \).

Step 6: Verify Total 150W with Parallel Circuit

Checking both resistances in parallel, \( \frac{1}{R_{\text{total}}} = \frac{1}{R_1} + \frac{1}{R_2} \). Calculate \( R_{\text{total}} = \frac{1}{288} + \frac{1}{144} \).

Key Concepts

Three-Way Lightbulb

A three-way lightbulb is a clever little device that allows you to set different levels of brightness. It uses two separate filaments inside the bulb, each capable of producing a different amount of light. These filaments are like tiny wires in the bulb that glow when electricity flows through them. Each filament has its wattage it can support without burning out.
Three-way bulbs are usually set for low, medium, and high light levels, allowing a bulb to produce, for instance, 50W, 100W, or 150W. The magic happens with how these filaments are connected. You can turn on one filament for low, the other for medium, and both together for high light output. This gives three light settings all in one bulb, ideal for different settings or moods. The filament connections play a vital role in how a three-way bulb operates, ensuring all three brightness options are available.

Power Formula

The power formula is a very handy tool for understanding electric circuits and is represented by the equation: \[ P = \frac{V^2}{R} \]where:

• \( P \) is the power in watts (W),
• \( V \) is the voltage in volts (V), and
• \( R \) is the resistance in ohms (Ω).

This formula helps us understand how power, voltage, and resistance are related. You can rearrange this equation to discover the resistance if you know the power and voltage levels.

For example, a 50W filament operating at 120V will have a resistance calculated as follows:\[ R = \frac{V^2}{P} = \frac{120^2}{50} = 288 \, \Omega \]Knowing the power formula is crucial in determining how these elements of electrical circuits interrelate, helping illuminate why certain resistances are necessary to produce specific power outputs.

Resistance Calculation

In terms of circuits, calculating resistance is key to understanding how well a circuit conducts electricity. The resistance of an object determines how much it resists the flow of electricity. Lower resistance means more current flows, and vice versa.
When you have a component with known voltage and power levels, the formula for resistance can help figure out what that resistance is:\[ R = \frac{V^2}{P} \]For instance, for a lightbulb filament:
- If the power is 50W and the voltage is 120V, the resistance is 288 Ω.
- If the power is 100W at the same voltage, the resistance is 144 Ω.
Understanding and calculating resistance in electrical circuits is pivotal to design and functionality, ensuring each component operates within safe and effective parameters.

Parallel Circuits

Parallel circuits are a common way to connect electrical components. They allow multiple paths for current to flow. In a three-way bulb, both filaments might be connected in parallel to create the highest wattage setting.
When components are in parallel, they have the same voltage across them, which is advantageous for consistent performance. The total resistance in parallel circuits can be calculated using:\[ \frac{1}{R_{\text{total}}} = \frac{1}{R_1} + \frac{1}{R_2} + \ldots \]For the example of the lightbulb where both filaments are used together (50W and 100W in parallel), the combined resistance is found by:\[ \frac{1}{R_{\text{total}}} = \frac{1}{288} + \frac{1}{144} \]Parallel circuits efficiently manage more significant loads without each component needing a larger capacity. They are fundamental in household electrical configurations, helping to ensure appliances receive stable currents while maintaining safety standards.