Question

A 12 -V automotive storage battery is charged with a constant current of \(2 \mathrm{amp}\) for \(24 \mathrm{~h}\). If electricity costs \(\$ 0.08\) per \(\mathrm{kW} \cdot \mathrm{h}\), determine the cost of recharging the battery.

Short answer

The cost of recharging the battery is \$0.04608.

Step-by-step solution

Calculate the Total Charge

To find the total charge supplied to the battery, use the formula: \[ Q = I \times t \]where \( I \) is the current in amperes (A) and \( t \) is the time in hours (h).Given: \( I = 2 \text{ A} \) and \( t = 24 \text{ h} \),\[ Q = 2 \text{ A} \times 24 \text{ h} = 48 \text{ Ah} \]

Convert Charge to Energy

Convert the charge (Ampere-hours) to energy in Watt-hours using the voltage of the battery.Energy (in Watt-hours) is calculated as:\[ \text{Energy} = Q \times V \]where \( V \) is the voltage and \( Q \) is the charge in Ampere-hours.Given: \( Q = 48 \text{ Ah} \) and \( V = 12 \text{ V} \),\[ \text{Energy} = 48 \text{ Ah} \times 12 \text{ V} = 576 \text{ Wh} \]

Convert Energy to Kilowatt-hours

Convert the energy from Watt-hours to Kilowatt-hours (kWh).\[ 1 \text{ kWh} = 1000 \text{ Wh} \]So,\[ \text{Energy in kWh} = \frac{576 \text{ Wh}}{1000} = 0.576 \text{ kWh} \]

Calculate the Cost

Calculate the total cost of recharging the battery by multiplying the energy (in kWh) by the cost per kWh.Given: Cost per kWh = \$ 0.08,\[ \text{Cost} = 0.576 \text{ kWh} \times 0.08 \text{ \$} = 0.04608 \text{ \$} \]

Key concepts

Electrical Energy

Electrical energy is the capacity to do work using electric forces. It is measured in units of energy like Joules (J) or more commonly in practical applications, it is measured in Watt-hours (Wh) or Kilowatt-hours (kWh). Electrical energy can be calculated as the product of power and time. In our example, the energy is derived from a constant current delivered to a battery over a period of time. By understanding electrical energy, you can easily comprehend the units used and the importance of power and time in energy consumption. This will help you understand how devices and batteries use electrical energy, enabling efficient calculations for energy usage and costs.

Ampere-Hours

Ampere-hours (Ah) is a unit of electric charge that is often used to describe battery capacity. It represents the amount of charge a battery can deliver at one ampere for a period of one hour. For example, a battery with a capacity of 48 Ah can theoretically deliver 1 ampere for 48 hours, or 2 amperes for 24 hours. In our exercise, we calculate the total charge supplied to the battery using the formula: \( Q = I \times t \), where \( I \) is the current in amperes and \( t \) is the time in hours. Given that \( I = 2 \text{ A} \) and \( t = 24 \text{ h} \), we obtain \( Q = 48 \text{ Ah} \). This illustrates how long a battery can sustain a current before it depletes its charge, which is essential for understanding battery performance and lifespan.

Watt-Hours

Watt-hours (Wh) is a unit of energy that indicates the amount of power (in watts) used over a specific amount of time (in hours). It is calculated as the product of charge (in ampere-hours) and voltage (in volts). This conversion is useful because it connects electrical charge to energy consumption. In our exercise, we use the formula: \( \text{Energy} = Q \times V \), where \( Q \) is the charge in ampere-hours and \( V \) is the voltage. With \( Q = 48 \text{ Ah} \) and \( V = 12 \text{ V} \), the energy in watt-hours is \( 576 \text{ Wh} \). This step helps us understand how much energy is needed to charge a battery, which can directly relate to the cost of charging.

Kilowatt-Hours

Kilowatt-hours (kWh) is a larger unit of energy commonly used by electricity providers to bill consumers. One kilowatt-hour equals 1,000 watt-hours. In our example, converting to kilowatt-hours helps us to calculate the cost of recharge easily. Since \( 1 \text{ kWh} = 1000 \text{ Wh} \), we convert our energy from 576 Wh to kWh by dividing by 1000, yielding \( 0.576 \text{ kWh} \). To find the cost of recharging the battery, multiply the energy in kWh by the cost per kWh: \( \text{Cost} = 0.576 \text{ kWh} \times 0.08 \text{ \$} = 0.04608 \text{ \$} \). This conversion and calculation are essential in understanding your energy bills and managing energy use efficiently.