Problem 69 - Fundamentals of Physics

Question

How much electrical energy is transferred to thermal energy in \(2.00 \mathrm{~h}\) by an electrical resistance of \(400 \Omega\) when the potential applied across it is \(90.0 \mathrm{~V} ?\)

Step-by-Step Solution

Answer

145,800 Joules

Step 1: Understand the Given Values

We are given the resistance, \( R = 400 \, \Omega \), the potential difference, \( V = 90.0 \, \text{V} \), and the time duration, \( t = 2.00 \, \text{h} \). We need to convert the time into seconds. Since 1 hour = 3600 seconds, \( t = 2 \times 3600 = 7200 \text{ s} \).

Step 2: Use the Power Formula

The power consumed by the resistor can be calculated using the formula \( P = \frac{V^2}{R} \). Substituting the given values, we have \( P = \frac{(90.0)^2}{400} \).

Step 3: Calculate the Power

Compute the power: \( P = \frac{8100}{400} = 20.25 \, \text{W} \). This is the rate at which energy is transferred in watts.

Step 4: Calculate Electrical Energy Transferred

Energy transferred, \( E \), is given by using the formula \( E = P \times t \), where \( t \) is the time duration in seconds. Substituting the values, \( E = 20.25 \times 7200 \).

Step 5: Compute the Total Energy

Calculate the total energy: \( E = 145800 \, \text{J} \). This is the amount of electrical energy transferred to thermal energy.

Key Concepts

Electrical Resistance

Electrical resistance is a property of a material that restricts the flow of electric current through it. Think of resistance like a bottleneck in a road that slows down traffic. In circuits, resistance is measured in ohms (\( \Omega \)). It determines how much current flows for a given potential difference across a component. For a resistor like the one in our example with 400 ohms, this means the electric current will face a certain level of opposition as it passes through.

It's important to note:

• Higher resistance means less current for the same voltage.
• Resistance can be used to control the current within a circuit.
• Different materials and components have different resistance values.

When you have the resistance value and the potential difference, you can use these values to calculate power consumed and then determine the energy transfer over time.

Potential Difference

Potential difference, also known as voltage, is the "push" that drives the electric current through a circuit. It's much like water pressure in a hose that propels water through it, allowing us to do work. Measured in volts (\( V \)), potential difference is crucial as it dictates how much energy each charge carries through a circuit.

In the exercise, a potential difference of 90.0 V means that each coulomb of charge has 90 J of energy to transfer. This energy will ultimately be converted into either work or another form of energy like thermal energy in a resistor.

Key points about potential difference:

• It's the energy transferred per unit charge.
• The higher the voltage, the greater the energy per charge.
• Voltage can be supplied by batteries, generators, or other sources.

Understanding voltage is essential for calculating power and energy transfer in circuits.

Power Formula

The power formula is a fundamental tool in understanding how electrical energy is transferred in a circuit. Power, measured in watts (\( W \)), represents the rate at which energy is consumed or transferred. The formula used in the exercise is:

• \( P = \frac{V^2}{R} \)

This formula helps us determine the power consumed by the resistor in the circuit. By finding power, we can then track how much energy is converted over time.

The steps using this formula were:

• Substitute the given voltage (90.0 V) into the formula.
• Divide by the resistance (400 \( \Omega \)) to get power.
• Calculate power as 20.25 W, meaning 20.25 joules of energy is transferred every second.

Power formulas are central to analyzing circuits and understanding how different components consume energy.

Energy Conversion

Energy conversion is the process of changing energy from one form to another. In electrical circuits, it often involves converting electrical energy into thermal, mechanical, or other forms of energy that can be used efficiently.

In our exercise, the energy conversion occurs in the resistor, where the electrical energy is transformed into thermal energy. The resistor heats up as current passes through it, and this heating effect is the manifestation of energy conversion.

Here's a simple breakdown of the energy conversion process:

• The power formula tells us how much energy per second (20.25 W) is used.
• Over time, using the formula \( E = P \times t \), we find the total energy converted.
• In this case, 145,800 J of electrical energy is turned into heat over 2 hours.

Understanding how energy is converted helps in designing circuits and choosing the correct components to achieve desired functionalities. It ensures that energy moves fluidly and is used effectively in electronic systems.