Problem 20 - 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 \(200-W\) 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

17.28 MJ; Inductance is 5400 H.

Step 1: Calculate Energy Consumed by the Light Bulb

First, determine the total energy consumed by the light bulb over one day. The given power of the light bulb is \( 200 \, \text{W} \). Since power is the rate of energy consumption, \( E = P \times t \). The time \( t \) in one day is \( 24 \times 3600 \text{ seconds} \) or \( 86400 \text{ seconds} \). Thus, the energy \( E = 200 \, \text{W} \times 86400 \, \text{s} = 17,280,000 \, \text{J} \) or \( 17.28 \, \text{MJ} \).

Step 2: Apply the Energy Formula for an Inductor

Now, use the energy stored in an inductor formula: \( E = \frac{1}{2} L I^2 \). We are given \( I = 80.0 \, \text{A} \), and we need to find \( L \) where \( E = 17,280,000 \, \text{J} \). Rearrange the formula to solve for \( L \): \( L = \frac{2E}{I^2} \).

Step 3: Calculate the Inductance

Substitute the values into the formula: \( L = \frac{2 \times 17,280,000}{80^2} \). Calculate \( 80^2 = 6400 \). Thus, \( L = \frac{34,560,000}{6400} = 5400 \, \text{H} \).

Key Concepts

Electrical Energy Conversion

Electrical energy conversion is an essential concept explaining how electrical energy can be transformed into various forms, such as light or heat. This change occurs whenever electric devices, such as light bulbs, are powered. For example, a 200-Watt light bulb converts electrical energy into light and thermal energy effectively because it demonstrates this energy conversion practically.
To find out how much energy a light bulb consumes, you multiply the power (in watts) by the time it's turned on (in seconds).

• The power for our example light bulb is 200 Watts.
• Considering one whole day, we compute the time as 24 hours, multiplied by 3600 to convert hours into seconds, resulting in 86,400 seconds.

Multiplying these gives the total energy used. This showcases the process of converting electrical energy to light energy, perfectly illustrating how much energy a light bulb uses over a set period.

Inductor Energy Formula

Inductors are commonly used in electronics to store and release energy. Knowing how to calculate the energy stored in an inductor is key to understanding how they work. The energy storage capability is described by the formula: \[E = \frac{1}{2} L I^2 \]
where:

• \( E \) is the energy in joules (J).
• \( L \) is the inductance in henrys (H).
• \( I \) is the current through the inductor in amperes (A).

Using this formula, you can see that energy is dependent not only on the current but also on the inductance. Inductance determines how effectively an inductor can store energy. In our example, a current value of 80.0 A is given, and we know the amount of energy from a previous calculation. Rearranging the formula helps us find the inductance needed to store that particular amount of energy. This is an excellent way to grasp how inductors contribute to managing energy in electronic circuits.

Power and Energy Calculation

Calculating power and energy is crucial when working with different electrical appliances and devices. Power essentially indicates how fast energy is used or transformed. The relationship between power, energy, and time can be represented by the formula:\[E = P \times t\]
Where:

• \( E \) is the total energy (in joules).
• \( P \) is the power (in watts).
• \( t \) represents the time duration for which the power is consumed (in seconds).

This formula is handy when calculating how much total energy is used over time, such as how much energy a light bulb consumes in a day. Knowing the power of the device, you can multiply it by specific time intervals to find the entire energy usage. This helps in gaining insights about consumption and managing electricity more efficiently.