Problem 3 - Physics

Question

A battery charger is connected to a dead battery and delivers a current of 6.0 A for 5.0 hours, keeping the voltage across the battery terminals at \(12 \mathrm{~V}\) in the process. How much energy is delivered to the battery?

Step-by-Step Solution

Answer

The energy delivered to the battery is 1,296,000 Joules.

Step 1: Identify the known values

We are provided with the current \(I = 6.0 \, \text{A}\), the time \(t = 5.0 \, \text{hours}\), and the voltage \(V = 12 \, \text{V}\). We need to convert time to seconds for calculation. So, \(t = 5.0 \, \text{hours} = 5.0 \times 3600 \, \text{s} = 18000 \, \text{s}\).

Step 2: Calculate the total charge transferred

Use the formula \(Q = I \times t\) to calculate the charge. \(Q = 6.0 \, \text{A} \times 18000 \, \text{s} = 108000 \, \text{Coulombs}\).

Step 3: Calculate the total energy delivered

Use the formula \(E = V \times Q\) to find the energy. Substituting the known values, \(E = 12 \, \text{V} \times 108000 \, \text{C}\) gives \(E = 1296000 \, \text{Joules}\).

Key Concepts

Ohm's Law

Ohm's Law is a fundamental principle in circuit theory. It defines the relationship between voltage, current, and resistance in an electrical circuit.
Ohm's Law is represented by the equation: \( V = I \times R \), where:

• \( V \) is the voltage across the resistance (in volts).
• \( I \) is the current flowing through the resistance (in amperes).
• \( R \) is the resistance (in ohms).

This law tells us that the voltage across a conductor is directly proportional to the current flowing through it, as long as the temperature remains constant. In our exercise, although we don't use Ohm's Law directly to calculate energy, understanding this relationship helps us appreciate how current and voltage can create power and influence electrical energy.

Electric Charge

Electric charge is a fundamental property of matter that causes it to experience a force when placed in an electromagnetic field. In simple terms, it's what gives rise to electricity.
The charge, denoted by \( Q \), is measured in coulombs (C).

• In circuits, the flow of electric charge is what we call electric current.
• One coulomb is equivalent to the charge of approximately \(6.242 \times 10^{18} \) electrons.

In the original problem, we calculated the total charge transferred using the formula: \( Q = I \times t \). This expressed how much electric charge moved through the circuit while the battery was charging.

Electrical Energy

Electrical energy is the energy stored in charged particles within an electric field. This energy is an essential component of our everyday electrical devices and systems.
Electrical energy can be calculated using the formula \( E = V \times Q \), which represents:

• \( E \) as the energy in joules (J),
• \( V \) as the voltage (in volts), and
• \( Q \) as the electric charge (in coulombs).

In the problem, we used this formula to determine the total amount of energy delivered to the battery. The resulting energy expressed in joules helps us understand how much power was required to recharge the dead battery.

Current and Voltage

Current and voltage are two key aspects of electrical circuits. Current refers to the rate of flow of electric charge, and is measured in amperes (A). On the other hand, voltage is the electrical potential difference between two points and is measured in volts (V).

• Current (\( I \)) can be imagined as the flow of a water stream through a pipe.
• Voltage (\( V \)) is akin to the pressure of that water stream, pushing the water through.

In the exercise, the specified current was \(6.0 \, \text{A}\) and the voltage across the battery terminals was \(12 \, \text{V}\). These values play a critical role in providing the energy needed to recharge the battery.

Physics Problem-Solving

Solving physics problems involves a systematic approach to applying fundamental principles and appropriate formulas. This exercise illustrates a step-by-step methodology that can help.
A typical problem-solving strategy includes:

• Identifying known values and what you need to determine.
• Converting units if necessary.
• Applying relevant formulas based on the measurable quantities.
• Performing calculations carefully to avoid errors.

Working through this process not only gives the solution but also deepens the understanding of the concepts at play, such as the relationships between charge, current, voltage, and energy in circuits.