Question

A resistive heater is used to supply heat into an insulated box. The heater has current \(0.04 \mathrm{~A}\) and resistance \(1 \mathrm{k} \Omega,\) and it operates for one hour. Energy is either stored in the box or used to spin a shaft. If the box gains \(2,500 \mathrm{~J}\) of energy in that one hour, how much energy was used to turn the shaft?

Short answer

The energy used to turn the shaft is 3260 J.

Step-by-step solution

Identify Given Quantities

First, let's identify the given values from the problem:- Current, \( I = 0.04 \text{ A} \)- Resistance, \( R = 1000 \text{ } \Omega \)- Time, \( t = 1 \text{ hour } = 3600 \text{ seconds } \)- Energy stored in the box, \( E_\text{stored} = 2500 \text{ J} \)

Calculate Total Energy Supplied by the Heater

We need to calculate the total energy supplied by the heater using the formula for electrical energy:\[ E_\text{total} = I^2 \cdot R \cdot t \]Substituting the given values:\[ E_\text{total} = (0.04)^2 \cdot 1000 \cdot 3600 \]\[ E_\text{total} = 0.0016 \cdot 1000 \cdot 3600 \]\[ E_\text{total} = 5760 \text{ J} \]

Determine Energy Used to Turn the Shaft

The energy used to turn the shaft can be found by subtracting the energy stored in the box from the total energy supplied:\[ E_\text{shaft} = E_\text{total} - E_\text{stored} \]Substituting the values we computed:\[ E_\text{shaft} = 5760 \text{ J} - 2500 \text{ J} \]\[ E_\text{shaft} = 3260 \text{ J} \]

Final Answer

We have calculated that the amount of energy used to turn the shaft is \( 3260 \text{ J} \).

Key concepts

Electric Energy Calculation

Calculating electric energy is a critical aspect of understanding energy conservation in physics. When dealing with electric circuits, we use electrical energy, which is usually calculated using a specific formula. This formula relates current, resistance, and time to calculate the energy input of an electrical component. The formula is:

\[ E = I^2 \cdot R \cdot t \]
where:

• \( E \) is the electrical energy in Joules.
• \( I \) represents the current in Amperes.
• \( R \) is the resistance in Ohms.
• \( t \) is the time in seconds.

This formula allows us to understand how energy is transferred through resistive materials.

In practice, calculating the total energy supplied by a device like a heater involves substituting these values into the formula. For example, if a resistive heater has a current of 0.04 A, resistance of 1000 Ω, and operates for an hour, the energy calculation would be:

\[ E = (0.04)^2 \cdot 1000 \cdot 3600 = 5760 \text{ J} \].
This calculation shows the total amount of energy supplied by the heater over that period.

Resistive Heater

A resistive heater is a device that uses electrical energy and converts it into heat through the process of resistance. In this kind of device, when an electric current is passed through a resistive material, it generates heat due to the resistance opposing the current.

Key characteristics of a resistive heater include:

• Operating based on the principle of Joule heating, also known as resistive or ohmic heating.
• Transforming electrical energy entirely into thermal energy without any other form of energy conversion.
• The resultant heat is usually transferred to the surrounding environment or utilized to perform work, such as heating a space or an object.

The efficiency of a resistive heater is determined by its ability to transfer energy effectively. It is considered highly efficient when almost all the input electrical energy is converted into heat.

Understanding this principle is vital when calculating how electrical energy is used within systems involving resistive heaters, as seen in the exercise example. Here, the heater efficiently supplies energy to an insulated box and a shaft, clearly demonstrating energy transfer through resistance.

Energy Transfer in Systems

Energy transfer in systems, especially in the context of a closed system like in the exercise, showcases the law of conservation of energy. This law states that energy cannot be created or destroyed but can be transformed from one form to another or transferred between systems.

Within a closed system involving a resistive heater:

• The total energy supplied by the heater is distributed across different forms of energy based on the work it performs.
• Some energy is stored within the system, such as the insulated box gaining energy.
• Other portions of the energy are used externally, highlighted in this instance by the energy used to spin a shaft.

In the exercise, we calculated the total energy using the resistive heater's power to find it supplied 5760 J over an hour. Knowing that 2500 J was stored in the box allows us to deduce that the remaining energy, 3260 J, was transferred elsewhere, exemplifying the process of energy conservation.

This understanding of how energy is conserved and transferred is key in numerous applications, demonstrating how systems can be optimized for energy efficiency and resource management. Recognizing these transformations can enhance our comprehension of energy systems' dynamics in fields ranging from engineering to environmental science.