Problem 16 - University Physics with Modern Physics

Question

It has been proposed to use large inductors as energy storage devices. (a) How much electrical energy is converted to light and thermal energy by a \(200-\mathrm{W}\) light bulb 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

(a) 17,280,000 J, (b) 5400 H

Step 1: Calculate the Energy Consumed by the Light Bulb

To find the energy produced by the light bulb in one day, use the formula for energy \[ E = P imes t \]where \( E \) is the energy, \( P \) is the power (200 W for the light bulb), and \( t \) is the time in seconds (24 hours = 86400 seconds). Substitute the given values:\[ E = 200 \, \text{W} \times 86400 \, \text{s} \] \[ E = 17,280,000 \, \text{J} \] Thus, the light bulb converts 17,280,000 joules of electrical energy to light and thermal energy in one day.

Step 2: Use the Inductor Energy Storage Formula

An inductor stores energy based on its current and inductance using the formula \[ E = \frac{1}{2} L I^2 \]where \( E \) is the energy, \( L \) is the inductance, and \( I \) is the current. We know the energy \( E = 17,280,000 \, \text{J} \) and the current \( I = 80.0 \, \text{A} \). We need to solve for \( L \):\[ 17,280,000 = \frac{1}{2} L (80.0)^2 \]

Step 3: Solve for the Inductance

Rearrange the formula to solve for \( L \):\[ 17,280,000 = \frac{1}{2} L \times 6400 \]\[ 17,280,000 = 3200L \]\[ L = \frac{17,280,000}{3200} \]\[ L = 5400 \, \text{H} \]Thus, the inductance of the inductor needed to store the same amount of energy is 5400 Henrys.

Key Concepts

Inductor Energy Storage

Inductors are essential components in electrical systems, used for various purposes, including energy storage. At the core of an inductor's ability to store energy lies its magnetic field. When an electric current flows through an inductor, it generates a magnetic field around it. This magnetic field stores energy, effectively making the inductor a temporary energy storage device.

The energy stored in an inductor can later be converted back into electrical energy. This conversion mainly depends on the current flowing through the inductor and its inductance. The stored energy is described by the formula:

• \( E = \frac{1}{2} L I^2 \)

where \( E \) is the stored energy in joules, \( L \) is the inductance in henrys, and \( I \) is the current in amperes. Understanding this formula is crucial for engineering applications where energy management and efficiency are important. Inductors are ideal for applications like smoothing out fluctuations in electrical circuits and forming RF filters.

Knowing how much energy an inductor can store helps in designing effective energy storage solutions and improving the overall efficiency of electrical devices.

Electrical Energy Conversion

Electrical energy conversion is a process that involves changing electrical energy into other forms, such as light, thermal, or mechanical energy. This is a fundamental concept in how electrical systems function, ranging from small electronic gadgets to large power grids.

Consider a 200-Watt light bulb, which is a classic example of electrical energy conversion. Every second, the bulb converts 200 watts of electrical power into light and heat. Over a full day, the total energy converted can be calculated with the equation:

• \( E = P \times t \)

Here, \( P \) is 200 watts, representing the power, and \( t \) is the time, which can be converted into seconds for one full day (24 hours = 86400 seconds). Using this equation, the converted energy amounts to 17,280,000 joules over a day.

Understanding how electrical energy is converted is essential for designing energy-efficient appliances and systems. It helps in identifying opportunities to reduce energy consumption and enhance efficiency.

Inductance Calculation

Calculating inductance is crucial for determining an inductor's ability to store and manage energy in electrical circuits. It is defined as the property of an electrical conductor that causes an EMF (electromotive force) to be generated by a change in the current flowing.
To find the inductance required to store a specific amount of energy, such as the 17,280,000 joules calculated for one day's operation of a 200-W light bulb, we can rearrange the formula for stored energy in an inductor:

• \( E = \frac{1}{2} L I^2 \)
• \( L = \frac{2E}{I^2} \)

Given \( E = 17,280,000 \text{ J} \) and a current of \( I = 80.0 \text{ A} \), we can plug these values into the formula to find \( L \), the inductance:
\[ L = \frac{2 \times 17,280,000}{80.0^2} \]
This calculation results in an inductance of 5400 henrys. Understanding how to calculate inductance helps in designing circuits and selecting appropriate components to optimize energy consumption and circuitry efficiency.