Problem 37 - College Physics

Question

Lightbulbs. The wattage rating of a lightbulb is the power it consumes when it is connected across a 120 potential difference. For example, a 60 W lightbulb consumes 60.0 \(\mathrm{W}\) of electrical power only when it is connected across a 120 \(\mathrm{V}\) potential difference. (a) What is the resistance of a 60 \(\mathrm{W}\) lightbulb? (b) Without doing any calculations, would you expect a 100 \(\mathrm{W}\) bulb to have more or less resistance than a 60 \(\mathrm{W}\) bulb? Calculate and find out.

Step-by-Step Solution

Answer

The resistance of a 60 W bulb is 240 ohms. A 100 W bulb has less resistance (144 ohms).

Step 1: Understanding the Formula for Resistance

To calculate the resistance of a lightbulb, we use the formula derived from Ohm's Law and the power formula. The power formula is given as \( P = V^2 / R \), where \( P \) is the power, \( V \) is the voltage, and \( R \) is the resistance. Rearranging for \( R \), we have \( R = V^2 / P \).

Step 2: Calculating the Resistance of a 60 W Bulb

For a 60 W bulb with a voltage of 120 V, substitute these values into the formula: \( R = (120)^2 / 60 \). Calculating this gives \( R = 14400 / 60 = 240 \ \Omega \). The resistance of a 60 W bulb is 240 ohms.

Step 3: Predicting the Resistance of a 100 W Bulb

A higher wattage bulb consumes more power, which generally means it has a lower resistance when connected to the same voltage. Thus, without calculation, we would expect the 100 W bulb to have less resistance than the 60 W bulb.

Step 4: Calculating the Resistance of a 100 W Bulb

For a 100 W bulb, again using the formula \( R = V^2 / P \) with \( P = 100 \) and \( V = 120 \), we have: \( R = (120)^2 / 100 \). Calculating this gives \( R = 14400 / 100 = 144 \ \Omega \). Thus, the resistance of a 100 W bulb is 144 ohms.

Step 5: Conclusion: Comparison of Resistances

The calculation confirms the prediction. The 60 W bulb has a resistance of 240 ohms, while the 100 W bulb has a resistance of 144 ohms, which means the 100 W bulb has less resistance.

Key Concepts

Electrical Power

Electrical power is the rate at which electrical energy is transferred by an electric circuit. It's measured in watts (W), a common unit that indicates how much energy is consumed over a period of time by an electrical device like a lightbulb. In the context of lightbulbs, a 60 W lightbulb uses 60 watts of power, meaning it converts electrical energy into light and heat at this rate when connected to a certain voltage. The formula associated with electrical power is

• \( P = IV \)
• where \( P \) represents power, \( I \) is the current, and \( V \) is the voltage.

This relationship helps to understand the amount of power used by an electrical device and is crucial for calculating other electrical properties.

Resistance Calculation

Resistance is a measure of how much a component, like a lightbulb, resists the flow of electric current. It's measured in ohms (Ω). To find the resistance of a lightbulb, especially in exercises involving a specified power and voltage, you need to use the formula

• \( R = \frac{V^2}{P} \)

Here, \( R \) is resistance, \( V \) is voltage, and \( P \) is power. This formula is derived from rearranging the power formula and can be used to calculate the electrical resistance of devices like lightbulbs. When solving textbook problems or real-world applications, this calculation helps in knowing more about how different electrical components behave when connected to a power source.

Voltage and Current

Voltage and current are essential concepts in the study of electricity. Voltage, measured in volts (V), is the difference in electric potential between two points. It's what "pushes" the electric current through the circuit. On the other hand, current, measured in amperes (A), is the flow of electric charge. Understanding their relationship is key to understanding Ohm's Law, which states:

• \( V = IR \)
• where \( V \) is the voltage, \( I \) is the current, and \( R \) is the resistance.

This relationship means that for a given resistance, increasing the voltage increases the current flowing through a circuit, making these concepts vital for not only solving problems but also for designing real-world electrical systems.

Power Consumption

Power consumption reflects how much energy an electrical device uses over time. It's closely related to the concepts of voltage, current, and resistance. Devices like a lightbulb have a specific wattage rating indicating the power they consume under certain conditions, typically a specified potential difference or voltage. When calculating power consumption, the idea is to determine how much energy will be used over a period, which is crucial for both efficiency considerations and cost calculations.
In a practical sense, knowing a bulb's power consumption assists in designing circuits to ensure that it operates safely and efficiently, without overloading the circuit or wasting energy. Lower resistance bulbs tend to consume more power, as seen in the comparison between 60 W and 100 W bulbs with the same voltage, highlighting resistance's crucial role in managing power consumption.