Problem 46 - College Physics

Question

Large inductors have been proposed as energy-storage devices. (a) How much electrical energy is converted to light and thermal energy by a 200 \(\mathrm{W}\) lightbulb in one day? (b) If the amount of energy calculated in part (a) is stored in an inductor in which the current is \(80.0 \mathrm{A},\) what is the inductance?

Step-by-Step Solution

Answer

17,280,000 J; 5,400 H

Step 1: Understanding the Problem

To solve the problem, we need to find the amount of energy used by the lightbulb in one day and then calculate the inductance required to store the same amount of energy with a given current.

Step 2: Calculate Energy Consumed by Lightbulb

The power of the lightbulb is given as 200 W. Power is the rate of energy consumption, which means energy can be calculated by multiplying power by time. One day is equal to 24 hours or 86,400 seconds. Thus, the energy consumed is \( E = P \times t = 200 \text{ W} \times 86,400 \text{ s} = 17,280,000 \text{ J} \).

Step 3: Use Energy Formula for Inductor

The energy stored in an inductor is given by the formula \( E = \frac{1}{2} L I^2 \), where \( L \) is the inductance and \( I \) is the current. We know the energy \( E = 17,280,000 \text{ J} \) and current \( I = 80.0 \text{ A} \). We need to solve for \( L \).

Step 4: Solve for Inductance

Rearrange the energy formula: \( L = \frac{2E}{I^2} \). Substitute the known values: \( L = \frac{2 \times 17,280,000 \text{ J}}{(80.0 \text{ A})^2} = \frac{34,560,000}{6,400} = 5,400 \text{ H} \).

Step 5: Conclusion

The energy consumed by the lightbulb in one day is 17,280,000 J, and the required inductance to store this energy with a current of 80.0 A is 5,400 H.

Key Concepts

Electrical Energy Conversion

Electrical energy conversion is a fundamental concept in physics and engineering, focusing on how electrical energy transforms into other forms of energy. In this exercise, a 200-watt lightbulb serves as an example of converting electrical energy into both light and heat.

• Electrical devices like lightbulbs use electricity to produce light and sometimes heat.
• The rate of energy usage is quantified in watts, which measures how much energy a device consumes per second.
• For a 200-watt bulb, this means it converts 200 joules of electrical energy per second to other forms.

To find out how much energy is consumed over a day, we multiply the power (200 W) by the total seconds in 24 hours (86,400 seconds). Hence, the energy conversion calculates to 17,280,000 joules. This figure represents the total energy turned from electrical energy to light and heat by the bulb in a full day.

Inductance Calculation

Inductance is a property of an electrical component that resists changes in current, and it plays a crucial role in energy storage within inductors. An inductor can store and release energy in a magnetic field, which makes it a potential candidate for energy storage devices.

• Inductance, denoted by the letter \( L \), is measured in henrys (H).
• The energy stored in an inductor is described by the formula: \( E = \frac{1}{2} L I^2 \).
• This equation shows the relationship between energy \( E \), inductance \( L \), and current \( I \).

For the given problem, where the energy to be stored equals 17,280,000 joules with a current of 80.0 amperes, we need to find the inductance \( L \). By rearranging the equation, we get \( L = \frac{2E}{I^2} \). Substituting the known values results in: \( L = \frac{2 \times 17,280,000}{80^2} = 5,400 \text{ H} \). Hence, the inductor's inductance required to store this energy at the given current is 5,400 henrys.

Power Consumption

Power consumption is a critical idea in both understanding and managing energy efficiency in electrical devices. It represents the amount of energy used by an appliance over time and can be calculated if both the power and the duration of use are known.

• Power is measured in watts (W), indicating how much energy a device uses per second.
• Energy consumed over time can be determined by multiplying the power by the number of seconds used.

In the context of the lightbulb, it draws 200 watts of power continuously. Over the span of an entire day, this results in a substantial energy consumption, calculated in step-by-step detail to be 17,280,000 joules. Understanding these principles aids in assessing the efficiency and cost-effectiveness of using such devices over extended periods.