Problem 24 - Applied Physics

Question

An electric heater is used \(5.00 \mathrm{~h}\) each day. (a) If it draws \(15.0 \mathrm{~A}\) on a \(12 \overline{0}-\mathrm{V}\) line, how much power does it use? (b) In 30 days, how much energy in $$\mathrm{kWh}$$ does the heater use? (c) At \(\$ 0.11 /\) \mathrm{kWh}$, what does it cost to operate the heater for 30 days?

Step-by-Step Solution

Answer

(a) The heater uses 1800 W. (b) It uses 270 kWh in 30 days. (c) The cost is $29.70 for 30 days.

Step 1: Calculate the Power Usage

To find the power used by the heater, use the formula for electrical power: \( P = IV \). Here, \( I = 15.0 \, \text{A} \) and \( V = 120 \, \text{V} \). Substitute these values into the formula: \[ P = 15.0 \, \text{A} \times 120 \, \text{V} = 1800 \, \text{W} \]. Thus, the power used is 1800 watts.

Step 2: Convert Power to Kilowatts

Power is often expressed in kilowatts when calculating energy usage in kilowatt-hours. Since 1 kilowatt (kW) is equivalent to 1000 watts (W), convert the power from watts to kilowatts:\[ 1800 \, \text{W} = \frac{1800}{1000} = 1.8 \, \text{kW} \].

Step 3: Calculate Daily Energy Consumption

To find the daily energy consumption in kilowatt-hours, multiply the power by the time the heater is used: \[ \text{Energy per day} = 1.8 \, \text{kW} \times 5.00 \, \text{h} = 9.0 \, \text{kWh} \].

Step 4: Calculate Monthly Energy Consumption

Multiply the daily energy consumption by the number of days to find the monthly energy usage: \[ \text{Energy in 30 days} = 9.0 \, \text{kWh/day} \times 30 \text{ days} = 270 \, \text{kWh} \].

Step 5: Calculate the Cost

Finally, calculate the cost by multiplying the total energy consumption by the cost per kilowatt-hour: \[ \text{Cost} = 270 \, \text{kWh} \times 0.11 \, \text{\$/kWh} = 29.70 \, \text{USD} \].

Key Concepts

Electrical Energy Consumption

Understanding electrical energy consumption helps us determine how much energy an electrical device uses over time. This is important when performing calculations to estimate cost and energy usage.
Energy consumption is usually measured in kilowatt-hours (kWh), which reflects both the power drawn and the time the device is used. To calculate this, we use the formula:

• Energy Consumption = Power (in kW) × Time (in hours)

In our example, a heater uses 1.8 kW. Thus, if it's used for 5 hours, it consumes 9.0 kWh a day. By understanding these calculations, you can estimate how much energy and cost a device might incur over a given period.

Voltage and Current

Electrical devices work by consuming power that depends on both voltage and current. Voltage is the potential difference that drives electric current through a circuit. It's measured in volts (V).
Current, measured in amperes (A), is the flow of electric charge. Together, current and voltage define the power consumption using the formula:

• Power (P) = Current (I) × Voltage (V)

For the heater example, the voltage is 120 V, and the current is 15 A. So, the power calculated is:

• P = 15 A × 120 V = 1800 W

Understanding these concepts is crucial for evaluating how much energy a device consumes and how effectively it operates.

Kilowatt-hour (kWh)

A kilowatt-hour (kWh) is a unit of energy that helps measure electricity usage, which is essential to understanding electric bills and energy costs.
One kWh is equivalent to using 1000 watts of power for one hour. When you calculate the kWh for a device, you can determine how much energy it uses over a specific time.
Previously, we calculated that the heater uses 1800 W or 1.8 kW. For 5 hours of use, the consumption is:

• 1.8 kW × 5 hours = 9.0 kWh per day

Kilowatt-hours simplify tracking and billing energy usage, making it easier for households to manage electricity expenses.

Energy Cost Calculation

Energy cost calculation is essential for budgeting and understanding the financial implications of energy consumption. To find the cost, you multiply the total energy consumed in kilowatt-hours by the cost per kWh.
For devices like heaters, knowing daily and monthly costs helps manage expenses. Using our example:

• Total monthly energy use = 9 kWh/day × 30 days = 270 kWh
• Cost per kWh = $0.11
• Total cost = 270 kWh × $0.11/kWh = $29.70

By calculating energy costs, you can make informed decisions, such as reducing usage or improving energy efficiency to save money.