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

145800 J

Step 1: Identify known values

We have the following known values from the problem:- Resistance \( R = 400 \, \Omega \)- Voltage \( V = 90.0 \, \text{V} \)- Time \( t = 2.00 \, \text{h} = 2.00 \, \times \, 3600 \, \text{s} = 7200 \, \text{s} \)

Step 2: Write the formula for power

The power \( P \) dissipated by a resistor can be calculated using Ohm's Law and the formula for electrical power. The formula is:\[ P = \frac{V^2}{R} \]where \( P \) is power in watts, \( V \) is voltage, and \( R \) is resistance.

Step 3: Calculate power

Substitute the known values into the power formula:\[ P = \frac{(90.0 \, \text{V})^2}{400 \, \Omega} = \frac{8100 \, \text{V}^2}{400 \, \Omega} = 20.25 \, \text{W} \]So, the power being dissipated is \( 20.25 \, \text{W} \).

Step 4: Calculate the total energy transferred

The total energy \( E \) transferred is given by the formula:\[ E = P \times t \]Substitute the power and the time:\[ E = 20.25 \, \text{W} \times 7200 \, \text{s} = 145800 \, \text{J} \]Therefore, the total electrical energy transferred to thermal energy is \( 145800 \, \text{J} \).

Key Concepts

Ohm's Law

Ohm's Law is a fundamental principle in the study of electricity and helps to connect the main variables related to electrical circuits. It states that the current (I) through a conductor between two points is directly proportional to the voltage (V) across the two points, and inversely proportional to the resistance (R) of the conductor. This relationship can be expressed with the equation \( V = IR \).

• Voltage (V): This is the potential difference across the conductor and is measured in volts (V).
• Current (I): This is the flow of electric charge and is measured in amperes (A).
• Resistance (R): This opposes the flow of charge and is measured in ohms (Ω).

In this particular exercise, we are applying Ohm's Law to calculate power, which involves another handy formula: \[ P = \frac{V^2}{R} \]This derives from combining Ohm's Law with the power formula and is useful to find power when the resistance and voltage are known but the current isn't readily available.

Power Calculation

Power calculation is central in determining how energy is used or dissipated in electrical circuits. Power (P) is the rate at which energy is transferred or converted. The standard formula used for power in electrical circuits is \( P = VI \), which gives power in watts (W).However, since direct current calculations often provide voltage and resistance, using \( P = \frac{V^2}{R} \) is very useful. This formula gives a straightforward way to calculate power when voltage and resistance aren't directly accessible.

• Power is measured in watts (W).
• It is the rate of doing work or the rate of energy conversion.
• A higher power indicates a higher rate of energy conversion from electrical to other forms, such as thermal energy.

In this exercise, using the voltage of 90.0 V and the resistance of 400 Ω, the power was calculated to be 20.25 W, laying ground for further energy calculations.

Thermal Energy Conversion

Thermal energy conversion refers to how electrical energy is transformed into thermal energy, a common process in multiple applications including resistors, heaters, and many home appliances. When electricity passes through a resistor, due to the resistance, some of the electrical energy is stored as heat. This concept is crucial to solving exercises where you're required to find out how much energy is converted from electrical to thermal form over a set duration, such as the 7200 s in this problem.

• Total Energy Calculated: The power of 20.25 W is multiplied by the time of 7200 s, resulting in converted energy of 145800 J (joules).
• A resistor converts all the electrical energy it doesn't use for moving electrons into heat, which is why light bulbs and other electric devices can become warm or even hot while operating.
• Understanding energy conversion is essential for improving energy efficiency and optimizing energy use in technology.

Thus, the exercise demonstrates practical application of concepts, providing insight into everyday appliances and broader energy concerns.