Problem 7 - Physics

Question

The resistance of a bagel toaster is \(14 \Omega\). To prepare a bagel, the toaster is operated for one minute from a \(120-\mathrm{V}\) outlet. How much energy is delivered to the toaster?

Step-by-Step Solution

Answer

The energy delivered to the toaster is approximately 61714.2 joules.

Step 1: Identify the Given Information

We are given the resistance \( R = 14 \, \Omega \), the voltage \( V = 120 \, \text{volts} \), and the time \( t = 1 \, \text{minute} = 60 \, \text{seconds} \).

Step 2: Use the Formula for Power

We use the formula for power delivered, \( P = \frac{V^2}{R} \), because we know both the voltage and the resistance.

Step 3: Calculate the Power

Substitute the values into the formula: \[ P = \frac{120^2}{14} = \frac{14400}{14} \approx 1028.57 \, \text{watts} \].

Step 4: Use the Formula for Energy Delivered

Energy delivered is given by the formula \( E = P \times t \). Since we already have the power \( P \) and the time \( t \) in seconds, we can calculate the energy.

Step 5: Calculate the Energy Delivered

Substitute the values into the formula: \[ E = 1028.57 \times 60 \approx 61714.2 \, \text{joules} \].

Key Concepts

Ohm's Law

Ohm's Law is an essential principle in the study of electrical circuits, and it relates voltage, current, and resistance. The law is expressed with the formula:

• \( V = I \cdot R \)

In this formula, \( V \) stands for voltage (in volts), \( I \) represents current (in amperes), and \( R \) is the resistance (in ohms, \( \Omega \)).
If you know any two of these values, you can calculate the third. This makes Ohm's Law extremely useful for solving electric circuit problems.
In our exercise, Ohm's Law could help us find the current once we have voltage and resistance. But for this particular problem, it wasn't directly used since we utilized the power formula instead.

Power Formula

The power formula is crucial for calculating the rate at which energy is used in an electrical circuit. Power is a measure of how much energy is consumed per unit of time.
The formula used for power calculation is:

• \( P = \frac{V^2}{R} \)

Here, \( P \) is power in watts, \( V \) denotes voltage across the resistance, and \( R \) is the resistance itself.
This specific form of the power formula is handy when you know the voltage and resistance, which was the case in our toaster example. By calculating the power, we can later derive how much energy is used over a period, as power directly influences energy consumption.

Energy Conversion

Energy conversion is a key concept when dealing with electrical devices, as it explains how energy is transformed from electrical energy to other forms, such as heat, light, or motion.
In the context of our example, the toaster is converting electrical energy from the outlet into heat energy to toast a bagel.
The energy delivered can be calculated using the formula:

• \( E = P \times t \)

where \( E \) is energy in joules, \( P \) is power in watts, and \( t \) is time in seconds. This formula tells us that the energy consumed is a product of the power and the duration of time the device is operating.

Electrical Resistance

Electrical resistance is a fundamental property that quantifies how much a material opposes the flow of electric current.
Measured in ohms (\( \Omega \)), resistance depends on factors like material composition, temperature, and physical dimensions of the component.
In our example, the toaster has a resistance of \( 14 \Omega \). This value determines how much current will flow through the toaster when connected to a voltage source (in this case, 120 V).
High resistance means less current flows and, subsequently, less energy conversion occurs, impacting the performance of devices like heaters or toasters. Understanding this helps in designing circuits with desired power and energy characteristics.