Problem 102 - Physics for Scientists and Engineers with Modern Physics

Question

Electric energy units are often expressed in the form of "kilowatt-hours." (a) Show that one kilowatt-hour (kWh) is equal to \(3.6 \times 10^{6} \mathrm{~J} .\) (b) If a typical family of four uses electric energy at an average rate of \(580 \mathrm{~W}\), how many \(\mathrm{kWh}\) would their electric bill show for one month, and (c) how many joules would this be? ( \(d\) ) At a cost of \(\$ 0.12\) per \(\mathrm{kWh}\), what would their monthly bill be in dollars? Does the monthly bill depend on the rate at which they use the electric energy?

Step-by-Step Solution

Answer

1 kWh is 3.6 million J. Monthly consumption: 417.6 kWh or 1.5 billion J; cost: $50.11. Bill depends only on total energy used, not usage rate.

Step 1: Understanding Kilowatt-Hours

A kilowatt-hour is a unit of energy equal to one kilowatt of power used for one hour. To express this in joules, recognize that: \(1 \text{ kW} = 1000 \text{ watts}\) and \(1 \text{ hour} = 3600 \text{ seconds}\). Therefore, \(1 \text{ kWh} = 1000 \times 3600 \text{ J} = 3.6 \times 10^6 \text{ J}\).

Step 2: Calculate Energy Usage for a Month

The family's energy usage is given in watts (W), which we need to convert to kilowatt-hours. First, convert the power usage from watts to kilowatts: \(580 \text{ W} = 0.58 \text{ kW}\). Then calculate the energy for a month (30 days): \(30\text{ days} \times 24 \text{ hours/day} \times 0.58 \text{ kW} = 417.6 \text{ kWh}\).

Step 3: Convert Monthly Kilowatt-Hours to Joules

Now convert the monthly energy usage from kilowatt-hours to joules. Use the conversion factor from Part (a): \(417.6 \text{ kWh} \times 3.6 \times 10^6 \text{ J/kWh} = 1.50336 \times 10^{9} \text{ J}\).

Step 4: Calculate the Monthly Electric Bill

To find the cost, multiply the energy used by the rate: \(417.6 \text{ kWh} \times 0.12 \text{ dollars/kWh} = 50.112 \text{ dollars}\).

Step 5: Determine Dependence on Energy Usage Rate

The monthly bill is based on the total energy consumed, not how rapidly it was consumed, so as long as the total kWh remains the same, the bill does not depend on the rate at which the power was used.

Key Concepts

Kilowatt-Hour

The kilowatt-hour, commonly abbreviated as kWh, is a standard unit of energy used by electricity companies. It represents the amount of energy consumed when a device uses one kilowatt of power for one hour. Understanding its connection to other units of energy helps us grasp its significance.
To convert the kilowatt-hour into joules, the International System of Units (SI) for energy, remember:

• 1 kilowatt (kW) = 1,000 watts (W)
• 1 hour = 3600 seconds

When you compute, 1 kWh equals 1,000 W multiplied by 3,600 seconds, which converts to 3.6 million joules (\(1 ext{ kWh} = 3.6 imes 10^6 ext{ J}\)).
This conversion helps us translate energy consumption into universally accepted terms and facilitates understanding among various scientific fields.

Joules Conversion

Joules are the standard unit of energy. It's crucial to understand how to switch between kilowatt-hours and joules depending on the context, as both are commonly used.
The conversion from kilowatt-hour to joules is essential in many calculations concerning electrical energy. Given that 1 kWh is equivalent to 3.6 million joules, you can multiply your kWh value by this number to convert any energy amount into joules.
For example, if you use 2 kWh, the energy consumed in joules will be:\[ 2 ext{ kWh} imes 3.6 imes 10^6 ext{ J/kWh} = 7.2 imes 10^6 ext{ J} \].
Understanding these conversions is important for technical applications like engineering and physics, where precise energy measurements are required.

Energy Consumption Calculation

Calculating energy consumption involves understanding how much energy an appliance or household uses over a period.
Given an average power usage rate (in watts or kilowatts), you can determine the total energy consumed.
Start by converting the power usage from watts to kilowatts if necessary, as most energy usage statistics are given in kWh.
Consider a family with an average usage of 580 W. Converting this to kilowatts gives 0.58 kW. Over a month (30 days)

• Total energy consumption = number of days x hours per day x average power usage in kilowatts
• For 30 days, this would be: \( 30 ext{ days} imes 24 ext{ hours/day} imes 0.58 ext{ kW} = 417.6 ext{ kWh} \)

This method is essential for estimating energy use for billing and identifying high energy use patterns.

Electric Bill Cost

Calculating the cost of an electric bill involves understanding both your energy consumption in kilowatt-hours and the rate charged per kilowatt-hour.
This rate might vary depending on the provider or location.
To estimate the bill, multiply the total energy consumption by the rate charged.
If the family consumes 417.6 kWh for one month, and the energy company charges $0.12 per kWh, the bill can be calculated as:\[417.6 ext{ kWh} imes 0.12 ext{ dollars/kWh} = 50.112 ext{ dollars} \].
Knowing how to perform this calculation keeps you informed about your energy expenditure and allows you to plan your finances better.

Power Usage Rate

The power usage rate indicates how quickly energy is consumed. While it informs real-time energy use, the total cost on an electricity bill depends on total energy consumption, not the rate of consumption.
This means, even if power is used slowly or quickly, the billing cost solely depends on the final total kWh used over the billing period.

• For example, using 580 W continuously over a month results in a particular energy consumption.
• As long as the monthly total remains the same, the cost will also remain unchanged, regardless of consumption speed.

Understanding the power usage rate concept helps in learning about daily habits affecting energy use, although it does not directly influence the bill.