Problem 62 - Fundamentals of Physics

Question

A resistor with a potential difference of \(200 \mathrm{~V}\) across it transfers electrical energy to thermal energy at the rate of \(3000 \mathrm{~W}\). What is the resistance of the resistor?

Step-by-Step Solution

Answer

The resistance of the resistor is approximately \(13.33 \Omega\).

Step 1: Identify Known Values

We know from the problem that the potential difference \( V \) across the resistor is \( 200 \text{ V} \), and the power \( P \) transferred is \( 3000 \text{ W} \).

Step 2: Recall Power Formulas

The formula for electrical power is \( P = VI \), where \( V \) is the voltage, and \( I \) is the current. We also have \( P = I^2 R \) where \( R \) is resistance. We need to find \( R \) using the given values.

Step 3: Determine Current

First, calculate the current \( I \) using \( P = VI \). Rearrange to get \( I = \frac{P}{V} \). Substituting the known values: \( I = \frac{3000}{200} = 15 \text{ A} \).

Step 4: Calculate Resistance

Using the calculated current and the formula \( P = I^2 R \), rearrange it to \( R = \frac{P}{I^2} \). Substitute \( P = 3000 \text{ W} \) and \( I = 15 \text{ A} \) to get \( R = \frac{3000}{15^2} = 13.33 \Omega \).

Step 5: Conclude Calculation

The resistance of the resistor is determined to be approximately \( 13.33 \Omega \).

Key Concepts

Ohm's Law

Ohm's Law is a fundamental principle in electronics that relates voltage, current, and resistance in electrical circuits. This law is expressed with the formula:

• \( V = IR \)

In the formula, \( V \) represents voltage, \( I \) is the current, and \( R \) is the resistance. According to this law, the voltage across a resistor is proportional to the current flowing through it, with the resistance being the proportionality constant.
For students working with electrical circuits, Ohm's Law is crucial for understanding how changes in voltage or current affect each other when resistance is constant. For example, if you know the voltage and the resistance, you can easily find the current by rearranging the formula to \( I = \frac{V}{R} \).
Ohm’s Law can also be used to calculate the resistance if voltage and current are known. Simply rearrange the formula to \( R = \frac{V}{I} \). This simple relationship plays a vital role in various calculations in physics and engineering, making it a backbone principle for many electrical calculations.

Electrical Power

Electrical power describes the rate at which electrical energy is converted into another form of energy, such as heat or mechanical energy. The formula for calculating power is:

• \( P = VI \)
• \( P = I^2 R \)
• \( P = \frac{V^2}{R} \)

Here, \( P \) stands for power, \( V \) is the voltage, \( I \) is the current, and \( R \) is the resistance. These equations show different ways to calculate power depending on the known values. The unit of power is the watt (W).
In electrical circuits, understanding power is essential because it tells us how much energy is being used or converted per unit of time. For example, in a light bulb, electrical power converted into light and heat determines the bulb's brightness and energy efficiency.
In problems involving power, like the one we discussed, you can calculate unknown values when the power and one other quantity are known. In the process, students can better understand how energy flows and transforms within a circuit, helping them make necessary adjustments in practical applications.

Resistor Calculations

Calculating the resistance in a circuit is key to designing effective electrical systems and solving problems related to energy consumption. Resistance is the property of a material that resists the flow of electric current, measured in ohms (\(\Omega\)). To find resistance, various formulas can be applied, each suitable for different sets of known data.

• Using Ohm’s Law: \( R = \frac{V}{I} \)
• From Power: \( R = \frac{P}{I^2} \) or \( R = \frac{V^2}{P} \)

In the exercise above, we utilized the power formula \( R = \frac{P}{I^2} \) to determine the resistance of the resistor, given the values for power and current. Substituting gives \( R = \frac{3000}{15^2} \), which results in approximately \( 13.33 \Omega \).
Mastering resistor calculations involves selecting the correct formula based on available measurements, rearranging equations as needed, and mathematical operations with precision. This skill extends beyond textbook exercises to real-world problems where engineers need to ensure components meet safety and performance standards.